diff --git a/.gitignore b/.gitignore index e3a4901..cb7491d 100644 --- a/.gitignore +++ b/.gitignore @@ -130,4 +130,13 @@ dist .pnp.* # vscode -.vscode \ No newline at end of file +.vscode + +# analytics +analyze/ + +# heapsnapshot +*.heapsnapshot + +# turbo +.turbo/ \ No newline at end of file diff --git a/app/[[...mdxPath]]/page.tsx b/app/[[...mdxPath]]/page.tsx new file mode 100644 index 0000000..1f24ed6 --- /dev/null +++ b/app/[[...mdxPath]]/page.tsx @@ -0,0 +1,23 @@ +import { generateStaticParamsFor, importPage } from 'nextra/pages' +import { useMDXComponents as getMDXComponents } from '../../mdx-components' + +export const generateStaticParams = generateStaticParamsFor('mdxPath') + +export async function generateMetadata(props) { + const params = await props.params + const { metadata } = await importPage(params.mdxPath) + return metadata +} + +const Wrapper = getMDXComponents().wrapper + +export default async function Page(props) { + const params = await props.params + const result = await importPage(params.mdxPath) + const { default: MDXContent, toc, metadata } = result + return ( + + + + ) +} \ No newline at end of file diff --git a/app/layout.tsx b/app/layout.tsx new file mode 100644 index 0000000..dad73d2 --- /dev/null +++ b/app/layout.tsx @@ -0,0 +1,90 @@ +/* eslint-env node */ +import { Footer, Layout, Navbar } from 'nextra-theme-docs' +import { Banner, Head } from 'nextra/components' +import { getPageMap } from 'nextra/page-map' +import 'nextra-theme-docs/style.css' +import { SpeedInsights } from "@vercel/speed-insights/next" +import { Analytics } from "@vercel/analytics/react" + +export const metadata = { + metadataBase: new URL('https://notenextra.trance-0.com'), + title: { + template: '%s - NoteNextra' + }, + description: 'A static note sharing site for minimum care', + applicationName: 'NoteNextra', + generator: 'Next.js', + appleWebApp: { + title: 'NoteNextra' + }, + other: { + 'msapplication-TileImage': '/ms-icon-144x144.png', + 'msapplication-TileColor': '#fff' + }, + twitter: { + site: 'https://notenextra.trance-0.com' + } +} + +export default async function RootLayout({ children }) { + const navbar = ( + + + + + + NoteNextra + + + } + projectLink="https://github.com/Trance-0/NoteNextra" + /> + ) + const pageMap = await getPageMap() + return ( + + + + + + MIT {new Date().getFullYear()} ©{' '} + + Trance-0 + + . + + + } + editLink="Edit this page on GitHub" + docsRepositoryBase="https://github.com/Trance-0/NoteNextra/tree/main" + sidebar={{ defaultMenuCollapseLevel: 1 }} + pageMap={pageMap} + > + {children} + {/* SpeedInsights in vercel */} + + {/* Analytics in vercel */} + + + + + ) +} \ No newline at end of file diff --git a/pages/CSE332S/CSE332S_L1.md b/content/CSE332S/CSE332S_L1.md similarity index 100% rename from pages/CSE332S/CSE332S_L1.md rename to content/CSE332S/CSE332S_L1.md diff --git a/pages/CSE332S/CSE332S_L10.md b/content/CSE332S/CSE332S_L10.md similarity index 100% rename from pages/CSE332S/CSE332S_L10.md rename to content/CSE332S/CSE332S_L10.md diff --git a/pages/CSE332S/CSE332S_L11.md b/content/CSE332S/CSE332S_L11.md similarity index 100% rename from pages/CSE332S/CSE332S_L11.md rename to content/CSE332S/CSE332S_L11.md diff --git a/pages/CSE332S/CSE332S_L12.md b/content/CSE332S/CSE332S_L12.md similarity index 100% rename from pages/CSE332S/CSE332S_L12.md rename to content/CSE332S/CSE332S_L12.md diff --git a/pages/CSE332S/CSE332S_L13.md b/content/CSE332S/CSE332S_L13.md similarity index 100% rename from pages/CSE332S/CSE332S_L13.md rename to content/CSE332S/CSE332S_L13.md diff --git a/pages/CSE332S/CSE332S_L14.md b/content/CSE332S/CSE332S_L14.md similarity index 100% rename from pages/CSE332S/CSE332S_L14.md rename to content/CSE332S/CSE332S_L14.md diff --git a/pages/CSE332S/CSE332S_L15.md b/content/CSE332S/CSE332S_L15.md similarity index 100% rename from pages/CSE332S/CSE332S_L15.md rename to content/CSE332S/CSE332S_L15.md diff --git a/pages/CSE332S/CSE332S_L16.md b/content/CSE332S/CSE332S_L16.md similarity index 100% rename from pages/CSE332S/CSE332S_L16.md rename to content/CSE332S/CSE332S_L16.md diff --git a/pages/CSE332S/CSE332S_L17.md b/content/CSE332S/CSE332S_L17.md similarity index 100% rename from pages/CSE332S/CSE332S_L17.md rename to content/CSE332S/CSE332S_L17.md diff --git a/pages/CSE332S/CSE332S_L2.md b/content/CSE332S/CSE332S_L2.md similarity index 100% rename from pages/CSE332S/CSE332S_L2.md rename to content/CSE332S/CSE332S_L2.md diff --git a/pages/CSE332S/CSE332S_L3.md b/content/CSE332S/CSE332S_L3.md similarity index 100% rename from pages/CSE332S/CSE332S_L3.md rename to content/CSE332S/CSE332S_L3.md diff --git a/pages/CSE332S/CSE332S_L4.md b/content/CSE332S/CSE332S_L4.md similarity index 100% rename from pages/CSE332S/CSE332S_L4.md rename to content/CSE332S/CSE332S_L4.md diff --git a/pages/CSE332S/CSE332S_L5.md b/content/CSE332S/CSE332S_L5.md similarity index 100% rename from pages/CSE332S/CSE332S_L5.md rename to content/CSE332S/CSE332S_L5.md diff --git a/pages/CSE332S/CSE332S_L6.md b/content/CSE332S/CSE332S_L6.md similarity index 100% rename from pages/CSE332S/CSE332S_L6.md rename to content/CSE332S/CSE332S_L6.md diff --git a/pages/CSE332S/CSE332S_L7.md b/content/CSE332S/CSE332S_L7.md similarity index 100% rename from pages/CSE332S/CSE332S_L7.md rename to content/CSE332S/CSE332S_L7.md diff --git a/pages/CSE332S/CSE332S_L8.md b/content/CSE332S/CSE332S_L8.md similarity index 100% rename from pages/CSE332S/CSE332S_L8.md rename to content/CSE332S/CSE332S_L8.md diff --git a/pages/CSE332S/CSE332S_L9.md b/content/CSE332S/CSE332S_L9.md similarity index 100% rename from pages/CSE332S/CSE332S_L9.md rename to content/CSE332S/CSE332S_L9.md diff --git a/pages/CSE332S/_meta.js b/content/CSE332S/_meta.js similarity index 93% rename from pages/CSE332S/_meta.js rename to content/CSE332S/_meta.js index b906529..d2e0bb1 100644 --- a/pages/CSE332S/_meta.js +++ b/content/CSE332S/_meta.js @@ -1,5 +1,5 @@ export default { - index: "Course Description", + index: {type:"page",title:"Course Description",href:"/CSE332S/index.mdx"}, "---":{ type: 'separator' }, diff --git a/pages/CSE332S/index.md b/content/CSE332S/index.mdx similarity index 100% rename from pages/CSE332S/index.md rename to content/CSE332S/index.mdx diff --git a/pages/CSE347/CSE347_L1.md b/content/CSE347/CSE347_L1.md similarity index 97% rename from pages/CSE347/CSE347_L1.md rename to content/CSE347/CSE347_L1.md index 73b0dab..145dcba 100644 --- a/pages/CSE347/CSE347_L1.md +++ b/content/CSE347/CSE347_L1.md @@ -1,245 +1,245 @@ -# Lecture 1 - -## Greedy Algorithms - -* Builds up a solution by making a series of small decisions that optimize some objective. -* Make one irrevocable choice at a time, creating smaller and smaller sub-problems of the same kind as the original problem. -* There are many potential greedy strategies and picking the right one can be challenging. - -### A Scheduling Problem - -You manage a giant space telescope. - -* There are $n$ research projects that want to use it to make observations. -* Only one project can use the telescope at a time. -* Project $p_i$ needs the telescope starting at time $s_i$ and running for a length of time $t_i$. -* Goal: schedule as many as possible - -Formally - -Input: - -* Given a set $P$ of projects, $|P|=n$ -* Each request $p_i\in P$ occupies interval $[s_i,f_i)$, where $f_i=s_i+t_i$ - -Goal: Choose a subset $\Pi\sqsubseteq P$ such that - -1. No two projects in $\Pi$ have overlapping intervals. -2. The number of selected projects $|\Pi|$ is maximized. - -#### Shortest Interval - -Counter-example: `[1,10],[9,12],[11,20]` - -#### Earliest start time - -Counter-example: `[1,10],[2,3],[4,5]` - -#### Fewest Conflicts - -Counter-example: `[1,2],[1,4],[1,4],[3,6],[7,8],[5,8],[5,8]` - -#### Earliest finish time - -Correct... but why - -#### Theorem of Greedy Strategy (Earliest Finishing Time) - -Say this greedy strategy (Earliest Finishing Time) picks a set $\Pi$ of intervals, some other strategy picks a set $O$ of intervals. - -Assume sorted by finishing time - -* $\Pi=\{i_1,i_2,...,i_k\},|\Pi|=k$ -* $O=\{j_1,j_2,...,j_m\},|O|=m$ - -We want to show that $|\Pi|\geq|O|,k>m$ - -#### Lemma: For all $r1. - Since $\Pi_r$ is the earliest finish time, so for any set in $O_r$, $f_{i_{r-1}}\leq f_{j_{r-1}}$, for any $j_r$ inserted to $O_r$, it can also be inserted to $\Pi_r$. So $O_r$ cannot pick an interval with earlier finish time than $Pi$ since it will also be picked by definition if $O_r$ is the optimal solution $OPT$. - -#### Problem of “Greedy Stays Ahead” Proof - -* Every problem has very different theorem. -* It can be challenging to even write down the correct statement that you must prove. -* We want a systematic approach to prove the correctness of greedy algorithms. - -### Road Map to Prove Greedy Algorithm - -#### 1. Make a Choice - -Pick an interval based on greedy choice, say $q$ - -Proof: **Greedy Choice Property**: Show that using our first choice is not "fatal" – at least one optimal solution makes this choice. - -Techniques: **Exchange Argument**: "If an optimal solution does not choose $q$, we can turn it into an equally good solution that does." - -Let $\Pi^*$ be any optimal solution for project set $P$. -- If $q\in \Pi^*$, we are done. -- Otherwise, let $x$ be the optimal solution from $\Pi^*$ that does not pick $q$. We create another solution $\bar{\Pi^*}$ that replace $x$ with $q$, and prove that the $\bar{\Pi^*}$ is as optimal as $\Pi^*$ - -#### 2. Create a smaller instance $P'$ of the original problem - -$P'$ has the same optimization criteria. - -Proof: **Inductive Structure**: Show that after making the first choice, we're left with a smaller version of the same problem, whose solution we can safely combine with the first choice. - -Let $P'$ be the subproblem left after making first choice $q$ in problem $P$ and let $\Pi'$ be an optimal solution to $P'$. Then $\Pi=\Pi^*\cup\{q\}$ is an optimal solution to $P$. - -$P'=P-\{q\}-\{$projects conflicting with $q\}$ - -#### 3. Solution: Union of choices that we made - -Union of choices that we made. - -Proof: **Optimal Substructure**: Show that if we solve the subproblem optimally, adding our first choice creates an optimal solution to the *whole* problem. - -Let $q$ be the first choice, $P'$ be the subproblem left after making $q$ in problem $P$, $\Pi'$ be an optimal solution to $P'$. We claim that $\Pi=\Pi'\cup \{q\}$ is an optimal solution to $P$. - -We proceed the proof by contradiction. - -Assume that $\Pi=\Pi'+\{q\}$ is not optimal. - - -By Greedy choice property $GCP$. we already know that $\exists$ an optimal solution $\Pi^*$ for problem $P$ that contains $q$. If $\Pi$ is not optimal, $cost(\Pi^*)cost(\Pi-q)=\Pi'$ which leads to contradiction that $\Pi'$ is an optimal solution to $P'$. - -#### 4. Put 1-3 together to write an inductive proof of the Theorem - -This is independent of problem, same for every problem. - -Use scheduling problem as an example: - -Theorem: given a scheduling problem $P$, if we repeatedly choose the remaining feasible project with the earliest finishing time, we will construct an optimal feasible solution to $P$. - -Proof: We proceed by induction on $|P|$. (based on the size of problem $P$). - -- Base case: $|P|=1$. -- Inductive step. - - Inductive hypothesis: For all problems of size $|\Pi|$ $\Pi^*-q$ is also feasible solution to $P'$. $|\Pi^*-q|>|\Pi-q|=\Pi'$ which is an optimal solution to $P'$ which leads to contradiction. - -Proof: **Smaller problem size** - -After merging the smallest two files into one, we have strictly less files waiting to merge. - -#### Optimal Substructure - -* We can combine optimal solution to the subproblem $P'$ with the greedy choice to get a optimal solution to $P$ - -Step 4 ignored, same for all greedy problems. - -### Conclusion: Greedy Algorithm - -* Algorithm -* Runtime Complexity -* Proof - * Greedy Choice Property - * Construct part of the solution by making a locally good decision. - * Inductive Structure - * We can combine feasible solution to the subproblem $P'$ with the greedy choice to get a feasible solution to $P$ - * After making greedy choice $q$, we are left with a strictly smaller subproblem $P'$ with the same optimality criteria of the original problem - * Optimal Substructure - * We can combine optimal solution to the subproblem $P'$ with the greedy choice to get a optimal solution to $P$ -* Standard Contradiction Argument simplifies it - -## Review: - -### Essence of master method - -Let $a\geq 1$ and $b>1$ be constants, let $f(n)$ be a function, and let $T(n)$ be defined on the nonnegative integers by the recurrence - -$$ -T(n)=aT(\frac{n}{b})+f(n) -$$ - -where we interpret $n/b$ to mean either ceiling or floor of $n/b$. $c_{crit}=\log_b a$ Then $T(n)$ has to following asymptotic bounds. - -* Case I: if $f(n) = O(n^{c})$ ($f(n)$ "dominates" $n^{\log_b a-c}$) where $c-1$ - - $T(n)=\Theta(n^{critical\_value}*(\log n)^{k+1})$ - - * if $k=-1$ - - $T(n)=\Theta(n^{critical\_value}*\log \log n)$ - - * if $k<-1$ - - $T(n)=\Theta(n^{critical\_value})$ - -* Case III: if $f(n) = \Omega(n^{log_b a+c})$ ($n^{log_b a-c}$ "dominates" $f(n)$) for some constant $c >0$, and if a $f(n/b)<= c f(n)$ for some constant $c <1$ then for all sufficiently large $n$, $T(n) = \Theta(n^{log_b a+c})$ - +# Lecture 1 + +## Greedy Algorithms + +* Builds up a solution by making a series of small decisions that optimize some objective. +* Make one irrevocable choice at a time, creating smaller and smaller sub-problems of the same kind as the original problem. +* There are many potential greedy strategies and picking the right one can be challenging. + +### A Scheduling Problem + +You manage a giant space telescope. + +* There are $n$ research projects that want to use it to make observations. +* Only one project can use the telescope at a time. +* Project $p_i$ needs the telescope starting at time $s_i$ and running for a length of time $t_i$. +* Goal: schedule as many as possible + +Formally + +Input: + +* Given a set $P$ of projects, $|P|=n$ +* Each request $p_i\in P$ occupies interval $[s_i,f_i)$, where $f_i=s_i+t_i$ + +Goal: Choose a subset $\Pi\sqsubseteq P$ such that + +1. No two projects in $\Pi$ have overlapping intervals. +2. The number of selected projects $|\Pi|$ is maximized. + +#### Shortest Interval + +Counter-example: `[1,10],[9,12],[11,20]` + +#### Earliest start time + +Counter-example: `[1,10],[2,3],[4,5]` + +#### Fewest Conflicts + +Counter-example: `[1,2],[1,4],[1,4],[3,6],[7,8],[5,8],[5,8]` + +#### Earliest finish time + +Correct... but why + +#### Theorem of Greedy Strategy (Earliest Finishing Time) + +Say this greedy strategy (Earliest Finishing Time) picks a set $\Pi$ of intervals, some other strategy picks a set $O$ of intervals. + +Assume sorted by finishing time + +* $\Pi=\{i_1,i_2,...,i_k\},|\Pi|=k$ +* $O=\{j_1,j_2,...,j_m\},|O|=m$ + +We want to show that $|\Pi|\geq|O|,k>m$ + +#### Lemma: For all $r1. + Since $\Pi_r$ is the earliest finish time, so for any set in $O_r$, $f_{i_{r-1}}\leq f_{j_{r-1}}$, for any $j_r$ inserted to $O_r$, it can also be inserted to $\Pi_r$. So $O_r$ cannot pick an interval with earlier finish time than $Pi$ since it will also be picked by definition if $O_r$ is the optimal solution $OPT$. + +#### Problem of “Greedy Stays Ahead” Proof + +* Every problem has very different theorem. +* It can be challenging to even write down the correct statement that you must prove. +* We want a systematic approach to prove the correctness of greedy algorithms. + +### Road Map to Prove Greedy Algorithm + +#### 1. Make a Choice + +Pick an interval based on greedy choice, say $q$ + +Proof: **Greedy Choice Property**: Show that using our first choice is not "fatal" – at least one optimal solution makes this choice. + +Techniques: **Exchange Argument**: "If an optimal solution does not choose $q$, we can turn it into an equally good solution that does." + +Let $\Pi^*$ be any optimal solution for project set $P$. +- If $q\in \Pi^*$, we are done. +- Otherwise, let $x$ be the optimal solution from $\Pi^*$ that does not pick $q$. We create another solution $\bar{\Pi^*}$ that replace $x$ with $q$, and prove that the $\bar{\Pi^*}$ is as optimal as $\Pi^*$ + +#### 2. Create a smaller instance $P'$ of the original problem + +$P'$ has the same optimization criteria. + +Proof: **Inductive Structure**: Show that after making the first choice, we're left with a smaller version of the same problem, whose solution we can safely combine with the first choice. + +Let $P'$ be the subproblem left after making first choice $q$ in problem $P$ and let $\Pi'$ be an optimal solution to $P'$. Then $\Pi=\Pi^*\cup\{q\}$ is an optimal solution to $P$. + +$P'=P-\{q\}-\{$projects conflicting with $q\}$ + +#### 3. Solution: Union of choices that we made + +Union of choices that we made. + +Proof: **Optimal Substructure**: Show that if we solve the subproblem optimally, adding our first choice creates an optimal solution to the *whole* problem. + +Let $q$ be the first choice, $P'$ be the subproblem left after making $q$ in problem $P$, $\Pi'$ be an optimal solution to $P'$. We claim that $\Pi=\Pi'\cup \{q\}$ is an optimal solution to $P$. + +We proceed the proof by contradiction. + +Assume that $\Pi=\Pi'+\{q\}$ is not optimal. + + +By Greedy choice property $GCP$. we already know that $\exists$ an optimal solution $\Pi^*$ for problem $P$ that contains $q$. If $\Pi$ is not optimal, $cost(\Pi^*)cost(\Pi-q)=\Pi'$ which leads to contradiction that $\Pi'$ is an optimal solution to $P'$. + +#### 4. Put 1-3 together to write an inductive proof of the Theorem + +This is independent of problem, same for every problem. + +Use scheduling problem as an example: + +Theorem: given a scheduling problem $P$, if we repeatedly choose the remaining feasible project with the earliest finishing time, we will construct an optimal feasible solution to $P$. + +Proof: We proceed by induction on $|P|$. (based on the size of problem $P$). + +- Base case: $|P|=1$. +- Inductive step. + - Inductive hypothesis: For all problems of size $|\Pi|$ $\Pi^*-q$ is also feasible solution to $P'$. $|\Pi^*-q|>|\Pi-q|=\Pi'$ which is an optimal solution to $P'$ which leads to contradiction. + +Proof: **Smaller problem size** + +After merging the smallest two files into one, we have strictly less files waiting to merge. + +#### Optimal Substructure + +* We can combine optimal solution to the subproblem $P'$ with the greedy choice to get a optimal solution to $P$ + +Step 4 ignored, same for all greedy problems. + +### Conclusion: Greedy Algorithm + +* Algorithm +* Runtime Complexity +* Proof + * Greedy Choice Property + * Construct part of the solution by making a locally good decision. + * Inductive Structure + * We can combine feasible solution to the subproblem $P'$ with the greedy choice to get a feasible solution to $P$ + * After making greedy choice $q$, we are left with a strictly smaller subproblem $P'$ with the same optimality criteria of the original problem + * Optimal Substructure + * We can combine optimal solution to the subproblem $P'$ with the greedy choice to get a optimal solution to $P$ +* Standard Contradiction Argument simplifies it + +## Review: + +### Essence of master method + +Let $a\geq 1$ and $b>1$ be constants, let $f(n)$ be a function, and let $T(n)$ be defined on the nonnegative integers by the recurrence + +$$ +T(n)=aT(\frac{n}{b})+f(n) +$$ + +where we interpret $n/b$ to mean either ceiling or floor of $n/b$. $c_{crit}=\log_b a$ Then $T(n)$ has to following asymptotic bounds. + +* Case I: if $f(n) = O(n^{c})$ ($f(n)$ "dominates" $n^{\log_b a-c}$) where $c-1$ + + $T(n)=\Theta(n^{critical\_value}*(\log n)^{k+1})$ + + * if $k=-1$ + + $T(n)=\Theta(n^{critical\_value}*\log \log n)$ + + * if $k<-1$ + + $T(n)=\Theta(n^{critical\_value})$ + +* Case III: if $f(n) = \Omega(n^{log_b a+c})$ ($n^{log_b a-c}$ "dominates" $f(n)$) for some constant $c >0$, and if a $f(n/b)<= c f(n)$ for some constant $c <1$ then for all sufficiently large $n$, $T(n) = \Theta(n^{log_b a+c})$ + diff --git a/pages/CSE347/CSE347_L10.md b/content/CSE347/CSE347_L10.md similarity index 100% rename from pages/CSE347/CSE347_L10.md rename to content/CSE347/CSE347_L10.md diff --git a/pages/CSE347/CSE347_L11.md b/content/CSE347/CSE347_L11.md similarity index 100% rename from pages/CSE347/CSE347_L11.md rename to content/CSE347/CSE347_L11.md diff --git a/pages/CSE347/CSE347_L2.md b/content/CSE347/CSE347_L2.md similarity index 95% rename from pages/CSE347/CSE347_L2.md rename to content/CSE347/CSE347_L2.md index a4f35fc..7244fed 100644 --- a/pages/CSE347/CSE347_L2.md +++ b/content/CSE347/CSE347_L2.md @@ -1,334 +1,334 @@ -# Lecture 2 - -## Divide and conquer - -Review of CSE 247 - -1. Divide the problem into (generally equal) smaller subproblems -2. Recursively solve the subproblems -3. Combine the solutions of subproblems to get the solution of the original problem - - Examples: Merge Sort, Binary Search - -Recurrence - -Master Method: - -$$ -T(n)=aT(\frac{n}{b})+\Theta(f(n)) -$$ - -### Example 1: Multiplying 2 numbers - -Normal Algorithm: - -```python -def multiply(x,y): - p=0 - for i in y: - p+=x*y - return p -``` - -divide and conquer approach - -```python -def multiply(x,y): - n=max(len(x),len(y)) - if n==1: - return x*y - xh,xl=x>>(n/2),x&((1<>(n/2),y&((1<>(n/2),x&((1<>(n/2),y&((1<p_y +\delta$ - -```python -# d is distance function -def closestP(P,d): - Px=sorted(P,key=lambda x:x[0]) - def closestPRec(P,d): - n=len(P) - if n==1: - return float('inf') - if n==2: - return d(P[0],P[1]) - Q,R=Px[:n//2],Px[n//2:] - midx=R[0][0] - dl,dr=closestP(Q),closestP(R) - dc=min(dl,dr) - ys=[i if midx-dcdc: - break - dc=min(dc,curd) - return dc - return closestPRec(Px,d): -``` - -Runtime analysis: - -$$ -T(n)=2T(n/2)+\Theta(n\log n)=\Theta(n\log^2 n) -$$ - -We can do even better by presorting Y - -1. Divide $P_x$ into 2 halves using the mid point -2. Recursively computer the $d_l$ and $d_r$, take $\delta=min(d_l,d_r)$. -3. Filter points into y-strip: points which are within $(mid_x-\delta,mid_x+\delta)$ by visiting presorted $P_y$ - -```python -# d is distance function -def closestP(P,d): - Px=sorted(P,key=lambda x:x[0]) - Py=sorted(P,key=lambda x:x[1]) - def closestPRec(P,d): - n=len(P) - if n==1: - return float('inf') - if n==2: - return d(P[0],P[1]) - Q,R=Px[:n//2],Px[n//2:] - midx=R[0][0] - dl,dr=closestP(Q),closestP(R) - dc=min(dl,dr) - ys=[i if midx-dcdc: - break - dc=min(dc,curd) - return dc - return closestPRec(Px,d): -``` - -Runtime analysis: - -$$ -T(n)=2T(n/2)+\Theta(n)=\Theta(n\log n) -$$ - -## In-person lectures - -$$ -T(n)=aT(n/b)+f(n) -$$ - -$a$ is number of sub problems, $n/b$ is size of subproblems, $f(n)$ is the cost of divide and combine cost. - -### Example 3: Max Contiguous Subsequence Sum (MCSS) - -Given: array of integers (positive or negative), $S=[s_1,s_2,...,s_n]$ - -Return: $max\{\sum^i_{k=i} s_k|1\leq i\leq n, i\leq j\leq n\}$ - -Trivial solution: - -brute force -$O(n^3)$ - -A bit better solution: - -$O(n^2)$ use prefix sum to reduce cost for sum. - -Divide and conquer solution. - -```python -def MCSS(S): - def MCSSMid(S,i,j,mid): - res=S[j] - for l in range(i,j): - curS=0 - for r in range(l,j): - curS+=S[r] - res=max(res,curS) - return res - def MCSSRec(i,j): - if i==j: - return S[i] - mid=(i+j)//2 - L,R=MCSSRec(i,mid),MCSSRec(mid,j) - C=MCSSMid(i,j) - return min([L,C,R]) - return MCSSRec(0,len(S)) -``` - -If `MCSSMid(S,i,j,mid)` use trivial solution, the running time is: - -$$ -T(n)=2T(n/2)+O(n^2)=\Theta(n^2) -$$ - -and we did nothing. - -Observations: Any contiguous subsequence that starts on the left and ends on the right can be split into two parts as `sum(S[i:j])=sum(S[i:mid])+sum(S[mid,j])` - -and let $LS$ be the subsequence that has the largest sum that ends at mid, and $RS$ be the subsequence that has the largest sum on the right that starts at mid. - -**Lemma:** Biggest subsequence that contains `S[mid]` is $LS+RP$ - -Proof: - -By contradiction, - -Assume for the sake of contradiction that $y=L'+R'$ is a sum of such a subsequence that is larger than $x$ ($y>x$). - -Let $z=LS+R'$, since $LS\geq L'$, by definition of $LS$, then $z\geq y$, WOLG, $RS\geq R'$, $x\geq y$, which contradicts that $y>x$. - -Optimized function as follows: - -```python -def MCSS(S): - def MCSSMid(S,i,j,mid): - res=S[mid] - LS,RS=0,0 - cl,cr=0,0 - for l in range(mid-1,i-1,-1): - cl+=S[l] - LS=max(LS,cl) - for r in range(mid+1,j): - cr+=S[r] - RS=max(RS,cr) - return res+LS+RS - def MCSSRec(i,j): - if i==j: - return S[i] - mid=(i+j)//2 - L,R=MCSSRec(i,mid),MCSSRec(mid,j) - C=MCSSMid(i,j) - return min([L,C,R]) - return MCSSRec(0,len(S)) -``` - -The running time is: - -$$ -T(n)=2T(n/2)+O(n)=\Theta(n\log n) -$$ - -Strengthening the recusions: - -```python -def MCSS(S): - def MCSSRec(i,j): - if i==j: - return S[i],S[i],S[i],S[i] - mid=(i+j)//2 - L,lp,ls,sl=MCSSRec(i,mid) - R,rp,rs,sr=MCSSRec(mid,j) - return min([L,R,ls+rp]),max(lp,sl+rp),max(rs,sr+ls),sl+sr - return MCSSRec(0,len(S)) -``` - -Pre-computer version: - -```python -def MCSS(S): - pfx,sfx=[0],[S[-1]] - n=len(S) - for i in range(n-1): - pfx.append(pfx[-1]+S[i]) - sfx.insert(sfx[0]+S[n-i-2],0) - def MCSSRec(i,j): - if i==j: - return S[i],pfx[i],sfx[i] - mid=(i+j)//2 - L,lp,ls=MCSSRec(i,mid) - R,rp,rs=MCSSRec(mid,j) - return min([L,R,ls+rp]),max(lp,sfx[mid]-sfx[i]+rp),max(rs,sfx[j]-sfx[mid]+ls) - return MCSSRec(0,n) -``` - -$$ -T(n)=2T(n/2)+O(1)=\Theta(n) +# Lecture 2 + +## Divide and conquer + +Review of CSE 247 + +1. Divide the problem into (generally equal) smaller subproblems +2. Recursively solve the subproblems +3. Combine the solutions of subproblems to get the solution of the original problem + - Examples: Merge Sort, Binary Search + +Recurrence + +Master Method: + +$$ +T(n)=aT(\frac{n}{b})+\Theta(f(n)) +$$ + +### Example 1: Multiplying 2 numbers + +Normal Algorithm: + +```python +def multiply(x,y): + p=0 + for i in y: + p+=x*y + return p +``` + +divide and conquer approach + +```python +def multiply(x,y): + n=max(len(x),len(y)) + if n==1: + return x*y + xh,xl=x>>(n/2),x&((1<>(n/2),y&((1<>(n/2),x&((1<>(n/2),y&((1<p_y +\delta$ + +```python +# d is distance function +def closestP(P,d): + Px=sorted(P,key=lambda x:x[0]) + def closestPRec(P,d): + n=len(P) + if n==1: + return float('inf') + if n==2: + return d(P[0],P[1]) + Q,R=Px[:n//2],Px[n//2:] + midx=R[0][0] + dl,dr=closestP(Q),closestP(R) + dc=min(dl,dr) + ys=[i if midx-dcdc: + break + dc=min(dc,curd) + return dc + return closestPRec(Px,d): +``` + +Runtime analysis: + +$$ +T(n)=2T(n/2)+\Theta(n\log n)=\Theta(n\log^2 n) +$$ + +We can do even better by presorting Y + +1. Divide $P_x$ into 2 halves using the mid point +2. Recursively computer the $d_l$ and $d_r$, take $\delta=min(d_l,d_r)$. +3. Filter points into y-strip: points which are within $(mid_x-\delta,mid_x+\delta)$ by visiting presorted $P_y$ + +```python +# d is distance function +def closestP(P,d): + Px=sorted(P,key=lambda x:x[0]) + Py=sorted(P,key=lambda x:x[1]) + def closestPRec(P,d): + n=len(P) + if n==1: + return float('inf') + if n==2: + return d(P[0],P[1]) + Q,R=Px[:n//2],Px[n//2:] + midx=R[0][0] + dl,dr=closestP(Q),closestP(R) + dc=min(dl,dr) + ys=[i if midx-dcdc: + break + dc=min(dc,curd) + return dc + return closestPRec(Px,d): +``` + +Runtime analysis: + +$$ +T(n)=2T(n/2)+\Theta(n)=\Theta(n\log n) +$$ + +## In-person lectures + +$$ +T(n)=aT(n/b)+f(n) +$$ + +$a$ is number of sub problems, $n/b$ is size of subproblems, $f(n)$ is the cost of divide and combine cost. + +### Example 3: Max Contiguous Subsequence Sum (MCSS) + +Given: array of integers (positive or negative), $S=[s_1,s_2,...,s_n]$ + +Return: $max\{\sum^i_{k=i} s_k|1\leq i\leq n, i\leq j\leq n\}$ + +Trivial solution: + +brute force +$O(n^3)$ + +A bit better solution: + +$O(n^2)$ use prefix sum to reduce cost for sum. + +Divide and conquer solution. + +```python +def MCSS(S): + def MCSSMid(S,i,j,mid): + res=S[j] + for l in range(i,j): + curS=0 + for r in range(l,j): + curS+=S[r] + res=max(res,curS) + return res + def MCSSRec(i,j): + if i==j: + return S[i] + mid=(i+j)//2 + L,R=MCSSRec(i,mid),MCSSRec(mid,j) + C=MCSSMid(i,j) + return min([L,C,R]) + return MCSSRec(0,len(S)) +``` + +If `MCSSMid(S,i,j,mid)` use trivial solution, the running time is: + +$$ +T(n)=2T(n/2)+O(n^2)=\Theta(n^2) +$$ + +and we did nothing. + +Observations: Any contiguous subsequence that starts on the left and ends on the right can be split into two parts as `sum(S[i:j])=sum(S[i:mid])+sum(S[mid,j])` + +and let $LS$ be the subsequence that has the largest sum that ends at mid, and $RS$ be the subsequence that has the largest sum on the right that starts at mid. + +**Lemma:** Biggest subsequence that contains `S[mid]` is $LS+RP$ + +Proof: + +By contradiction, + +Assume for the sake of contradiction that $y=L'+R'$ is a sum of such a subsequence that is larger than $x$ ($y>x$). + +Let $z=LS+R'$, since $LS\geq L'$, by definition of $LS$, then $z\geq y$, WOLG, $RS\geq R'$, $x\geq y$, which contradicts that $y>x$. + +Optimized function as follows: + +```python +def MCSS(S): + def MCSSMid(S,i,j,mid): + res=S[mid] + LS,RS=0,0 + cl,cr=0,0 + for l in range(mid-1,i-1,-1): + cl+=S[l] + LS=max(LS,cl) + for r in range(mid+1,j): + cr+=S[r] + RS=max(RS,cr) + return res+LS+RS + def MCSSRec(i,j): + if i==j: + return S[i] + mid=(i+j)//2 + L,R=MCSSRec(i,mid),MCSSRec(mid,j) + C=MCSSMid(i,j) + return min([L,C,R]) + return MCSSRec(0,len(S)) +``` + +The running time is: + +$$ +T(n)=2T(n/2)+O(n)=\Theta(n\log n) +$$ + +Strengthening the recusions: + +```python +def MCSS(S): + def MCSSRec(i,j): + if i==j: + return S[i],S[i],S[i],S[i] + mid=(i+j)//2 + L,lp,ls,sl=MCSSRec(i,mid) + R,rp,rs,sr=MCSSRec(mid,j) + return min([L,R,ls+rp]),max(lp,sl+rp),max(rs,sr+ls),sl+sr + return MCSSRec(0,len(S)) +``` + +Pre-computer version: + +```python +def MCSS(S): + pfx,sfx=[0],[S[-1]] + n=len(S) + for i in range(n-1): + pfx.append(pfx[-1]+S[i]) + sfx.insert(sfx[0]+S[n-i-2],0) + def MCSSRec(i,j): + if i==j: + return S[i],pfx[i],sfx[i] + mid=(i+j)//2 + L,lp,ls=MCSSRec(i,mid) + R,rp,rs=MCSSRec(mid,j) + return min([L,R,ls+rp]),max(lp,sfx[mid]-sfx[i]+rp),max(rs,sfx[j]-sfx[mid]+ls) + return MCSSRec(0,n) +``` + +$$ +T(n)=2T(n/2)+O(1)=\Theta(n) $$ \ No newline at end of file diff --git a/pages/CSE347/CSE347_L3.md b/content/CSE347/CSE347_L3.md similarity index 96% rename from pages/CSE347/CSE347_L3.md rename to content/CSE347/CSE347_L3.md index a09f94a..6d9498e 100644 --- a/pages/CSE347/CSE347_L3.md +++ b/content/CSE347/CSE347_L3.md @@ -1,161 +1,161 @@ -# Lecture 3 - -## Dynamic programming - -When we cannot find a good Greedy Choice, the only thing we can do is to iterate all choices. - -### Example 1: Edit distance - -Input: 2 sequences of some character set, e.g. - -$S=ABCADA$, $T=ABADC$ - -Goal: Computer the minimum number of **insertions or deletions** you could do to convert $S$ into $T$ - -We will call it `Edit Distance(S[1...n],T[1...m])`. where `n` and `m` be the length of `S` and `T` respectively. - -Idea: computer difference between the sequences. - -Observe: The difference we observed appears at index 3, and in this example where the sequences are short, it is obvious that it is better to delete 'C'. But for long sequence, we donot know that the later sequence looks like so it is hard to make a decision on whether to insert 'A' or delete 'C'. - -Use branching algorithm: - -```python -def editDist(S,T,i,j): - if len(S)<=i: - return len(T) - if len(T)<=j: - return len(S) - if S[i]==T[j]: - return editDist(S,T,i+1,j+1) - else: - return min(editDist(S,T,i+1,j),editDist(S,T,i,j+1)) -``` - -Correctness Proof Outline: - -- ~~Greedy Choice Property~~ - -- Complete Choice Property: - - The optimal solution makes **one** of the choices that we consider -- Inductive Structure: - - Once you make **any** choice, you are left with a smaller problem of the same type. **Any** first choice + **feasible** solution to the subproblem = feasible solution to the entire problem. -- Optimal Substructure: - - If we optimally solve the subproblem for **a particular choice c**, and combine it with c, resulting solution is the **optimal solution that makes choice c**. - -Correctness Proof: - -Claim: For any problem $P$, the branking algorithm finds the optimal solution. - -Proof: Induct on problem size - -- Base case: $|S|=0$ or $|T|=0$, obvious -- Inductive Case: By inductive hypothesis: Branching algorithm works for all smaller problems, either $S$ is smaller or $T$ is smaller or both - - For each choice we make, we got a strictly smaller problem: by inductive structure, and the answer is correct by inductive hypothesis. - - By Optimal substructure, we know for any choice, the solution of branching algorithm for subproblem and the choice we make is an optimal solution for that problem. - - Using Complete choice property, we considered all the choices. - -Using tree graph, the left and right part of the tree has height n, but the middle part of the tree has height 2n. So the running time is $\Omega(2^n)$, at least $2^n$. - -#### How could we reduce the complexity? - -There are **overlapping subproblems** that we compute more than once! Number of distinct subproblems is polynomial, we can **share the solution** that we have already computed! - -**store the result of subprolem in 2D array** - -Use dp: - -```python -def editDist(S,T,i,j): - m,n=len(S),len(T) - dp=[[0]*(n+1) for _ in range(m+1)] - for i in range(n): - dp[i][m]=n-i - for i in range(m): - dp[n][j]=m-i - for i in range(m): - for j in range(n): - if S[i]==T[j]: - dp[i][j]=dp[i+1][j+1] - else: - # assuming the cost of insertion and deletion is 1 - dp[i][j]=min(1+dp[i][j+1],1+dp[i+1][j]) -``` - -We can use backtracking to find out how do we reach our final answer. Then the new runtime will be the time used to complete the table, which is $T(n,m)=\Theta(mn)$ - -### Example 2: Weighted Interval Scheduling (IS) - -Input: $P=\{p_1,p_2,...,p_n\}$, $p_i=\{s_i,f_i,w_i\}$ -$s_i$ is the start time, $f_i$ is the finish time, $w_i$ is the weight of the task for job $i$ - -Goal: Pick a set of **non-overlapping** intervals $\Pi$ such that $\sum_{p_i\in \Pi} w_i$ is maximized. - -Trivial solution ($T(n)=O(2^n)$) - -```python -# p=[[s_i,f_i,w_i],...] -p=[] -p.sort() -n=len(p) -def intervalScheduling(idx): - res=0 - if i>=n: - return res - for i in range(idx,n): - # pick when end - if p[idx][1]>p[i][0]: - continue - res=max(intervalScheduling(i+1)+p[i][2],res) -return intervalScheduling(0) -``` - -Using dp ($T(n)=O(n^2)$) - -```python -def intervalScheduling(p): - p.sort() - n=len(p) - dp=[0]*(n+1) - for i in range(n-1,-1,-1): - # load initial best case: do nothing - dp[i]=dp[i+1] - _,e,w=p[i] - for j in range(bisect.bisect_left(p,e,key=lambda x:x[0]),n+1): - dp[i]=max(dp[i],w+dp[j]) - return dp[0] -``` - -### Example 3: Subset sums - -Input: a set $S$ of positive and unique integers and another integer $K$. - -Problem: Is there a subset $X\subseteq S$ such that $sum(X)=K$ - -Brute force takes $O(2^n)$. - -```python -def subsetSum(arr,i,k)->bool: - if i>=len(arr): - if k==0: - return True - return False - return subsetSum(i+1,k-arr[i]) or subsetSum(i+1,k) -``` - -Using dp $O(nk)$ - -```python -def subsetSum(arr,k)->bool: - n=len(arr) - dp=[False]*(k+1) - dp[0]=True - for e in arr: - ndp=[] - for i in range(k+1): - ndp.append(dp[i]) - if i-e>=0: - ndp[i]|=dp[i-e] - dp=ndp - return dp[-1] -``` +# Lecture 3 + +## Dynamic programming + +When we cannot find a good Greedy Choice, the only thing we can do is to iterate all choices. + +### Example 1: Edit distance + +Input: 2 sequences of some character set, e.g. + +$S=ABCADA$, $T=ABADC$ + +Goal: Computer the minimum number of **insertions or deletions** you could do to convert $S$ into $T$ + +We will call it `Edit Distance(S[1...n],T[1...m])`. where `n` and `m` be the length of `S` and `T` respectively. + +Idea: computer difference between the sequences. + +Observe: The difference we observed appears at index 3, and in this example where the sequences are short, it is obvious that it is better to delete 'C'. But for long sequence, we donot know that the later sequence looks like so it is hard to make a decision on whether to insert 'A' or delete 'C'. + +Use branching algorithm: + +```python +def editDist(S,T,i,j): + if len(S)<=i: + return len(T) + if len(T)<=j: + return len(S) + if S[i]==T[j]: + return editDist(S,T,i+1,j+1) + else: + return min(editDist(S,T,i+1,j),editDist(S,T,i,j+1)) +``` + +Correctness Proof Outline: + +- ~~Greedy Choice Property~~ + +- Complete Choice Property: + - The optimal solution makes **one** of the choices that we consider +- Inductive Structure: + - Once you make **any** choice, you are left with a smaller problem of the same type. **Any** first choice + **feasible** solution to the subproblem = feasible solution to the entire problem. +- Optimal Substructure: + - If we optimally solve the subproblem for **a particular choice c**, and combine it with c, resulting solution is the **optimal solution that makes choice c**. + +Correctness Proof: + +Claim: For any problem $P$, the branking algorithm finds the optimal solution. + +Proof: Induct on problem size + +- Base case: $|S|=0$ or $|T|=0$, obvious +- Inductive Case: By inductive hypothesis: Branching algorithm works for all smaller problems, either $S$ is smaller or $T$ is smaller or both + - For each choice we make, we got a strictly smaller problem: by inductive structure, and the answer is correct by inductive hypothesis. + - By Optimal substructure, we know for any choice, the solution of branching algorithm for subproblem and the choice we make is an optimal solution for that problem. + - Using Complete choice property, we considered all the choices. + +Using tree graph, the left and right part of the tree has height n, but the middle part of the tree has height 2n. So the running time is $\Omega(2^n)$, at least $2^n$. + +#### How could we reduce the complexity? + +There are **overlapping subproblems** that we compute more than once! Number of distinct subproblems is polynomial, we can **share the solution** that we have already computed! + +**store the result of subprolem in 2D array** + +Use dp: + +```python +def editDist(S,T,i,j): + m,n=len(S),len(T) + dp=[[0]*(n+1) for _ in range(m+1)] + for i in range(n): + dp[i][m]=n-i + for i in range(m): + dp[n][j]=m-i + for i in range(m): + for j in range(n): + if S[i]==T[j]: + dp[i][j]=dp[i+1][j+1] + else: + # assuming the cost of insertion and deletion is 1 + dp[i][j]=min(1+dp[i][j+1],1+dp[i+1][j]) +``` + +We can use backtracking to find out how do we reach our final answer. Then the new runtime will be the time used to complete the table, which is $T(n,m)=\Theta(mn)$ + +### Example 2: Weighted Interval Scheduling (IS) + +Input: $P=\{p_1,p_2,...,p_n\}$, $p_i=\{s_i,f_i,w_i\}$ +$s_i$ is the start time, $f_i$ is the finish time, $w_i$ is the weight of the task for job $i$ + +Goal: Pick a set of **non-overlapping** intervals $\Pi$ such that $\sum_{p_i\in \Pi} w_i$ is maximized. + +Trivial solution ($T(n)=O(2^n)$) + +```python +# p=[[s_i,f_i,w_i],...] +p=[] +p.sort() +n=len(p) +def intervalScheduling(idx): + res=0 + if i>=n: + return res + for i in range(idx,n): + # pick when end + if p[idx][1]>p[i][0]: + continue + res=max(intervalScheduling(i+1)+p[i][2],res) +return intervalScheduling(0) +``` + +Using dp ($T(n)=O(n^2)$) + +```python +def intervalScheduling(p): + p.sort() + n=len(p) + dp=[0]*(n+1) + for i in range(n-1,-1,-1): + # load initial best case: do nothing + dp[i]=dp[i+1] + _,e,w=p[i] + for j in range(bisect.bisect_left(p,e,key=lambda x:x[0]),n+1): + dp[i]=max(dp[i],w+dp[j]) + return dp[0] +``` + +### Example 3: Subset sums + +Input: a set $S$ of positive and unique integers and another integer $K$. + +Problem: Is there a subset $X\subseteq S$ such that $sum(X)=K$ + +Brute force takes $O(2^n)$. + +```python +def subsetSum(arr,i,k)->bool: + if i>=len(arr): + if k==0: + return True + return False + return subsetSum(i+1,k-arr[i]) or subsetSum(i+1,k) +``` + +Using dp $O(nk)$ + +```python +def subsetSum(arr,k)->bool: + n=len(arr) + dp=[False]*(k+1) + dp[0]=True + for e in arr: + ndp=[] + for i in range(k+1): + ndp.append(dp[i]) + if i-e>=0: + ndp[i]|=dp[i-e] + dp=ndp + return dp[-1] +``` diff --git a/pages/CSE347/CSE347_L4.md b/content/CSE347/CSE347_L4.md similarity index 97% rename from pages/CSE347/CSE347_L4.md rename to content/CSE347/CSE347_L4.md index 3461451..bfb21fc 100644 --- a/pages/CSE347/CSE347_L4.md +++ b/content/CSE347/CSE347_L4.md @@ -1,321 +1,321 @@ -# Lecture 4 - -## Maximum Flow - -### Example 1: Ship cement from factory to building - -Input $s$: source, $t$: destination - -Graph with **directed** edges weights on each edge: **capacity** - -**Goal:** Ship as much stuff as possible while obeying capacity constrains. - -Graph: $(V,E)$ directed and weighted - -- Unique source and sink nodes $\to s, t$ -- Each edge has capacity $c(e)$ [Integer] - -A valid flow assignment assigns an integer $f(e)$ to each edge s.t. - -Capacity constraint: $0\leq f(e)\leq c(e)$ - -Flow conservation: - -$$ -\sum_{e\in E_{in}(v)}f(e)=\sum_{e\in E_{out}(v)}f(e),\forall v\in V-{s,t} -$$ - -$E_{in}(v)$: set of incoming edges to $v$ -$E_{out}(v)$: set of outgoing edges from $v$ - -Compute: Maximum Flow: Find a valid flow assignment to - -Maximize $|F|=\sum_{e\in E_{in}(t)}f(e)=\sum_{e\in E_{out}(s)}f(e)$ (total units received by end and sent by source) - -Additional assumptions - -1. $s$ has no incoming edges, $t$ has no outgoing edges -2. You do not have a cycle of 2 nodes - -A proposed algorithm: - -1. Find a path from $s$ to $t$ -2. Push as much flow along the path as possible -3. Adjust the capacities -4. Repeat until we cannot find a path - -**Residual Graph:** If there is an edge $e=(u,v)$ in $G$, we will add a back edge $\bar{e}=(v,u)$. Capacity of $\bar{e}=$ flow on $e$. Call this graph $G_R$. - -Algorithm: - -- Find an "augmenting path" $P$. - - $P$ can contain forward or backward edges! -- Say the smallest residual capacity along the path is $k$. -- Push $k$ flow on the path ($f(e) =f(e) + k$ for all edges on path $P$) - - Reduce the capacity of all edges on the path $P$ by $k$ - - **Increase** the capacity of the corresponding mirror/back edges -- Repeat until there are no augmenting paths - -### Formalize: Ford-Fulkerson (FF) Algorithm - -1. Initialize the residual graph $G_R=G$ -2. Find an augmenting path $P$ with capacity $k$ (min capacity of any edge on $P$) -3. Fix up the residual capacities in $G_R$ - - $c(e)=c(e)-k,\forall e\in P$ - - $c(\bar{e})=c(\bar{e})+k,\forall \bar{e}\in P$ -4. Repeat 2 and 3 until no augmenting path can be found in $G_R$. - -```python -def ford_fulkerson_algo(G,n,s,t): - """ - Args: - G: is the graph for max_flow - n: is the number of vertex in the graph - s: start vertex of flow - t: end vertex of flow - Returns: - the max flow in graph from s to t - """ - # Initialize the residual graph $G_R=G$ - GR=[defaultdict(int) for i in range(n)] - for i in range(n): - for v,_ in enumerate(G[i]): - # weight w is unused - GR[v][i]=0 - path=set() - def augP(cur): - # Find an augumentting path $P$ with capacity $k$ (min capacity of any edge on $P$) - if cur==t: return True - # true for edge in residual path, false for edge in graph - for v,w in G[cur]: - if w==0 or (cur,v,False) in path: continue - path.add((cur,v,False)) - if augP(v): return True - path.remove((cur,v,False)) - for v,w in GR[cur]: - if w==0 or (cur,v,True) in path: continue - path.add((cur,v,True)) - if augP(v): return True - path.remove((cur,v,True)) - return False - while augP(s): - k=min([GR[a][b] if isR else G[a][b] for a,b,isR in path]) - # Fix up the residual capacities in $G_R$ - # - $c(e)=c(e)-k,\forall e\in P$ - # - $c(\bar{e})=c(\bar{e})+k,\forall \bar{e}\in P$ - for a,b,isR in path: - if isR: - GR[a][b]+=k - else: - G[a][b]-=k - return sum(GR[s].values()) -``` - -#### Proof of Correctness: Valid Flow - -**Lemma 1:** FF finds a valid flow - -- Capacity and conservation constrains are not violated -- Capacity constraint: $0\leq f(e)\leq c(e)$ -- Flow conservation: $\sum_{e\in E_{in}(v)}f(e)=\sum_{e\in E_{out}(v)}f(e),\forall v\in V-\{s,t\}$ - -Proof: We proceed by induction on **augmenting paths** - -##### Base Case - -$f(e)=0$ on all edges - -##### Inductive Case - -By inductive hypothesis, we have a valid flow and the corresponding residual graph $G_R$. - -Inductive Step: - -Now we find an augmented path $P$ in $GR$, pushed $k$ (which is the smallest edge capacity on $P$). Argue that the constraints are not violated. - -**Capacity Constrains:** Consider an edge $e$ in $P$. - -- If $e$ is an forward edge (in the original graph) - - by construction of $G_R$, it had left over capacities. -- If $e$ is an back edge with residual capacity $\geq k$ - - flow on real edge reduces, but the real capacity is still $\geq 0$, no capacity constrains violation. - -**Conservation Constrains:** Consider a vertex $v$ on path $P$ - -1. Both forward edges - - No violation, push $k$ flow into $v$ and out. -2. Both back edges - - No violation, push $k$ less flow into $v$ and out. -3. Redirecting flow - - No violation, change of $0$ by $k-k$ on $v$. - -#### Proof of Correctness: Termination - -**Lemma 2:** FF terminate - -Proof: - -Every time it finds an augmenting path that increases the total flow. - -Must terminate either when it finds a max flow or before. - -Each iteration we use $\Theta(m+n)$ to find a valid path. - -The number of iteration $\leq |F|$, the total is $\Theta(|F|(m+n))$ (not polynomial time) - -#### Proof of Correctness: Optimality - -From Lemma 1 and 2, we know that FF returns a feasible solution, but does it return the **maximum** flow? - -##### Max-flow Min-cut Theorem - -Given a graph $G(V,E)$, a **graph cut** is a partition of vertices into 2 subsets. - -- $S$: $s$ + maybe some other vertices -- $V-S$: $t$ + maybe some other vertices - -Define capacity of the cut be the sum of capacity of edges that go from a vertex in $S$ to a vertex in $T$. - -**Lemma 3:** For all valid flows $f$, $|f|\leq C(S)$ for all cut $S$ (Max-flow $\leq$ Min-cut) - -Proof: all flow must go through one of the cut edges. - -**Min-cut:** cut of smallest capacity, $S^*$. $|f|\leq C(S^*)$ - -**Lemma 4:** FF produces a flow $=C(S^*)$ - -Proof: Let $\hat{f}$ be the flow found by FF. Mo augmenting paths in $G_R$. - -Let $\hat{S}$ be all vertices that can be reached from $s$ using edges with capacities $>0$. - -and all the forward edges going out of the cut are saturated. Since back edges have capacity 0, no flow is going into the cut $S$. - -If some flow was coming from $V-\hat{S}$, then there must be some edges with capacity $>0$. So, $|f|\leq C(S^*)$ - -### Example 2: Bipartite Matching - -input: Given $n$ classes and $n$ rooms; we want to match classes to rooms. - -Bipartite graph $G=(V,E)$ (unweighted and undirected) - -- Vertices are either in set $L$ or $R$ -- Edges only go between vertices of different sets - -Matching: A subset of edges $M\subseteq E$ s.t. - -- Each vertex has at most one edge from $M$ incident on it. - -Maximum Matching: matching of the largest size. - -We will reduce the problem to the problem of finding the maximum flow - -#### Reduction - -Given a bipartite graph $G=(V,E)$, construct a graph $G'=(V',E')$ such that - -$$ -|max-flow (G')|=|max-flow(G)| -$$ - -Let $s$ connects to all vertices in $L$ and all vertex in $R$ connects to $t$. - -$G'=G+s+t+$added edges form $S$ to $T$ and added capacities. - -#### Proof of correctness - -Claim: $G'$ has a flow of $k$ iff $G$ has a matching of size $k$ - -Proof: Two directions: - -1. Say $G$ has a matching of size $k$, we want to prove $G'$ has a flow of size $k$. -2. Say $G'$ has a flow of size $k$, we want to prove $G$ has a matching of size $k$. - -## Conclusion: Maximum Flow - -Problem input and target - -Ford-Fulkerson Algorithm - -- Execution: residual graph -- Runtime - -FF correctness proof - -- Max-flow Min-cut Theorem -- Graph Cut definition -- Capacity of cut - -Reduction to Bipartite Matching - -### Example 3: Image Segmentation: (reduction from min-cut) - -Given: - -- Image consisting of an object and a background. -- the object occupies some set of pixels $A$, while the background occupies the remaining pixels $B$. - -Required: - -- Separate $A$ from $B$ but if doesn't know which pixels are each. -- For each pixel $i,p_i$ is the probability that $i\in A$ -- For each pair of adjacent pixels $i,j,c_{ij}$ is the cost of placing the object boundary between them. i.e. putting $i$ in $A$ and $j$ in $B$. -- A segmentation of the image is an assignment of each pixel to $A$ or $B$. -- The goal is to find a segmentation that maximizes - -$$ -\sum_{i\in A}p_i+\sum_{i\in B}(1-p_i)-\sum_{i,j\ on \ boundary}c_{ij} -$$ - -Solution: - -- Let's turn our maximization into a minimization -- If the image has $N$ pixels, then we can rewrite the objective as - -$$ -N-\sum_{i\in A}(1-p_i)-\sum_{i\in B}p_i-\sum_{i,j\ on \ boundary}c_{ij} -$$ - -because $N=\sum_{i\in A}p_i+\sum_{i\in A}(1-p_i)+\sum_{i\in B}p_i+\sum_{i\in B}(1-p_i)$ boundary - -New maximization problem: - -$$ -Max\left( N-\sum_{i\in A}(1-p_i)-\sum_{i\in B}p_i-\sum_{i,j\ on \ boundary}c_{ij}\right) -$$ - -Now, this is equivalent ot minimizing - -$$ -\sum_{i\in A}(1-p_i)+\sum_{i\in B}p_i+\sum_{i,j\ on \ boundary}c_{ij} -$$ - -Second steps - -- Form a graph with $n$ vertices, $v_i$ on for each pixel -- Add vertices $s$ and $t$ -- For each $v_i$, add edges $S-T$ cut of $G$ assigned each $v_i$ to either $S$ side or $T$ side. -- The $S$ side of an $S-T$ is the $A$ side, while the $T$ side of the cur is the $B$ side. -- Observer that if $v_i$ goes on the $S$ side, it becomes part of $A$, so the cut increases by $1-p$. Otherwise, it become part of $B$, so the cut increases by $p_i$ instead. -- Now add edges $v_i\to v_j$ with capacity $c_{ij}$ for all adjacent pixels pairs $i,j$ -- If $v_i$ and $v_j$ end up on opposite sides of the cut (boundary), then the cut increases by $c_{ij}$. -- Conclude that any $S-T$ cut that assigns $S\subseteq V$ to the $A$ side and $V\backslash S$ to the $B$ side pays a total of - 1. $1-p_i$ for each $v_i$ on the $A$ side - 2. $p_i$ for each $v_i$ on the $B$ side - 3. $c_{ij}$ for each adjacent pair $i,j$ that is at the boundary. i.e. $i\in S\ and\ j\in V\backslash S$ -- Conclude that a cut with a capacity $c$ implies a segmentation with objective value $cs$. -- The converse can (and should) be also checked: a segmentation with subjective value $c$ implies a $S-T$ cut with capacity $c$. - -#### Algorithm - -- Given an image with $N$ pixels, build the graph $G$ as desired. -- Use the FF algorithm to find a minimum $S-T$ cut of $G$ -- Use this cut to assign each pixel to $A$ or $B$ as described, i.e pixels that correspond to vertices on the $S$ side are assigned to $A$ and those corresponding to vertices on the $T$ side to $B$. -- Minimizing the cut capacity minimizes our transformed minimization objective function. - -#### Running time - -The graph $G$ contains $\Theta(N)$ edges, because each pixel is adjacent to a maximum of of 4 neighbors and $S$ and $T$. - -FF algorithm has running time $O((m+n)|F|)$, where $|F|\leq |n|$ is the size of set of min-cut. The edge count is $m=6n$. - +# Lecture 4 + +## Maximum Flow + +### Example 1: Ship cement from factory to building + +Input $s$: source, $t$: destination + +Graph with **directed** edges weights on each edge: **capacity** + +**Goal:** Ship as much stuff as possible while obeying capacity constrains. + +Graph: $(V,E)$ directed and weighted + +- Unique source and sink nodes $\to s, t$ +- Each edge has capacity $c(e)$ [Integer] + +A valid flow assignment assigns an integer $f(e)$ to each edge s.t. + +Capacity constraint: $0\leq f(e)\leq c(e)$ + +Flow conservation: + +$$ +\sum_{e\in E_{in}(v)}f(e)=\sum_{e\in E_{out}(v)}f(e),\forall v\in V-{s,t} +$$ + +$E_{in}(v)$: set of incoming edges to $v$ +$E_{out}(v)$: set of outgoing edges from $v$ + +Compute: Maximum Flow: Find a valid flow assignment to + +Maximize $|F|=\sum_{e\in E_{in}(t)}f(e)=\sum_{e\in E_{out}(s)}f(e)$ (total units received by end and sent by source) + +Additional assumptions + +1. $s$ has no incoming edges, $t$ has no outgoing edges +2. You do not have a cycle of 2 nodes + +A proposed algorithm: + +1. Find a path from $s$ to $t$ +2. Push as much flow along the path as possible +3. Adjust the capacities +4. Repeat until we cannot find a path + +**Residual Graph:** If there is an edge $e=(u,v)$ in $G$, we will add a back edge $\bar{e}=(v,u)$. Capacity of $\bar{e}=$ flow on $e$. Call this graph $G_R$. + +Algorithm: + +- Find an "augmenting path" $P$. + - $P$ can contain forward or backward edges! +- Say the smallest residual capacity along the path is $k$. +- Push $k$ flow on the path ($f(e) =f(e) + k$ for all edges on path $P$) + - Reduce the capacity of all edges on the path $P$ by $k$ + - **Increase** the capacity of the corresponding mirror/back edges +- Repeat until there are no augmenting paths + +### Formalize: Ford-Fulkerson (FF) Algorithm + +1. Initialize the residual graph $G_R=G$ +2. Find an augmenting path $P$ with capacity $k$ (min capacity of any edge on $P$) +3. Fix up the residual capacities in $G_R$ + - $c(e)=c(e)-k,\forall e\in P$ + - $c(\bar{e})=c(\bar{e})+k,\forall \bar{e}\in P$ +4. Repeat 2 and 3 until no augmenting path can be found in $G_R$. + +```python +def ford_fulkerson_algo(G,n,s,t): + """ + Args: + G: is the graph for max_flow + n: is the number of vertex in the graph + s: start vertex of flow + t: end vertex of flow + Returns: + the max flow in graph from s to t + """ + # Initialize the residual graph $G_R=G$ + GR=[defaultdict(int) for i in range(n)] + for i in range(n): + for v,_ in enumerate(G[i]): + # weight w is unused + GR[v][i]=0 + path=set() + def augP(cur): + # Find an augumentting path $P$ with capacity $k$ (min capacity of any edge on $P$) + if cur==t: return True + # true for edge in residual path, false for edge in graph + for v,w in G[cur]: + if w==0 or (cur,v,False) in path: continue + path.add((cur,v,False)) + if augP(v): return True + path.remove((cur,v,False)) + for v,w in GR[cur]: + if w==0 or (cur,v,True) in path: continue + path.add((cur,v,True)) + if augP(v): return True + path.remove((cur,v,True)) + return False + while augP(s): + k=min([GR[a][b] if isR else G[a][b] for a,b,isR in path]) + # Fix up the residual capacities in $G_R$ + # - $c(e)=c(e)-k,\forall e\in P$ + # - $c(\bar{e})=c(\bar{e})+k,\forall \bar{e}\in P$ + for a,b,isR in path: + if isR: + GR[a][b]+=k + else: + G[a][b]-=k + return sum(GR[s].values()) +``` + +#### Proof of Correctness: Valid Flow + +**Lemma 1:** FF finds a valid flow + +- Capacity and conservation constrains are not violated +- Capacity constraint: $0\leq f(e)\leq c(e)$ +- Flow conservation: $\sum_{e\in E_{in}(v)}f(e)=\sum_{e\in E_{out}(v)}f(e),\forall v\in V-\{s,t\}$ + +Proof: We proceed by induction on **augmenting paths** + +##### Base Case + +$f(e)=0$ on all edges + +##### Inductive Case + +By inductive hypothesis, we have a valid flow and the corresponding residual graph $G_R$. + +Inductive Step: + +Now we find an augmented path $P$ in $GR$, pushed $k$ (which is the smallest edge capacity on $P$). Argue that the constraints are not violated. + +**Capacity Constrains:** Consider an edge $e$ in $P$. + +- If $e$ is an forward edge (in the original graph) + - by construction of $G_R$, it had left over capacities. +- If $e$ is an back edge with residual capacity $\geq k$ + - flow on real edge reduces, but the real capacity is still $\geq 0$, no capacity constrains violation. + +**Conservation Constrains:** Consider a vertex $v$ on path $P$ + +1. Both forward edges + - No violation, push $k$ flow into $v$ and out. +2. Both back edges + - No violation, push $k$ less flow into $v$ and out. +3. Redirecting flow + - No violation, change of $0$ by $k-k$ on $v$. + +#### Proof of Correctness: Termination + +**Lemma 2:** FF terminate + +Proof: + +Every time it finds an augmenting path that increases the total flow. + +Must terminate either when it finds a max flow or before. + +Each iteration we use $\Theta(m+n)$ to find a valid path. + +The number of iteration $\leq |F|$, the total is $\Theta(|F|(m+n))$ (not polynomial time) + +#### Proof of Correctness: Optimality + +From Lemma 1 and 2, we know that FF returns a feasible solution, but does it return the **maximum** flow? + +##### Max-flow Min-cut Theorem + +Given a graph $G(V,E)$, a **graph cut** is a partition of vertices into 2 subsets. + +- $S$: $s$ + maybe some other vertices +- $V-S$: $t$ + maybe some other vertices + +Define capacity of the cut be the sum of capacity of edges that go from a vertex in $S$ to a vertex in $T$. + +**Lemma 3:** For all valid flows $f$, $|f|\leq C(S)$ for all cut $S$ (Max-flow $\leq$ Min-cut) + +Proof: all flow must go through one of the cut edges. + +**Min-cut:** cut of smallest capacity, $S^*$. $|f|\leq C(S^*)$ + +**Lemma 4:** FF produces a flow $=C(S^*)$ + +Proof: Let $\hat{f}$ be the flow found by FF. Mo augmenting paths in $G_R$. + +Let $\hat{S}$ be all vertices that can be reached from $s$ using edges with capacities $>0$. + +and all the forward edges going out of the cut are saturated. Since back edges have capacity 0, no flow is going into the cut $S$. + +If some flow was coming from $V-\hat{S}$, then there must be some edges with capacity $>0$. So, $|f|\leq C(S^*)$ + +### Example 2: Bipartite Matching + +input: Given $n$ classes and $n$ rooms; we want to match classes to rooms. + +Bipartite graph $G=(V,E)$ (unweighted and undirected) + +- Vertices are either in set $L$ or $R$ +- Edges only go between vertices of different sets + +Matching: A subset of edges $M\subseteq E$ s.t. + +- Each vertex has at most one edge from $M$ incident on it. + +Maximum Matching: matching of the largest size. + +We will reduce the problem to the problem of finding the maximum flow + +#### Reduction + +Given a bipartite graph $G=(V,E)$, construct a graph $G'=(V',E')$ such that + +$$ +|max-flow (G')|=|max-flow(G)| +$$ + +Let $s$ connects to all vertices in $L$ and all vertex in $R$ connects to $t$. + +$G'=G+s+t+$added edges form $S$ to $T$ and added capacities. + +#### Proof of correctness + +Claim: $G'$ has a flow of $k$ iff $G$ has a matching of size $k$ + +Proof: Two directions: + +1. Say $G$ has a matching of size $k$, we want to prove $G'$ has a flow of size $k$. +2. Say $G'$ has a flow of size $k$, we want to prove $G$ has a matching of size $k$. + +## Conclusion: Maximum Flow + +Problem input and target + +Ford-Fulkerson Algorithm + +- Execution: residual graph +- Runtime + +FF correctness proof + +- Max-flow Min-cut Theorem +- Graph Cut definition +- Capacity of cut + +Reduction to Bipartite Matching + +### Example 3: Image Segmentation: (reduction from min-cut) + +Given: + +- Image consisting of an object and a background. +- the object occupies some set of pixels $A$, while the background occupies the remaining pixels $B$. + +Required: + +- Separate $A$ from $B$ but if doesn't know which pixels are each. +- For each pixel $i,p_i$ is the probability that $i\in A$ +- For each pair of adjacent pixels $i,j,c_{ij}$ is the cost of placing the object boundary between them. i.e. putting $i$ in $A$ and $j$ in $B$. +- A segmentation of the image is an assignment of each pixel to $A$ or $B$. +- The goal is to find a segmentation that maximizes + +$$ +\sum_{i\in A}p_i+\sum_{i\in B}(1-p_i)-\sum_{i,j\ on \ boundary}c_{ij} +$$ + +Solution: + +- Let's turn our maximization into a minimization +- If the image has $N$ pixels, then we can rewrite the objective as + +$$ +N-\sum_{i\in A}(1-p_i)-\sum_{i\in B}p_i-\sum_{i,j\ on \ boundary}c_{ij} +$$ + +because $N=\sum_{i\in A}p_i+\sum_{i\in A}(1-p_i)+\sum_{i\in B}p_i+\sum_{i\in B}(1-p_i)$ boundary + +New maximization problem: + +$$ +Max\left( N-\sum_{i\in A}(1-p_i)-\sum_{i\in B}p_i-\sum_{i,j\ on \ boundary}c_{ij}\right) +$$ + +Now, this is equivalent ot minimizing + +$$ +\sum_{i\in A}(1-p_i)+\sum_{i\in B}p_i+\sum_{i,j\ on \ boundary}c_{ij} +$$ + +Second steps + +- Form a graph with $n$ vertices, $v_i$ on for each pixel +- Add vertices $s$ and $t$ +- For each $v_i$, add edges $S-T$ cut of $G$ assigned each $v_i$ to either $S$ side or $T$ side. +- The $S$ side of an $S-T$ is the $A$ side, while the $T$ side of the cur is the $B$ side. +- Observer that if $v_i$ goes on the $S$ side, it becomes part of $A$, so the cut increases by $1-p$. Otherwise, it become part of $B$, so the cut increases by $p_i$ instead. +- Now add edges $v_i\to v_j$ with capacity $c_{ij}$ for all adjacent pixels pairs $i,j$ +- If $v_i$ and $v_j$ end up on opposite sides of the cut (boundary), then the cut increases by $c_{ij}$. +- Conclude that any $S-T$ cut that assigns $S\subseteq V$ to the $A$ side and $V\backslash S$ to the $B$ side pays a total of + 1. $1-p_i$ for each $v_i$ on the $A$ side + 2. $p_i$ for each $v_i$ on the $B$ side + 3. $c_{ij}$ for each adjacent pair $i,j$ that is at the boundary. i.e. $i\in S\ and\ j\in V\backslash S$ +- Conclude that a cut with a capacity $c$ implies a segmentation with objective value $cs$. +- The converse can (and should) be also checked: a segmentation with subjective value $c$ implies a $S-T$ cut with capacity $c$. + +#### Algorithm + +- Given an image with $N$ pixels, build the graph $G$ as desired. +- Use the FF algorithm to find a minimum $S-T$ cut of $G$ +- Use this cut to assign each pixel to $A$ or $B$ as described, i.e pixels that correspond to vertices on the $S$ side are assigned to $A$ and those corresponding to vertices on the $T$ side to $B$. +- Minimizing the cut capacity minimizes our transformed minimization objective function. + +#### Running time + +The graph $G$ contains $\Theta(N)$ edges, because each pixel is adjacent to a maximum of of 4 neighbors and $S$ and $T$. + +FF algorithm has running time $O((m+n)|F|)$, where $|F|\leq |n|$ is the size of set of min-cut. The edge count is $m=6n$. + So the total running time is $O(n^2)$ \ No newline at end of file diff --git a/pages/CSE347/CSE347_L5.md b/content/CSE347/CSE347_L5.md similarity index 97% rename from pages/CSE347/CSE347_L5.md rename to content/CSE347/CSE347_L5.md index fa16035..eeed02a 100644 --- a/pages/CSE347/CSE347_L5.md +++ b/content/CSE347/CSE347_L5.md @@ -1,341 +1,341 @@ -# Lecture 5 - -## Takeaway from Bipartite Matching - -- We saw how to solve a problem (bi-partite matching and others) by reducing it to another problem (maximum flow). -- In general, we can design an algorithm to map instances of a new problem to instances of known solvable problem (e.g., max-flow) to solve this new problem! -- Mapping from one problem to another which preserves solutions is called reduction. - -## Reduction: Basic Ideas - -Convert solutions to the known problem to the solutions to the new problem - -- Instance of new problem -- Instance of known problem -- Solution of known problem -- Solution of new problem - -## Reduction: Formal Definition - -Problems $L,K$. - -$L$ reduces to $K$ ($L\leq K$) if there is a mapping $\phi$ from **any** instance $l\in L$ to some instance $\phi(l)\in K'\subset K$, such that the solution for $\phi(l)$ yields a solution for $l$. - -This means that **L is no harder than K** - -### Using reduction to design algorithms - -In the example of reduction to solve Bipartite Matching: - -$L:$ Bipartite Matching - -$K:$ Max-flow Problem - -Efficiency: - -1. Reduction: $\phi:l\to\phi(l)$ (Polynomial time reduction $\phi(l)$) -2. Solve prom $\phi(l)$ (Polynomial time to solve $poly(g)$) -3. Convert the solution for $\phi(l)$ to a solution to $l$ (Polynomial time to solve $poly(g)$) - -### Efficient Reduction - -A reduction $\phi:l\to\phi(l)$ is efficient ($L\leq p(k)$) if for any $l\in L$: - -1. $\phi(l)$ is computable from $l$ in polynomial ($|l|$) time. -2. Solution to $l$ is computable from solution of $\phi(l)$ in polynomial ($|l|$) time. - -We call $L$ is **poly-time reducible** to $K$, or $L$ poly-time -reduces to $K$. - -### Which problem is harder? - -Theorem: If $L\leq p(k)$ and there is a polynomial time algorithm to solve $K$, then there is a polynomial time algorithm to solve $L$. - -Proof: Given an instance of $l\in L$ If we can convert the problem in polynomial time with respect to the original problem $l$. - -1. Compute $\phi(l)$: $p(l)$ -2. Solve $\phi(l)$: $p(\phi(l))$ -3. Convert solution: $p(\phi(l))$ - -Total time: $p(l)+p(\phi(l))+p(\phi(l))=p(l)+p(\phi(l))$ -Need to show: $|\phi(l)|=poly(|l|)$ - -Proof: - -Since we can convert $\phi(l)$ in $p(l)$ time, and on every time step, (constant step) we can only write constant amount of data. - -So $|\phi(l)|=poly(|l|)$ - -## Hardness Problems - -Reductions show the relationship between problem hardness! - -Question: Could you solve a problem in polynomial time? - -Easy: polynomial time solution -Hard: No polynomial time solution (as far as we know) - -### Types of Problems - -Decision Problem: Yes/No answer - -Examples: Subset sums - -1. Is the there a flow of size $F$ -2. Is there a shortest path of length $L$ from vertex $u$ to vertex $v$. -3. Given a set of intercal, can you schedule $k$ of them. - -Optimization Problem: What is the value of an optimal feasible solution of a problem? - -- Minimization: Minimize cost - - min cut - - minimal spanning tree - - shortest path -- Maximization: Maximize profit - - interval scheduling - - maximum flow - - maximum matching - -#### Canonical Decision Problem - -Does the instance $l\in L$ (an optimization problem) have a feasible solution with objective value $k$: - -Objective value $\geq k$ (maximization) $\leq k$ (minimization) - -$DL$ is the reduced Canonical Decision problem $L$ - -##### Hardness of Canonical Decision Problems - -Lemma 1: $DL\leq p(L)$ ($DL$ is no harder than $L$) - -Proof: Assume $L$ **maximization** problem $DL(l)$: does have a solution $\geq k$. - -Example: Does graph $G$ have flow $\geq k$. - -Let $v^∗$ be the maximum objective on $l$ by solving $l$. - -Let the instance of $DL:(l,k)$ and $l$ be the problem and $k$ be the objective - -1. $l\to \phi(l)\in L$ (optimization problem) $\phi(l,k)=l$ -2. Is $v^*(l)\geq k$? If so, return true, else return false. - -Lemma 2: If $v^* =O(c^{|l|})$ for any constant $c$, then $L\leq p(DL)$. - -Proof: First we could show $L\leq DL$. Suppose maximization problem, canonical decision problem is is there a solution $\geq k$. - -Naïve Linear Search: Ask $DL(l,k)$, if returns false, ask $DL(l,k+1)$ until returns true - -Runtime: At most $k$ search to iterate all possibilities. - -This is exponential! How to reduce it? - -Our old friend Binary (exponential) Search is back! - -You gets a no at some value: try power of 2 until you get a no, then do binary search - -\# questions: $=log_2(v^*(l))=poly(l)$ - -Binary search in area: from last yes to first no. - -Runtime: Binary search ($O(n)=\log(v^*(l))$) - -### Reduction for Algorithm Design vs Hardness - -For problems $L,K$ - -If $K$ is “easy” (exists a poly-time solution), then $L$ is also easy. - -If $L$ is “hard” (no poly-time solution), then $k$ is also hard. - -Every problem that we worked on so far, $K$ is “easy”, so we reduce from new problem to known problem (e.g., max-flow). - -#### Reduction for Hardness: Independent Set (ISET) - -Input: Given an undirected graph $G = (V,E)$, - -A subset of vertices $S\subset V$ is called an **independent set** if no two vertices of are connected by an edge. - -Problem: Does $G$ contain an independent set of size $\geq k$? - -$ISET(G,k)$ returns true if $G$ contains an independent set of size $\geq k$, and false otherwise. - -Algorithm? NO! We think that this is a hard problem. - -A lot of pQEDle have tried and could not find a poly-time solution - -### Example: Vertex Cover (VC) - -Input: Given an undirected graph $G = (V,E)$ - -A subset of vertices $C\subset V$ is called a **vertex cover** if contains at least one end point of every edge. - -Formally, for all edges $(u,v)\in E$, either $u\in C$, or $v\in C$. - -Problem: $VC(G,j)$ returns true if has a vertex cover of size $\leq j$, and false otherwise (minimization problem) - -Example: - -#### How hard is Vertex Cover? - -Claim: $ISET\leq p(VC)$ -Side Note: when we prove $VC$ is hard, we prove it is no easier than $ISET$. - -DO NOT: $VC\leq p(ISET)$ - -Proof: Show that $G=(V,E)$ has an independent set of $k$ **if and only if** the same graph (not always!) has a vertex cover of size $|V|-k$. - -Map: - -$$ -ISET(G,k)\to VC(g,|v|-k) -$$ - -$G'=G$ - -##### Proof of reduction: Direction 1 - -Claim 1: $ISET$ of size $k\to$ $VC$ of size $|V|-k$ - -Proof: Assume $G$ has an $ISET$ of size $k:S$, consider $C = V-S,|C|=|V|-k$ - -Claim: $C$ is a vertex cover - -##### Proof of reduction: Direction 2 - - -Claim 2: $VC$ of size $|V|-k\to ISET$ of size $k$ - -Proof: Assume $G$ has an $VC$ of size $|V| −k:C$, consider $S = V − C, |S| =k$ - -Claim: $S$ is an independent set - -### What does poly-time mean? - -Algorithm runs in time polynomial to input size. - -- If the input has items, algorithm runs in $\Theta(n^c)$ for any constant is poly-time. - - Examples: intervals to schedule, number of integers to sort, # vertices + # edges in a graph -- Numerical Value (Integer $n$), what is the input size? - - Examples: weights, capacity, total time, flow constraints - - It is not straightforward! - -### Real time complexity of F-F? - -In class: $O(F( |V| + |E|))$ - -- $|V| + |E|$ = this much space to represent the graph -- $F$ : size of the maximum flow. - -If every edge has capacity , then $F = O(CE)$ -Running time:$O(C|E|(|V| + |E| )))$ - -### What is the actual input size? - -Each edge ($|E|$ edges): - -- 2 vertices: $|V|$ distinct symbol, $\log |V|$ bits per symbol -- 1 capacity: $\log C$ - -Size of graph: - -- $O(|E|(|V| + \log C))$ - - $p( |E| , |V| , \log C)$ - -Running time: - -- $P( |E| , |V| , |C| )$ - - Exponential if is exponential in $|V|+|E|$ - -### Pseudo-polynomial - -Naïve Ford-Fulkerson is bad! - -Problem ’s inputs contain some numerical values, say $|W|$. We need only log bits to store . If algorithms runs in $p(W)$, then it is exponential, or **pseudopolynomial**. - -In homework, you improved F-F to make it work in -$p( |V| ,|E| , \log C)$, to make it a real polynomial algorithm. - -## Conclusion: Reductions - -- Reduction - - Construction of mapping with runtime - - Bidirectional proof -- Efficient Reduction $L\leq p(K)$ - - Which problem is harder? - - If $L$ is hard, then $K$ is hard. $\to$ Used to show hardness - - If $K$ is easy, then $L$ is easy. $\to$ Used for design algorithms -- Canonical Decision Problem - - Reduction to and from the optimization problem -- Reduction for hardness - - Independent Set$leq p$ Vertex Cover - -## On class - -Reduction: $V^* = O(c^k)$ - -OPT: Find max flow of at least one instance $(G,s,t)$ - -DEC: Is there a flow of size $pK$, given $G,s,t \implies$ the instance is defined by the tuple $(G,s,t,k)$ - -Yes, if there exists one -No, otherwise - -Forget about F-F and assume that you have an oracle that solves the decision problem. - -First solution (the naive solution): iterate over $k = 1, 2, \dots$ until the oracle returns false and the last one returns true would be the max flow. - -Time complexity: $K\cdot X$, where $X$ is the time complexity of the oracle -Input size: $poly(||V|,|E|, |E|log(max-capacity))$, and $V^* \leq \sum$ capacities - -A better solution: do a binary search. If there is no upper bound, we use exponential binary search instead. Then, - -$$ -\begin{aligned} -log(V^*) &\leq X\cdot log(\sum capacities)\\ -&\leq X\cdot log(|E|\cdot maxCapacity)\\ -&\leq X\cdot (log(|E| + log(maxCapacity))) -\end{aligned} -$$ -As $\log(maxCapacity)$ is linear in the size of the input, the running time is polynomial to the solution of the original problem. - -Assume that ISET is a hard problem, i.e. we don't know of any polynomial time solution. We want to show that vertex cover is also a hard problem here: - -$ISET \leq_{p} VC$ - -1. Given an instance of ISET, construct an instance of VC -2. Show that the construction can be done in polynomial time -3. Show that if the ISET instance is true than the CV instance is true -4. Show that if the VC instance is true then the ISET instance is true. - -> ISET: given $(G,K)$, is there a set of vertices that do not share edges of size $K$ -> VC: given $(G,K)$, is there a set of vertices that cover all edges of size $K$ - -1. Given $l: (G,K)$ being an instance of ISET, we construct $\phi(l): (G',K')$ as an instance of VC. $\phi(l): (G, |V|-K), \textup{i.e., } G' = G \cup K' = |V| - K$ -2. It is obvious that it is a polynomial time construction since copying the graph is linear, in the size of the graph and the subtraction of integers is constant time. - -**Direction 1**: ISET of size k $\implies$ VC of size $|V| - K$ Assume that ISET(G,K) returns true, show that $VC(G, |V|-K)$ returns true - -Let $S$ be an independent set of size $K$ and $C = V-S$ - -We claim that $C$ is a vertex cover of size $|V|-K$ - -Proof: - -We proceed by contradiction. Assume that $C$ is NOT a vertex cover, and it means that there is an edge $(u,v)$ such that $u\notin c , v\notin C$. And it implies that $u\in S , v\in S$, which contradicts with the assumption that S is an independent set. -Therefore, $c$ is an vertex cover - -**Direction 2**: VC of size $|V|-K \implies$ ISET of size $K$ - -Let $C$ be a vertex cover of size $|V|-K$ , let $s = |v| - c$ - -We claim that $S$ is an independent set of size $K$. - -Again, assume, for the sake of contradiction, that $S$ is not an independent set. And we get - -$\exists (u,v) \textup{such that } u\in S, v \in S$ - -$u,v \notin C$ - -$C \textup{ is not a vertex cover}$ - -And this is a contradiction with our assumption. +# Lecture 5 + +## Takeaway from Bipartite Matching + +- We saw how to solve a problem (bi-partite matching and others) by reducing it to another problem (maximum flow). +- In general, we can design an algorithm to map instances of a new problem to instances of known solvable problem (e.g., max-flow) to solve this new problem! +- Mapping from one problem to another which preserves solutions is called reduction. + +## Reduction: Basic Ideas + +Convert solutions to the known problem to the solutions to the new problem + +- Instance of new problem +- Instance of known problem +- Solution of known problem +- Solution of new problem + +## Reduction: Formal Definition + +Problems $L,K$. + +$L$ reduces to $K$ ($L\leq K$) if there is a mapping $\phi$ from **any** instance $l\in L$ to some instance $\phi(l)\in K'\subset K$, such that the solution for $\phi(l)$ yields a solution for $l$. + +This means that **L is no harder than K** + +### Using reduction to design algorithms + +In the example of reduction to solve Bipartite Matching: + +$L:$ Bipartite Matching + +$K:$ Max-flow Problem + +Efficiency: + +1. Reduction: $\phi:l\to\phi(l)$ (Polynomial time reduction $\phi(l)$) +2. Solve prom $\phi(l)$ (Polynomial time to solve $poly(g)$) +3. Convert the solution for $\phi(l)$ to a solution to $l$ (Polynomial time to solve $poly(g)$) + +### Efficient Reduction + +A reduction $\phi:l\to\phi(l)$ is efficient ($L\leq p(k)$) if for any $l\in L$: + +1. $\phi(l)$ is computable from $l$ in polynomial ($|l|$) time. +2. Solution to $l$ is computable from solution of $\phi(l)$ in polynomial ($|l|$) time. + +We call $L$ is **poly-time reducible** to $K$, or $L$ poly-time +reduces to $K$. + +### Which problem is harder? + +Theorem: If $L\leq p(k)$ and there is a polynomial time algorithm to solve $K$, then there is a polynomial time algorithm to solve $L$. + +Proof: Given an instance of $l\in L$ If we can convert the problem in polynomial time with respect to the original problem $l$. + +1. Compute $\phi(l)$: $p(l)$ +2. Solve $\phi(l)$: $p(\phi(l))$ +3. Convert solution: $p(\phi(l))$ + +Total time: $p(l)+p(\phi(l))+p(\phi(l))=p(l)+p(\phi(l))$ +Need to show: $|\phi(l)|=poly(|l|)$ + +Proof: + +Since we can convert $\phi(l)$ in $p(l)$ time, and on every time step, (constant step) we can only write constant amount of data. + +So $|\phi(l)|=poly(|l|)$ + +## Hardness Problems + +Reductions show the relationship between problem hardness! + +Question: Could you solve a problem in polynomial time? + +Easy: polynomial time solution +Hard: No polynomial time solution (as far as we know) + +### Types of Problems + +Decision Problem: Yes/No answer + +Examples: Subset sums + +1. Is the there a flow of size $F$ +2. Is there a shortest path of length $L$ from vertex $u$ to vertex $v$. +3. Given a set of intercal, can you schedule $k$ of them. + +Optimization Problem: What is the value of an optimal feasible solution of a problem? + +- Minimization: Minimize cost + - min cut + - minimal spanning tree + - shortest path +- Maximization: Maximize profit + - interval scheduling + - maximum flow + - maximum matching + +#### Canonical Decision Problem + +Does the instance $l\in L$ (an optimization problem) have a feasible solution with objective value $k$: + +Objective value $\geq k$ (maximization) $\leq k$ (minimization) + +$DL$ is the reduced Canonical Decision problem $L$ + +##### Hardness of Canonical Decision Problems + +Lemma 1: $DL\leq p(L)$ ($DL$ is no harder than $L$) + +Proof: Assume $L$ **maximization** problem $DL(l)$: does have a solution $\geq k$. + +Example: Does graph $G$ have flow $\geq k$. + +Let $v^∗$ be the maximum objective on $l$ by solving $l$. + +Let the instance of $DL:(l,k)$ and $l$ be the problem and $k$ be the objective + +1. $l\to \phi(l)\in L$ (optimization problem) $\phi(l,k)=l$ +2. Is $v^*(l)\geq k$? If so, return true, else return false. + +Lemma 2: If $v^* =O(c^{|l|})$ for any constant $c$, then $L\leq p(DL)$. + +Proof: First we could show $L\leq DL$. Suppose maximization problem, canonical decision problem is is there a solution $\geq k$. + +Naïve Linear Search: Ask $DL(l,k)$, if returns false, ask $DL(l,k+1)$ until returns true + +Runtime: At most $k$ search to iterate all possibilities. + +This is exponential! How to reduce it? + +Our old friend Binary (exponential) Search is back! + +You gets a no at some value: try power of 2 until you get a no, then do binary search + +\# questions: $=log_2(v^*(l))=poly(l)$ + +Binary search in area: from last yes to first no. + +Runtime: Binary search ($O(n)=\log(v^*(l))$) + +### Reduction for Algorithm Design vs Hardness + +For problems $L,K$ + +If $K$ is “easy” (exists a poly-time solution), then $L$ is also easy. + +If $L$ is “hard” (no poly-time solution), then $k$ is also hard. + +Every problem that we worked on so far, $K$ is “easy”, so we reduce from new problem to known problem (e.g., max-flow). + +#### Reduction for Hardness: Independent Set (ISET) + +Input: Given an undirected graph $G = (V,E)$, + +A subset of vertices $S\subset V$ is called an **independent set** if no two vertices of are connected by an edge. + +Problem: Does $G$ contain an independent set of size $\geq k$? + +$ISET(G,k)$ returns true if $G$ contains an independent set of size $\geq k$, and false otherwise. + +Algorithm? NO! We think that this is a hard problem. + +A lot of pQEDle have tried and could not find a poly-time solution + +### Example: Vertex Cover (VC) + +Input: Given an undirected graph $G = (V,E)$ + +A subset of vertices $C\subset V$ is called a **vertex cover** if contains at least one end point of every edge. + +Formally, for all edges $(u,v)\in E$, either $u\in C$, or $v\in C$. + +Problem: $VC(G,j)$ returns true if has a vertex cover of size $\leq j$, and false otherwise (minimization problem) + +Example: + +#### How hard is Vertex Cover? + +Claim: $ISET\leq p(VC)$ +Side Note: when we prove $VC$ is hard, we prove it is no easier than $ISET$. + +DO NOT: $VC\leq p(ISET)$ + +Proof: Show that $G=(V,E)$ has an independent set of $k$ **if and only if** the same graph (not always!) has a vertex cover of size $|V|-k$. + +Map: + +$$ +ISET(G,k)\to VC(g,|v|-k) +$$ + +$G'=G$ + +##### Proof of reduction: Direction 1 + +Claim 1: $ISET$ of size $k\to$ $VC$ of size $|V|-k$ + +Proof: Assume $G$ has an $ISET$ of size $k:S$, consider $C = V-S,|C|=|V|-k$ + +Claim: $C$ is a vertex cover + +##### Proof of reduction: Direction 2 + + +Claim 2: $VC$ of size $|V|-k\to ISET$ of size $k$ + +Proof: Assume $G$ has an $VC$ of size $|V| −k:C$, consider $S = V − C, |S| =k$ + +Claim: $S$ is an independent set + +### What does poly-time mean? + +Algorithm runs in time polynomial to input size. + +- If the input has items, algorithm runs in $\Theta(n^c)$ for any constant is poly-time. + - Examples: intervals to schedule, number of integers to sort, # vertices + # edges in a graph +- Numerical Value (Integer $n$), what is the input size? + - Examples: weights, capacity, total time, flow constraints + - It is not straightforward! + +### Real time complexity of F-F? + +In class: $O(F( |V| + |E|))$ + +- $|V| + |E|$ = this much space to represent the graph +- $F$ : size of the maximum flow. + +If every edge has capacity , then $F = O(CE)$ +Running time:$O(C|E|(|V| + |E| )))$ + +### What is the actual input size? + +Each edge ($|E|$ edges): + +- 2 vertices: $|V|$ distinct symbol, $\log |V|$ bits per symbol +- 1 capacity: $\log C$ + +Size of graph: + +- $O(|E|(|V| + \log C))$ + - $p( |E| , |V| , \log C)$ + +Running time: + +- $P( |E| , |V| , |C| )$ + - Exponential if is exponential in $|V|+|E|$ + +### Pseudo-polynomial + +Naïve Ford-Fulkerson is bad! + +Problem ’s inputs contain some numerical values, say $|W|$. We need only log bits to store . If algorithms runs in $p(W)$, then it is exponential, or **pseudopolynomial**. + +In homework, you improved F-F to make it work in +$p( |V| ,|E| , \log C)$, to make it a real polynomial algorithm. + +## Conclusion: Reductions + +- Reduction + - Construction of mapping with runtime + - Bidirectional proof +- Efficient Reduction $L\leq p(K)$ + - Which problem is harder? + - If $L$ is hard, then $K$ is hard. $\to$ Used to show hardness + - If $K$ is easy, then $L$ is easy. $\to$ Used for design algorithms +- Canonical Decision Problem + - Reduction to and from the optimization problem +- Reduction for hardness + - Independent Set$leq p$ Vertex Cover + +## On class + +Reduction: $V^* = O(c^k)$ + +OPT: Find max flow of at least one instance $(G,s,t)$ + +DEC: Is there a flow of size $pK$, given $G,s,t \implies$ the instance is defined by the tuple $(G,s,t,k)$ + +Yes, if there exists one +No, otherwise + +Forget about F-F and assume that you have an oracle that solves the decision problem. + +First solution (the naive solution): iterate over $k = 1, 2, \dots$ until the oracle returns false and the last one returns true would be the max flow. + +Time complexity: $K\cdot X$, where $X$ is the time complexity of the oracle +Input size: $poly(||V|,|E|, |E|log(max-capacity))$, and $V^* \leq \sum$ capacities + +A better solution: do a binary search. If there is no upper bound, we use exponential binary search instead. Then, + +$$ +\begin{aligned} +log(V^*) &\leq X\cdot log(\sum capacities)\\ +&\leq X\cdot log(|E|\cdot maxCapacity)\\ +&\leq X\cdot (log(|E| + log(maxCapacity))) +\end{aligned} +$$ +As $\log(maxCapacity)$ is linear in the size of the input, the running time is polynomial to the solution of the original problem. + +Assume that ISET is a hard problem, i.e. we don't know of any polynomial time solution. We want to show that vertex cover is also a hard problem here: + +$ISET \leq_{p} VC$ + +1. Given an instance of ISET, construct an instance of VC +2. Show that the construction can be done in polynomial time +3. Show that if the ISET instance is true than the CV instance is true +4. Show that if the VC instance is true then the ISET instance is true. + +> ISET: given $(G,K)$, is there a set of vertices that do not share edges of size $K$ +> VC: given $(G,K)$, is there a set of vertices that cover all edges of size $K$ + +1. Given $l: (G,K)$ being an instance of ISET, we construct $\phi(l): (G',K')$ as an instance of VC. $\phi(l): (G, |V|-K), \textup{i.e., } G' = G \cup K' = |V| - K$ +2. It is obvious that it is a polynomial time construction since copying the graph is linear, in the size of the graph and the subtraction of integers is constant time. + +**Direction 1**: ISET of size k $\implies$ VC of size $|V| - K$ Assume that ISET(G,K) returns true, show that $VC(G, |V|-K)$ returns true + +Let $S$ be an independent set of size $K$ and $C = V-S$ + +We claim that $C$ is a vertex cover of size $|V|-K$ + +Proof: + +We proceed by contradiction. Assume that $C$ is NOT a vertex cover, and it means that there is an edge $(u,v)$ such that $u\notin c , v\notin C$. And it implies that $u\in S , v\in S$, which contradicts with the assumption that S is an independent set. +Therefore, $c$ is an vertex cover + +**Direction 2**: VC of size $|V|-K \implies$ ISET of size $K$ + +Let $C$ be a vertex cover of size $|V|-K$ , let $s = |v| - c$ + +We claim that $S$ is an independent set of size $K$. + +Again, assume, for the sake of contradiction, that $S$ is not an independent set. And we get + +$\exists (u,v) \textup{such that } u\in S, v \in S$ + +$u,v \notin C$ + +$C \textup{ is not a vertex cover}$ + +And this is a contradiction with our assumption. diff --git a/pages/CSE347/CSE347_L6.md b/content/CSE347/CSE347_L6.md similarity index 97% rename from pages/CSE347/CSE347_L6.md rename to content/CSE347/CSE347_L6.md index 83db0ee..f2cf6ee 100644 --- a/pages/CSE347/CSE347_L6.md +++ b/content/CSE347/CSE347_L6.md @@ -1,287 +1,287 @@ -# Lecture 6 - -## NP-completeness - -### $P$: Polynomial-time Solvable - -$P$: Class of decision problems $L$ such that there is a polynomial-time algorithm that correctly answers yes or not for every instance $l\in L$. - -Algorithm "$A$ decides $L$". If algorithm $A$ always correctly answers for any instance $l\in L$. - -Example: - -Is the number $n$ prime? Best algorithm so far: $O(\log^6 n)$, 2002 - -## Introduction to NP - -- NP$\neq$ Non-polynomial (Non-deterministic polynomial time) -- Let $L$ be a decision problem. -- Let $l$ be an instance of the problem that the answer happens to be "yes". -- A **certificate** c(l) for $l$ is a "proof" that the answer for $l$ is true. [$l$ is a true instance] - - For canonical decision problems for optimization problems, the certificate is often a feasible solution for the corresponding optimization problem. - -### Example of certificates - -- Problem: Is there a path from $s$ to $t$ - - Instance: graph $G(V,E),s,t$. - - Certificate: path from $s$ to $t$. -- Problem: Can I schedule $k$ intervals in the room so that they do not conflict. - - Instance: $l:(I,k)$ - - Certificate: set of $k$ non-conflicting intervals. -- Problem: ISET - - Instance: $G(V,E),k$. - - Certificate: $k$ vertices with no edges between them. - -If the answer to the problem is NO, you don't need to provide anything to prove that. - -### Useful certificates - -For a problem to be in NP, the problem need to have "useful" certificates. What is considered a good certificate? - -- Easy to check - - Verifying algorithm which can check a YES answer and a certificate in $poly(l)$ -- Not too long: [$poly(l)$] - -### Verifier Algorithm - -**Verifier algorithm** is one that takes an instance $l\in L$ and a certificate $c(l)$ and says yes if the certificate proves that $l$ is a true instance and false otherwise. - -$V$ is a poly-time verifier for $L$ is it is a verifier and runs in $poly(|l|,|c|)$ time. (c=$poly(l)$) - -- The runtime must be polynomial -- Must check **every** problem constraint -- Not always trivial - -## Class NP - -**NP:** A class of decision problems such that exists a certificate schema $c$ and a verifier algorithm $V$ such that: - -1. certificate is $poly(l)$ in size. -2. $V:poly(l)$ in time. - -**P:** is a class of problems that you can **solve** in polynomial time - -**NP:** is a class of problems that you can **verify** TRUE instances in polynomial time given a poly-size certificate - -**Millennium question** - -$P\subseteq NP$? $NP\subseteq P$? - -$P\subseteq NP$ is true. - -Proof: Let $L$ be a problem in $P$, we want to show that there is a polynomial size certificate with a poly-time verifier. - -There is an algorithm $A$ which solves $L$ in polynomial time. - -**Certificate:** empty thing. - -**Verifier:** $(l,c)$ - -1. Discard $c$. -2. Run $A$ on $l$ and return the answer. - -Nobody knows the solution $NP\subseteq P$. Sad. - -### Class of problem: NP complete - -Informally: hardest problem in NP - -Consider a problem $L$. - -- We want to show if $L\subseteq P$, then $NP\subseteq P$ - -**NP-hard**: A decision problem $L$ is NP-hard if for any problem $K\in NP$, $K\leq_p L$. - -$L$ is at least as hard as all the problems in NP. If we have an algorithm for $L$, we have an algorithm for any problem in NP with only polynomial time extra cost. - -MindMap: - -$K\implies L\implies sol(L)\implies sol(K)$ - -#### Lemma $P=NP$ - -Let $L$ be an NP-hard problem. If $L\in P$, then $P=NP$. - -Proof: - -Say $L$ has a poly-time solution, some problem $K$ in $NP$. - -For any $K\in NP$, $NP\subset P$, $P\subset NP$, then $P=NP$. - -**NP-complete:** $L$ is **NP-complete** if it is both NP-hard and $L\in NP$. - -**NP-optimization:** $L$ is **NP-optimization** problem if the canonical decision problem is NP-complete. - -**Claim:** If any NP-optimization problem have polynomial-time solution, then $P=NP$. - -### Is $P=NP$? - -- Answering this problem is hard. -- But for any NP-complete problem, if you could find a poly-time algorithm for $L$, then you would have answered this question. -- Therefore, finding a poly-time algorithm for $L$ is hard. - -## NP-Complete problem - -### Satisfiability (SAT) - -Boolean Formulas: - -A set of Boolean variables: - -$x,y,a,b,c,w,z,...$ they take values true or false. - -A boolean formula is a formula of Boolean variables with and, or and not. - -Examples: - -$\phi:x\land (\neg y \lor z)\land\neg(y\lor w)$ - -$x=1,y=0,z=1,w=0$, the formula is $1$. - -**SAT:** given a formula $\phi$, is there a setting $M$ of variables such that the $\phi$ evaluates to True under this setting. - -If there is such assignment, then $\phi$ is satisfiable. Otherwise, it is not. - -Example: $x\land y\land \neg(x\lor y)$ is not satisfiable. - -A seminar paper by Cook and Levin in 1970 showed that SAT is NP-complete. - -1. SAT is in NP - Proof: - $\exists$ a certificate schema and a poly-time verifier. - $c$ satisfying assignment $M$ and $v$ check that $M$ makes $\phi$ true. -2. SAT is NP-hard. we can just accept it has a fact. - -#### How to show a problem is NP-complete? - -Say we have a problem $L$. - -1. Show that $L\in NP$. - Exists certificate schema and verification algorithm in polynomial time. -2. Prove that we can reduce SAT to $L$. $SAT\leq_p L$ **(NOT $L\leq_p SAT$)** - Solving $L$ also solve SAT. - -### CNF-SAT - -**CNF:** Conjugate normal form of SAT - -The formula $\phi$ must be an "and of ors" - -$$ -\phi=\land_{i=1}^n(\lor^{m_i}_{j=1}l_{i,j}) -$$ - -$l_{i,j}$: clause - -### 3-CNF-SAT - -**3-CNF-SAT:** where every clauses has exactly 3 literals. - -is NP complete [not all version of them are, 2-CNF-SAT is in P] - -Input: 3-CNF expression with $n$ variables and $m$ clauses in the form: - -number of total literals: $3m$ - -Output: An assignment of the $n$ variables such that at least one literal from each clauses evaluates to true. - -Note: - -1. One variable can be used to satisfy multiple clauses. -2. $x_i$ and $\neg x_i$ cannot both evaluate to true. - -Example: ISET is NP-complete. - -Proof: - -Say we have a problem $L$ - -1. Show that $ISET\in NP$ - Certificate: set of $k$ vertices: $|S|=k\in poly(g)$\ - Verifier: checks that there are no edges between them $O(E k^2)$ -2. ISET is NP-hard. We need to prove $3SAT\leq_p ISET$ - - Construct a reduction from $3SAT$ to $ISET$. - - Show that $ISET$ is harder than $3SAT$. - -We need to prove $\phi\in 3SAT$ is satisfiable if and only if the constructed $G$ has an $ISET$ of size $\geq k=m$ - -#### Reduction mapping construction - -We construct an ISET instance from $3-SAT$. - -Suppose the formula has $n$ variables and $m$ clauses - -1. for each clause, we construct vertex for each literal and connect them (for $x\lor \neg y\lor z$, we connect $x,\neg y,z$ together) -2. then we connect all the literals with their negations (connects $x$ and $\neg x$) - -$\implies$ - -If $\phi$ has a satisfiable assignment, then $G$ has an independent set of size $\geq m$, - -For a set $S$ we pick exactly one true literal from every clause and take the corresponding vertex to that clause, $|S|=m$ - -Must also argue that $S$ is an independent set. - -Example: picked a set of vertices $|S|=4$. - -A literal has edges: - -- To all literals in the same clause: We never pick two literals form the same clause. -- To its negation. - -Since it is a satisfiable 3-SAT assignment, $x$ and $\neg x$ cannot both evaluate to true, those edges are not a problem, so $S$ is an independent set. - -$\impliedby$ - -If $G$ has an independent set of size $\geq m$, then $\phi$ is satisfiable. - -Say that $S$ is an independent set of $m$, we need to construct a satisfiable assignment for the original $\phi$. - -- If $S$ contains a vertex corresponding to literal $x_i$, then set $x_i$ to true. -- If contains a vertex corresponding to literal $\neg x_i$, then set $x_i$ to false. -- Other variables can be set arbitrarily - -Question: Is it a valid 3-SAT assignment? - -Your ISET $S$ can contain at most $1$ vertex from each clause. Since vertices in a clause are connected by edges. - -- Since $S$ contains $m$ vertices, it must contain exactly $1$ vertex from each clause. -- Therefore, we will make at least $1$ literals form each clause to be true. -- Therefore, all the clauses are true and $\phi$ is satisfied. - -## Conclusion: NP-completeness - -- Prove NP-Complete: - - If NP-optimization, convert to canonical decision problem - - Certificate, Verification algorithm - - Prove NP-hard: reduce from existing NP-Complete - problems -- 3-SAT Problem: - - Input, output, constraints - - A well-known NP-Complete problem - - Reduce from 3-SAT to ISET to show ISET is NP-Complete - -## On class - -### NP-complete - -$p\in NP$, if we have a certificate schema and a verifier algorithm. - -### NP-complete proof - -#### P is in NP - -what a certificate would looks like, show that if has a polynomial time o the problem size. - -design a verifier algorithm that checks a certificate if it indeed prove tha the answer is YES and has a polynomial time complexity. Inputs: certificate and the problem input $poly(|l|,|c|)=poly(|p|)$ - -#### P is NP hard - -select an already known NP-hard problem: eg. 3-SAT, ISET, VC,... - -show that $3-SAT\leq_p p$ - -- present an algorithm that given any instance of 3-SAT (on the chosen NP hard problem) to an instance of $p$. -- show that the construction is done in polynomial time. -- show that if $p$'s instance answer is YES, then the instance of 3-SAT is YES. +# Lecture 6 + +## NP-completeness + +### $P$: Polynomial-time Solvable + +$P$: Class of decision problems $L$ such that there is a polynomial-time algorithm that correctly answers yes or not for every instance $l\in L$. + +Algorithm "$A$ decides $L$". If algorithm $A$ always correctly answers for any instance $l\in L$. + +Example: + +Is the number $n$ prime? Best algorithm so far: $O(\log^6 n)$, 2002 + +## Introduction to NP + +- NP$\neq$ Non-polynomial (Non-deterministic polynomial time) +- Let $L$ be a decision problem. +- Let $l$ be an instance of the problem that the answer happens to be "yes". +- A **certificate** c(l) for $l$ is a "proof" that the answer for $l$ is true. [$l$ is a true instance] + - For canonical decision problems for optimization problems, the certificate is often a feasible solution for the corresponding optimization problem. + +### Example of certificates + +- Problem: Is there a path from $s$ to $t$ + - Instance: graph $G(V,E),s,t$. + - Certificate: path from $s$ to $t$. +- Problem: Can I schedule $k$ intervals in the room so that they do not conflict. + - Instance: $l:(I,k)$ + - Certificate: set of $k$ non-conflicting intervals. +- Problem: ISET + - Instance: $G(V,E),k$. + - Certificate: $k$ vertices with no edges between them. + +If the answer to the problem is NO, you don't need to provide anything to prove that. + +### Useful certificates + +For a problem to be in NP, the problem need to have "useful" certificates. What is considered a good certificate? + +- Easy to check + - Verifying algorithm which can check a YES answer and a certificate in $poly(l)$ +- Not too long: [$poly(l)$] + +### Verifier Algorithm + +**Verifier algorithm** is one that takes an instance $l\in L$ and a certificate $c(l)$ and says yes if the certificate proves that $l$ is a true instance and false otherwise. + +$V$ is a poly-time verifier for $L$ is it is a verifier and runs in $poly(|l|,|c|)$ time. (c=$poly(l)$) + +- The runtime must be polynomial +- Must check **every** problem constraint +- Not always trivial + +## Class NP + +**NP:** A class of decision problems such that exists a certificate schema $c$ and a verifier algorithm $V$ such that: + +1. certificate is $poly(l)$ in size. +2. $V:poly(l)$ in time. + +**P:** is a class of problems that you can **solve** in polynomial time + +**NP:** is a class of problems that you can **verify** TRUE instances in polynomial time given a poly-size certificate + +**Millennium question** + +$P\subseteq NP$? $NP\subseteq P$? + +$P\subseteq NP$ is true. + +Proof: Let $L$ be a problem in $P$, we want to show that there is a polynomial size certificate with a poly-time verifier. + +There is an algorithm $A$ which solves $L$ in polynomial time. + +**Certificate:** empty thing. + +**Verifier:** $(l,c)$ + +1. Discard $c$. +2. Run $A$ on $l$ and return the answer. + +Nobody knows the solution $NP\subseteq P$. Sad. + +### Class of problem: NP complete + +Informally: hardest problem in NP + +Consider a problem $L$. + +- We want to show if $L\subseteq P$, then $NP\subseteq P$ + +**NP-hard**: A decision problem $L$ is NP-hard if for any problem $K\in NP$, $K\leq_p L$. + +$L$ is at least as hard as all the problems in NP. If we have an algorithm for $L$, we have an algorithm for any problem in NP with only polynomial time extra cost. + +MindMap: + +$K\implies L\implies sol(L)\implies sol(K)$ + +#### Lemma $P=NP$ + +Let $L$ be an NP-hard problem. If $L\in P$, then $P=NP$. + +Proof: + +Say $L$ has a poly-time solution, some problem $K$ in $NP$. + +For any $K\in NP$, $NP\subset P$, $P\subset NP$, then $P=NP$. + +**NP-complete:** $L$ is **NP-complete** if it is both NP-hard and $L\in NP$. + +**NP-optimization:** $L$ is **NP-optimization** problem if the canonical decision problem is NP-complete. + +**Claim:** If any NP-optimization problem have polynomial-time solution, then $P=NP$. + +### Is $P=NP$? + +- Answering this problem is hard. +- But for any NP-complete problem, if you could find a poly-time algorithm for $L$, then you would have answered this question. +- Therefore, finding a poly-time algorithm for $L$ is hard. + +## NP-Complete problem + +### Satisfiability (SAT) + +Boolean Formulas: + +A set of Boolean variables: + +$x,y,a,b,c,w,z,...$ they take values true or false. + +A boolean formula is a formula of Boolean variables with and, or and not. + +Examples: + +$\phi:x\land (\neg y \lor z)\land\neg(y\lor w)$ + +$x=1,y=0,z=1,w=0$, the formula is $1$. + +**SAT:** given a formula $\phi$, is there a setting $M$ of variables such that the $\phi$ evaluates to True under this setting. + +If there is such assignment, then $\phi$ is satisfiable. Otherwise, it is not. + +Example: $x\land y\land \neg(x\lor y)$ is not satisfiable. + +A seminar paper by Cook and Levin in 1970 showed that SAT is NP-complete. + +1. SAT is in NP + Proof: + $\exists$ a certificate schema and a poly-time verifier. + $c$ satisfying assignment $M$ and $v$ check that $M$ makes $\phi$ true. +2. SAT is NP-hard. we can just accept it has a fact. + +#### How to show a problem is NP-complete? + +Say we have a problem $L$. + +1. Show that $L\in NP$. + Exists certificate schema and verification algorithm in polynomial time. +2. Prove that we can reduce SAT to $L$. $SAT\leq_p L$ **(NOT $L\leq_p SAT$)** + Solving $L$ also solve SAT. + +### CNF-SAT + +**CNF:** Conjugate normal form of SAT + +The formula $\phi$ must be an "and of ors" + +$$ +\phi=\land_{i=1}^n(\lor^{m_i}_{j=1}l_{i,j}) +$$ + +$l_{i,j}$: clause + +### 3-CNF-SAT + +**3-CNF-SAT:** where every clauses has exactly 3 literals. + +is NP complete [not all version of them are, 2-CNF-SAT is in P] + +Input: 3-CNF expression with $n$ variables and $m$ clauses in the form: + +number of total literals: $3m$ + +Output: An assignment of the $n$ variables such that at least one literal from each clauses evaluates to true. + +Note: + +1. One variable can be used to satisfy multiple clauses. +2. $x_i$ and $\neg x_i$ cannot both evaluate to true. + +Example: ISET is NP-complete. + +Proof: + +Say we have a problem $L$ + +1. Show that $ISET\in NP$ + Certificate: set of $k$ vertices: $|S|=k\in poly(g)$\ + Verifier: checks that there are no edges between them $O(E k^2)$ +2. ISET is NP-hard. We need to prove $3SAT\leq_p ISET$ + - Construct a reduction from $3SAT$ to $ISET$. + - Show that $ISET$ is harder than $3SAT$. + +We need to prove $\phi\in 3SAT$ is satisfiable if and only if the constructed $G$ has an $ISET$ of size $\geq k=m$ + +#### Reduction mapping construction + +We construct an ISET instance from $3-SAT$. + +Suppose the formula has $n$ variables and $m$ clauses + +1. for each clause, we construct vertex for each literal and connect them (for $x\lor \neg y\lor z$, we connect $x,\neg y,z$ together) +2. then we connect all the literals with their negations (connects $x$ and $\neg x$) + +$\implies$ + +If $\phi$ has a satisfiable assignment, then $G$ has an independent set of size $\geq m$, + +For a set $S$ we pick exactly one true literal from every clause and take the corresponding vertex to that clause, $|S|=m$ + +Must also argue that $S$ is an independent set. + +Example: picked a set of vertices $|S|=4$. + +A literal has edges: + +- To all literals in the same clause: We never pick two literals form the same clause. +- To its negation. + +Since it is a satisfiable 3-SAT assignment, $x$ and $\neg x$ cannot both evaluate to true, those edges are not a problem, so $S$ is an independent set. + +$\impliedby$ + +If $G$ has an independent set of size $\geq m$, then $\phi$ is satisfiable. + +Say that $S$ is an independent set of $m$, we need to construct a satisfiable assignment for the original $\phi$. + +- If $S$ contains a vertex corresponding to literal $x_i$, then set $x_i$ to true. +- If contains a vertex corresponding to literal $\neg x_i$, then set $x_i$ to false. +- Other variables can be set arbitrarily + +Question: Is it a valid 3-SAT assignment? + +Your ISET $S$ can contain at most $1$ vertex from each clause. Since vertices in a clause are connected by edges. + +- Since $S$ contains $m$ vertices, it must contain exactly $1$ vertex from each clause. +- Therefore, we will make at least $1$ literals form each clause to be true. +- Therefore, all the clauses are true and $\phi$ is satisfied. + +## Conclusion: NP-completeness + +- Prove NP-Complete: + - If NP-optimization, convert to canonical decision problem + - Certificate, Verification algorithm + - Prove NP-hard: reduce from existing NP-Complete + problems +- 3-SAT Problem: + - Input, output, constraints + - A well-known NP-Complete problem + - Reduce from 3-SAT to ISET to show ISET is NP-Complete + +## On class + +### NP-complete + +$p\in NP$, if we have a certificate schema and a verifier algorithm. + +### NP-complete proof + +#### P is in NP + +what a certificate would looks like, show that if has a polynomial time o the problem size. + +design a verifier algorithm that checks a certificate if it indeed prove tha the answer is YES and has a polynomial time complexity. Inputs: certificate and the problem input $poly(|l|,|c|)=poly(|p|)$ + +#### P is NP hard + +select an already known NP-hard problem: eg. 3-SAT, ISET, VC,... + +show that $3-SAT\leq_p p$ + +- present an algorithm that given any instance of 3-SAT (on the chosen NP hard problem) to an instance of $p$. +- show that the construction is done in polynomial time. +- show that if $p$'s instance answer is YES, then the instance of 3-SAT is YES. - show that if 3-SAT's instance answer is YES then the instance of $p$ is YES. \ No newline at end of file diff --git a/pages/CSE347/CSE347_L7.md b/content/CSE347/CSE347_L7.md similarity index 96% rename from pages/CSE347/CSE347_L7.md rename to content/CSE347/CSE347_L7.md index a9ee80a..1c66380 100644 --- a/pages/CSE347/CSE347_L7.md +++ b/content/CSE347/CSE347_L7.md @@ -1,312 +1,312 @@ -# Lecture 7 - -## Known NP-Complete Problems - -- SAT and 3-SAT -- Vertex Cover -- Independent Set - -## How to show a problem $L$ is NP-Complete - -- Show $L \in$ NP - - Give a polynomial time certificate - - Give a polynomial time verifier -- Show $L$ is NP-Hard: for some known NP-Complete problem $K$, show $K \leq_p L$ - - Construct a mapping $\phi$ from instance in $K$ to instance in $L$, given an instance $k\in K$, $\phi(k)\in L$. - - Show that you can compute $\phi(k)$ in polynomial time. - - Show that $k \in K$ is true if and only if $\phi(k) \in L$ is true. - -### Example 1: Subset Sum - -Input: A set $S$ of integers and a target positive integer $t$. - -Problem: Determine if there exists a subset $S' \subseteq S$ such that $\sum_{a_i\in S'} a_i = t$. - -We claim that Subset Sum is NP-Complete. - -Step 1: Subset Sum $\in$ NP - -- Certificate: $S' \subseteq S$ -- Verifier: Check that $\sum_{a_i\in S'} a_i = t$ - -Step 2: Subset Sum is NP-Hard - -We claim that 3-SAT $\leq_p$ Subset Sum - -Given any $3$-CNF formula $\Psi$, we will construct an instance $(S, t)$ of Subset Sum such that $\Psi$ is satisfiable if and only if there exists a subset $S' \subseteq S$ such that $\sum_{a_i\in S'} a_i = t$. - -#### How to construct $\Psi$? - -Reduction construction: - -Assumption: No clause contains both a literal and its negation. - -3-SAT problem: $\Psi$ has $n$ variables and $m$ clauses. - -Need to: construct $S$ of positive numbers and a target $t$ - -Ideas of construction: - -For 3-SAT instance $\Psi$: - -- At least one literal in each clause is true -- A variable and its negation cannot both be true - -$S$ contains integers with $n+m$ digits (base 10) - -$$ -p_1p_2\cdots p_n q_1 q_2 \cdots q_m -$$ - -where $p_i$ are representations of variables that are either $0$ or $1$ and $q_j$ are representations of clauses. - -For each variable $x_i$, we will have two integers in $S$, called $v_i$ and $\overline{v_i}$. - -- For each variable $x_i$, both $v_i$ and $\overline{v_i}$ have digits $p_i=1$. all other $p$ positions are zero - -- Each digit $q_j$ in $v_i$ is $1$ if $x_i$ appears in clause $j$; otherwise $q_j=0$ - -For example: - -$\Psi=(x_1\lor \neg x_2 \lor x_3) \land (\neg x_1 \lor x_2 \lor x_3)$ - -| | $p_1$ | $p_2$ | $p_3$ | $q_1$ | $q_2$ | -| ---------------- | ----- | ----- | ----- | ----- | ----- | -| $v_1$ | 1 | 0 | 0 | 1 | 0 | -| $\overline{v_1}$ | 1 | 0 | 0 | 0 | 1 | -| $v_2$ | 0 | 1 | 0 | 0 | 1 | -| $\overline{v_2}$ | 0 | 1 | 0 | 1 | 0 | -| $v_3$ | 0 | 0 | 1 | 1 | 1 | -| $\overline{v_3}$ | 0 | 0 | 1 | 0 | 0 | -| t | 1 | 1 | 1 | 1 | 1 | - -Let's try to prove correctness of the reduction. - -Direction 1: Say subset sum has a solution $S'$. - -We must prove that there is a satisfying assignment for $\Psi$. - -Set $x_i=1$ if $v_i\in S'$ - -Set $x_i=0$ if $\overline{v_i}\in S'$ - -1. We want set $x_i$ to be both true and false, we will pick (in $S'$) either $v_i$ or $\overline{v_i}$ -2. For each clause we have at least one literal that is true since $q_j$ has a $1$ in the clause. - -Direction 2: Say $\Psi$ has a satisfying assignment. - -We must prove that there is a subset $S'$ such that $\sum_{a_i\in S'} a_i = t$. - -If $x_i=1$, then $v_i\in S'$ - -If $x_i=0$, then $\overline{v_i}\in S'$ - -Problem: 1,2 or 3 literals in every clause can be true. - -Fix - -Add 2 numbers to $S$ for each clause $j$. We add $y_j,z_j$. - -- All $p$ digits are zero -- $q_j$ of $y_j$ is $1$, $q_j$ of $z_j$ is $2$, for all $j$, other digits are zero. -- Intuitively, these numbers account for the number of literals in clause $j$ that are true. - -New target are as follows: - -| | $p_1$ | $p_2$ | $p_3$ | $q_1$ | $q_2$ | -| ----- | ----- | ----- | ----- | ----- | ----- | -| $y_1$ | 0 | 0 | 0 | 1 | 0 | -| $z_1$ | 0 | 0 | 0 | 2 | 0 | -| $y_2$ | 0 | 0 | 0 | 0 | 1 | -| $z_2$ | 0 | 0 | 0 | 0 | 2 | -| $t$ | 1 | 1 | 1 | 4 | 4 | - -#### Time Complexity of construction for Subset Sum - -- $O(n+m)$ -- $n$ is the number of variables -- $m$ is the number of clauses - -How many integers are in $S$? - -- $2n$ for variables -- $2m$ for new numbers -- Total: $2n+2m$ integers - -How many digits are in each integer? - -- $n+m$ digits -- Time complexity: $O((n+m)^2)$ - -#### Proof of reduction for Subset Sum - -Claim 1: If Subset Sum has a solution, then $\Psi$ is satisfiable. - -Proof: - -Say $S'$ is a solution to Subset Sum. Then there exists a subset $S' \subseteq S$ such that $\sum_{a_i\in S'} a_i = t$. Here is an assignment of truth values to variables in $\Psi$ that satisfies $\Psi$: - -- Set $x_i=1$ if $v_i\in S'$ -- Set $x_i=0$ if $\overline{v_i}\in S'$ - -This is a valid assignment since: - -- We pick either $v_i$ or $\overline{v_i}$ -- For each clause, at least one literal is true - -QED - -Claim 2: If $\Psi$ is satisfiable, then Subset Sum has a solution. - -Proof: - -If $A$ is a satisfiable assignment for $\Psi$, then we can construct a subset $S'$ of $S$ such that $\sum_{a_i\in S'} a_i = t$. - -If $x_i=1$, then $v_i\in S'$ - -If $x_i=0$, then $\overline{v_i}\in S'$ - -Say $t=\sum$ elements we picked from $S$. - -- All $p_i$ in $t$ are $1$ -- All $q_j$ in $t$ are either $1$ or $2$ or $3$. - - If $q_j=1$, then $y_j,z_j\in S'$ - - If $q_j=2$, then $z_j\in S'$ - - If $q_j=3$, then $y_j\in S'$ - -QED - -### Example 2: 3 Color - -Input: Graph $G$ - -Problem: Determine if $G$ is 3-colorable. - -We claim that 3-Color is NP-Complete. - -#### Proof of NP for 3-Color - -Homework - -#### Proof of NP-Hard for 3-Color - -We claim that 3-SAT $\leq_p$ 3-Color - -Given a 3-CNF formula $\Psi$, we will construct a graph $G$ such that $\Psi$ is satisfiable if and only if $G$ is 3-colorable. - -Construction: - -1. Construct a core triangle (3 vertices for 3 colors) -2. 2 vertices for each variable $x_i:v_i,\overline{v_i}$ -3. Clause widget - -Clause widget: - -- 3 vertices for each clause $C_j:y_j,z_j,t_j$ (clause widget) -- 3 edges extended from clause widget -- variable vertex connected to extended edges - -Key for dangler design: - -Connect to all $v_i$ with true to the same color. and connect to all $v_i$ with false to another color. - -''' -TODO: Add dangler design image here. -''' - -#### Proof of reduction for 3-Color - -Direction 1: If $\Psi$ is satisfiable, then $G$ is 3-colorable. - -Proof: - -Say $\Psi$ is satisfiable. Then $v_i$ and $\overline{v_i}$ are in different colors. - -For the color in central triangle, we can pick any color. - -For each dangler color is connected to blue, all literals cannot be blue. - -... - -QED - -Direction 2: If $G$ is 3-colorable, then $\Psi$ is satisfiable. - -Proof: - -QED - -### Example 3:Hamiltonian cycle problem (HAMCYCLE) - -Input: $G(V,E)$ - -Output: Does $G$ have a Hamiltonian cycle? (A cycle that visits each vertex exactly once.) - -Proof is too hard. - -but it is an existing NP-complete problem. - -## On lecture - -### Example 4: Scheduling problem (SCHED) - -scheduling with release time, deadline and execution times. - -Given $n$ jobs, where job $i$ has release time $r_i$, deadline $d_i$, and execution time $t_i$. - -Example: - -$S=\{2,3,7,5,4\}$. we created 5 jobs release time is 0, deadline is 26, execution time is $1$. - -Problem: Can you schedule these jobs so that each job starts after its release time and finishes before its deadline, and executed for $t_i$ time units? - -#### Proof of NP-completeness - -Step 1: Show that the problem is in NP. - -Certificate: $\langle (h_i,j_i),(h_2,j_2),\cdots,(h_n,j_n)\rangle$, where $h_i$ is the start time of job $i$ and $j_i$ is the machine that job $i$ is assigned to. - -Verifier: Check that $h_i + t_i \leq d_i$ for all $i$. - -Step 2: Show that the problem is NP-hard. - -We proceed by proving that $SSS\leq_p$ Scheduling. - -Consider an instance of SSS: $\{ a_1,a_2,\cdots,a_n\}$ and sum $b$. We can create a scheduling instance with release time 0, deadline $b$, and execution time $1$. - -Then we prove that the scheduling instance is a "yes" instance if and only if the SSS instance is a "yes" instance. - -Ideas of proof: - -If there is a subset of $\{a_1,a_2,\cdots,a_n\}$ that sums to $b$, then we can schedule the jobs in that order on one machine. - -If there is a schedule where all jobs are finished by time $b$, then the sum of the scheduled jobs is exactly $b$. - -### Example 5: Component grouping problem (CG) - -Given an undirected graph which is not necessarily connected. (A component is a subgraph that is connected.) - -Problem: Component Grouping: Give a graph $G$ that is not connected, and a positive integer $k$, is there a subset of its components that sums up to $k$? - -Denoted as $CG(G,k)$. - -#### Proof of NP-completeness for Component Grouping - -Step 1: Show that the problem is in NP. - -Certificate: $\langle S\rangle$, where $S$ is the subset of components that sums up to $k$. - -Verifier: Check that the sum of the sizes of the components in $S$ is $k$. This can be done in polynomial time using breadth-first search. - -Step 2: Show that the problem is NP-hard. - -We proceed by proving that $SSS\leq_p CG$. (Subset Sum $\leq_p$ Component Grouping) - -Consider an instance of SSS: $\langle a_1,a_2,\cdots,a_n,b\rangle$. - -We construct an instance of CG as follows: - -For each $a_i\in S$, we create a chain of $a_i$ vertices. - -WARNING: this is not a valid proof for NP hardness since the reduction is not polynomial for $n$, where $n$ is the number of vertices in the SSS instance. - +# Lecture 7 + +## Known NP-Complete Problems + +- SAT and 3-SAT +- Vertex Cover +- Independent Set + +## How to show a problem $L$ is NP-Complete + +- Show $L \in$ NP + - Give a polynomial time certificate + - Give a polynomial time verifier +- Show $L$ is NP-Hard: for some known NP-Complete problem $K$, show $K \leq_p L$ + - Construct a mapping $\phi$ from instance in $K$ to instance in $L$, given an instance $k\in K$, $\phi(k)\in L$. + - Show that you can compute $\phi(k)$ in polynomial time. + - Show that $k \in K$ is true if and only if $\phi(k) \in L$ is true. + +### Example 1: Subset Sum + +Input: A set $S$ of integers and a target positive integer $t$. + +Problem: Determine if there exists a subset $S' \subseteq S$ such that $\sum_{a_i\in S'} a_i = t$. + +We claim that Subset Sum is NP-Complete. + +Step 1: Subset Sum $\in$ NP + +- Certificate: $S' \subseteq S$ +- Verifier: Check that $\sum_{a_i\in S'} a_i = t$ + +Step 2: Subset Sum is NP-Hard + +We claim that 3-SAT $\leq_p$ Subset Sum + +Given any $3$-CNF formula $\Psi$, we will construct an instance $(S, t)$ of Subset Sum such that $\Psi$ is satisfiable if and only if there exists a subset $S' \subseteq S$ such that $\sum_{a_i\in S'} a_i = t$. + +#### How to construct $\Psi$? + +Reduction construction: + +Assumption: No clause contains both a literal and its negation. + +3-SAT problem: $\Psi$ has $n$ variables and $m$ clauses. + +Need to: construct $S$ of positive numbers and a target $t$ + +Ideas of construction: + +For 3-SAT instance $\Psi$: + +- At least one literal in each clause is true +- A variable and its negation cannot both be true + +$S$ contains integers with $n+m$ digits (base 10) + +$$ +p_1p_2\cdots p_n q_1 q_2 \cdots q_m +$$ + +where $p_i$ are representations of variables that are either $0$ or $1$ and $q_j$ are representations of clauses. + +For each variable $x_i$, we will have two integers in $S$, called $v_i$ and $\overline{v_i}$. + +- For each variable $x_i$, both $v_i$ and $\overline{v_i}$ have digits $p_i=1$. all other $p$ positions are zero + +- Each digit $q_j$ in $v_i$ is $1$ if $x_i$ appears in clause $j$; otherwise $q_j=0$ + +For example: + +$\Psi=(x_1\lor \neg x_2 \lor x_3) \land (\neg x_1 \lor x_2 \lor x_3)$ + +| | $p_1$ | $p_2$ | $p_3$ | $q_1$ | $q_2$ | +| ---------------- | ----- | ----- | ----- | ----- | ----- | +| $v_1$ | 1 | 0 | 0 | 1 | 0 | +| $\overline{v_1}$ | 1 | 0 | 0 | 0 | 1 | +| $v_2$ | 0 | 1 | 0 | 0 | 1 | +| $\overline{v_2}$ | 0 | 1 | 0 | 1 | 0 | +| $v_3$ | 0 | 0 | 1 | 1 | 1 | +| $\overline{v_3}$ | 0 | 0 | 1 | 0 | 0 | +| t | 1 | 1 | 1 | 1 | 1 | + +Let's try to prove correctness of the reduction. + +Direction 1: Say subset sum has a solution $S'$. + +We must prove that there is a satisfying assignment for $\Psi$. + +Set $x_i=1$ if $v_i\in S'$ + +Set $x_i=0$ if $\overline{v_i}\in S'$ + +1. We want set $x_i$ to be both true and false, we will pick (in $S'$) either $v_i$ or $\overline{v_i}$ +2. For each clause we have at least one literal that is true since $q_j$ has a $1$ in the clause. + +Direction 2: Say $\Psi$ has a satisfying assignment. + +We must prove that there is a subset $S'$ such that $\sum_{a_i\in S'} a_i = t$. + +If $x_i=1$, then $v_i\in S'$ + +If $x_i=0$, then $\overline{v_i}\in S'$ + +Problem: 1,2 or 3 literals in every clause can be true. + +Fix + +Add 2 numbers to $S$ for each clause $j$. We add $y_j,z_j$. + +- All $p$ digits are zero +- $q_j$ of $y_j$ is $1$, $q_j$ of $z_j$ is $2$, for all $j$, other digits are zero. +- Intuitively, these numbers account for the number of literals in clause $j$ that are true. + +New target are as follows: + +| | $p_1$ | $p_2$ | $p_3$ | $q_1$ | $q_2$ | +| ----- | ----- | ----- | ----- | ----- | ----- | +| $y_1$ | 0 | 0 | 0 | 1 | 0 | +| $z_1$ | 0 | 0 | 0 | 2 | 0 | +| $y_2$ | 0 | 0 | 0 | 0 | 1 | +| $z_2$ | 0 | 0 | 0 | 0 | 2 | +| $t$ | 1 | 1 | 1 | 4 | 4 | + +#### Time Complexity of construction for Subset Sum + +- $O(n+m)$ +- $n$ is the number of variables +- $m$ is the number of clauses + +How many integers are in $S$? + +- $2n$ for variables +- $2m$ for new numbers +- Total: $2n+2m$ integers + +How many digits are in each integer? + +- $n+m$ digits +- Time complexity: $O((n+m)^2)$ + +#### Proof of reduction for Subset Sum + +Claim 1: If Subset Sum has a solution, then $\Psi$ is satisfiable. + +Proof: + +Say $S'$ is a solution to Subset Sum. Then there exists a subset $S' \subseteq S$ such that $\sum_{a_i\in S'} a_i = t$. Here is an assignment of truth values to variables in $\Psi$ that satisfies $\Psi$: + +- Set $x_i=1$ if $v_i\in S'$ +- Set $x_i=0$ if $\overline{v_i}\in S'$ + +This is a valid assignment since: + +- We pick either $v_i$ or $\overline{v_i}$ +- For each clause, at least one literal is true + +QED + +Claim 2: If $\Psi$ is satisfiable, then Subset Sum has a solution. + +Proof: + +If $A$ is a satisfiable assignment for $\Psi$, then we can construct a subset $S'$ of $S$ such that $\sum_{a_i\in S'} a_i = t$. + +If $x_i=1$, then $v_i\in S'$ + +If $x_i=0$, then $\overline{v_i}\in S'$ + +Say $t=\sum$ elements we picked from $S$. + +- All $p_i$ in $t$ are $1$ +- All $q_j$ in $t$ are either $1$ or $2$ or $3$. + - If $q_j=1$, then $y_j,z_j\in S'$ + - If $q_j=2$, then $z_j\in S'$ + - If $q_j=3$, then $y_j\in S'$ + +QED + +### Example 2: 3 Color + +Input: Graph $G$ + +Problem: Determine if $G$ is 3-colorable. + +We claim that 3-Color is NP-Complete. + +#### Proof of NP for 3-Color + +Homework + +#### Proof of NP-Hard for 3-Color + +We claim that 3-SAT $\leq_p$ 3-Color + +Given a 3-CNF formula $\Psi$, we will construct a graph $G$ such that $\Psi$ is satisfiable if and only if $G$ is 3-colorable. + +Construction: + +1. Construct a core triangle (3 vertices for 3 colors) +2. 2 vertices for each variable $x_i:v_i,\overline{v_i}$ +3. Clause widget + +Clause widget: + +- 3 vertices for each clause $C_j:y_j,z_j,t_j$ (clause widget) +- 3 edges extended from clause widget +- variable vertex connected to extended edges + +Key for dangler design: + +Connect to all $v_i$ with true to the same color. and connect to all $v_i$ with false to another color. + +''' +TODO: Add dangler design image here. +''' + +#### Proof of reduction for 3-Color + +Direction 1: If $\Psi$ is satisfiable, then $G$ is 3-colorable. + +Proof: + +Say $\Psi$ is satisfiable. Then $v_i$ and $\overline{v_i}$ are in different colors. + +For the color in central triangle, we can pick any color. + +For each dangler color is connected to blue, all literals cannot be blue. + +... + +QED + +Direction 2: If $G$ is 3-colorable, then $\Psi$ is satisfiable. + +Proof: + +QED + +### Example 3:Hamiltonian cycle problem (HAMCYCLE) + +Input: $G(V,E)$ + +Output: Does $G$ have a Hamiltonian cycle? (A cycle that visits each vertex exactly once.) + +Proof is too hard. + +but it is an existing NP-complete problem. + +## On lecture + +### Example 4: Scheduling problem (SCHED) + +scheduling with release time, deadline and execution times. + +Given $n$ jobs, where job $i$ has release time $r_i$, deadline $d_i$, and execution time $t_i$. + +Example: + +$S=\{2,3,7,5,4\}$. we created 5 jobs release time is 0, deadline is 26, execution time is $1$. + +Problem: Can you schedule these jobs so that each job starts after its release time and finishes before its deadline, and executed for $t_i$ time units? + +#### Proof of NP-completeness + +Step 1: Show that the problem is in NP. + +Certificate: $\langle (h_i,j_i),(h_2,j_2),\cdots,(h_n,j_n)\rangle$, where $h_i$ is the start time of job $i$ and $j_i$ is the machine that job $i$ is assigned to. + +Verifier: Check that $h_i + t_i \leq d_i$ for all $i$. + +Step 2: Show that the problem is NP-hard. + +We proceed by proving that $SSS\leq_p$ Scheduling. + +Consider an instance of SSS: $\{ a_1,a_2,\cdots,a_n\}$ and sum $b$. We can create a scheduling instance with release time 0, deadline $b$, and execution time $1$. + +Then we prove that the scheduling instance is a "yes" instance if and only if the SSS instance is a "yes" instance. + +Ideas of proof: + +If there is a subset of $\{a_1,a_2,\cdots,a_n\}$ that sums to $b$, then we can schedule the jobs in that order on one machine. + +If there is a schedule where all jobs are finished by time $b$, then the sum of the scheduled jobs is exactly $b$. + +### Example 5: Component grouping problem (CG) + +Given an undirected graph which is not necessarily connected. (A component is a subgraph that is connected.) + +Problem: Component Grouping: Give a graph $G$ that is not connected, and a positive integer $k$, is there a subset of its components that sums up to $k$? + +Denoted as $CG(G,k)$. + +#### Proof of NP-completeness for Component Grouping + +Step 1: Show that the problem is in NP. + +Certificate: $\langle S\rangle$, where $S$ is the subset of components that sums up to $k$. + +Verifier: Check that the sum of the sizes of the components in $S$ is $k$. This can be done in polynomial time using breadth-first search. + +Step 2: Show that the problem is NP-hard. + +We proceed by proving that $SSS\leq_p CG$. (Subset Sum $\leq_p$ Component Grouping) + +Consider an instance of SSS: $\langle a_1,a_2,\cdots,a_n,b\rangle$. + +We construct an instance of CG as follows: + +For each $a_i\in S$, we create a chain of $a_i$ vertices. + +WARNING: this is not a valid proof for NP hardness since the reduction is not polynomial for $n$, where $n$ is the number of vertices in the SSS instance. + diff --git a/pages/CSE347/CSE347_L8.md b/content/CSE347/CSE347_L8.md similarity index 96% rename from pages/CSE347/CSE347_L8.md rename to content/CSE347/CSE347_L8.md index 7e08ae7..0c225c1 100644 --- a/pages/CSE347/CSE347_L8.md +++ b/content/CSE347/CSE347_L8.md @@ -1,353 +1,353 @@ -# Lecture 8 - -## NP-optimization problem - -Cannot be solved in polynomial time. - -Example: - -- Maximum independent set -- Minimum vertex cover - -What can we do? - -- solve small instances -- hard instances are rare - average case analysis -- solve special cases -- find an approximate solution - -## Approximation algorithms - -We find a "good" solution in polynomial time, but may not be optimal. - -Example: - -- Minimum vertex cover: we will find a small vertex cover, but not necessarily the smallest one. -- Maximum independent set: we will find a large independent set, but not necessarily the largest one. - -Question: How do we quantify the quality of the solution? - -### Approximation ratio - -Intuition: - -How good is an algorithm $A$ compared to an optimal solution in the worst case? - -Definition: - -Consider algorithm $A$ for an NP-optimization problem $L$. Say for **any** instance $l$, $A$ finds a solution output $c_A(l)$ and the optimal solution is $c^*(l)$. - -Approximation ratio is either: - -$$ -\max_{l \in L} \frac{c_A(l)}{c^*(l)}=\alpha -$$ - -for maximization problems, or - -$$ -\min_{l \in L} \frac{c^A(l)}{c_*(l)}=\alpha -$$ - -for minimization problems. - -Example: - -Alice's Algorithm, $A$, finds a vertex cover of size $c_A(l)$ for instance $l(G)$. The optimal vertex cover has size $c^*(l)$. - -We want approximation ratio to be as close to 1 as possible. - -> Vertex cover: -> -> A vertex cover is a set of vertices that touches all edges. - -Let's try an approximation algorithm for the vertex cover problem, called Greedy cover. - -#### Greedy cover - -Pick any uncovered edge, both its endpoints are added to the cover $C$, until all edges are covered. - -Runtime: $O(m)$ - -Claim: Greedy cover is correct, and it finds a vertex cover. - -Proof: - -Algorithm only terminates when all edges are covered. - -Claim: Greedy cover is a 2-approximation algorithm. - -Proof: - -Look at the two edges we picked. - -Either it is covered by Greedy cover, or it is not. - -If it is not covered by Greedy cover, then we will add both endpoints to the cover. - -In worst case, Greedy cover will add both endpoints of each edge to the cover. (Consider the graph with disjoint edges.) - -Thus, the size of the vertex cover found by Greedy cover is at most twice the size of the optimal vertex cover. - -Thus, Greedy cover is a 2-approximation algorithm. - -> Min-cut: -> -> Given a graph $G$ and two vertices $s$ and $t$, find the minimum cut between $s$ and $t$. -> -> Max-cut: -> -> Given a graph $G$, find the maximum cut. - -#### Local cut - -Algorithm: - -- start with an arbitrary cut of $G$. -- While you can move a vertex from one side to the other side while increasing the size of the cut, do so. -- Return the cut found. - -We will prove its: - -- Runtime -- Feasibility -- Approximation ratio - -##### Runtime for local cut - -Since size of cut is at most $|E|$, the runtime is $O(m)$. - -When we move a vertex from one side to the other side, the size of the cut increases by at least 1. - -Thus, the algorithm terminates in at most $|V|$ steps. - -So the runtime is $O(|E||V|^2)$. - -##### Feasibility for local cut - -The algorithm only terminates when no more vertices can be moved. - -Thus, the cut found is a feasible solution. - -##### Approximation ratio for local cut - -This is a half-approximation algorithm. - -We need to show that the size of the cut found is at least half of the size of the optimal cut. - -We could first upper bound the size of the optimal cut is at most $|E|$. - -We will then prove that solution we found is at least half of the optimal cut $\frac{|E|}{2}$ for any graph $G$. - -Proof: - -When we terminate, no vertex could be moved - -Therefore, **The number of crossing edges is at least the number of non-crossing edges**. - -Let $d(u)$ be the degree of vertex $u\in V$. - -The total number of crossing edges for vertex $u$ is at least $\frac{1}{2}d(u)$. - -Summing over all vertices, the total number of crossing edges is at least $\frac{1}{2}\sum_{u\in V}d(u)=\frac{1}{2}|E|$. - -So the total number of non-crossing edges is at most $\frac{|E|}{2}$. - -QED - -#### Set cover - -Problem: - -You are collecting a set of magic cards. - -$X$ is the set of all possible cards. You want at least one of each card. - -Each dealer $j$ has a pack $S_j\subseteq X$ of cards. You have to buy entire pack or none from dealer $j$. - -Goal: What is the least number of packs you need to buy to get all cards? - -Formally: - -Input $X$ is a universe of $n$ elements, and a collection of subsets of $X$, $Y=\{S_1, S_2, \ldots, S_m\}\subseteq X$. - -Goal: Pick $C\subseteq Y$ such that $\bigcup_{S_i\in C}S_i=X$, and $|C|$ is minimized. - -Set cover is an NP-optimization problem. It is a generalization of the vertex cover problem. - -#### Greedy set cover - -Algorithm: - -- Start with empty set $C$. -- While there is an element $x$ in $X$ that is not covered, pick one such element $x\in S_i$ where $S_i$ is the set that has not been picked before. -- Add $S_i$ to $C$. -- Return $C$. - -```python -def greedy_set_cover(X, Y): - # X is the set of elements - # Y is the collection of sets, hashset by default - C = [] - def non_covered_elements(X, C): - # return the elements in X that are not covered by C - # O(|X|) - return [x for x in X if not any(x in c for c in C)] - non_covered = non_covered_elements(X, C) - # O(|X|) every loop reduce the size of non_covered by 1 - while non_covered: - max_cover,max_set = 0,None - # O(|Y|) - for S in Y: - # Intersection of two sets is O(min(|X|,|S|)) - cur_cover = len(set(non_covered) & set(S)) - if cur_cover > max_cover: - max_cover,max_set = cur_cover,S - C.append(max_set) - non_covered = non_covered_elements(X, C) - return C -``` - -It is not optimal. - -Need to prove its: - -- Correctness: - Keep picking until all elements are covered. -- Runtime: - $O(|X||Y|^2)$ -- Approximation ratio: - -##### Approximation ratio for greedy set cover - -> Harmonic number: -> -> $H_n=\sum_{i=1}^n\frac{1}{i}=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}=\Theta(\log n)$ - -We claim that the size of the set cover found is at most $H_n\log n$ times the size of the optimal set cover. - -###### First bound: - -Proof: - -If the optimal picks $k$ sets, then the size of the set cover found is at most $(1+\log n)k$ sets. - -Let $n=|X|$. - -Observe that - -For the first round, the elements that we not covered is $n$. -$$ -|U_0|=n -$$ - -In the second round, the elements that we not covered is at most $|U_0|-x$ where $x=|S_1|$ is the number of elements in the set picked in the first round. - -$$ -|U_1|=|U_0|-|S_1| -$$ - -... - -So $x_i\geq \frac{|U_{i-1}|}{k}$. - -We proceed by contradiction. - -Suppose all sets in the optimal solution are $< \frac{|U_0|}{k}$. Then the sum of the sizes of the sets in the optimal solution is $< |U_0|=n$. - -_There exists a least ratio of selection of sets determined by $k_i$. Otherwise the function (selecting the set cover) will not terminate (no such sets exists)_ - -> Some math magics: -> $$(1-\frac{1}{k})^k\leq \frac{1}{e}$$ - -So $n(1-\frac{1}{k})^{|C|-1}=1$, $|C|\leq 1+k\ln n$. - -So the size of the set cover found is at most $(1+\ln n)k$. - -QED - -So the greedy set cover is not too bad... - -###### Second bound: - -Greedy set cover is a $H_d$-approximation algorithm of set cover. - -Proof: - -Assign a cost to the elements of $X$ according to the decisions of the greedy set cover. - -Let $\delta(S^i)$ be the new number of elements covered by set $S^i$. - -$$ -\delta(S^i)=|S_i\cap U_{i-1}| -$$ - -If the element $x$ is added by step $i$, when set $S_i$ is picked, then the cost of $x$ to - -$$ -\frac{1}{\delta(S^i)}=\frac{1}{x_i} -$$ - -Example: - -$$ -\begin{aligned} -X&=\{A,B,C,D,E,F,G\}\\ -S_1&=\{A,C,E\}\\ -S_2&=\{B,C,F,G\}\\ -S_3&=\{B,D,F,G\}\\ -S_4&=\{D,G\} -\end{aligned} -$$ - -First we select $S_2$, then $cost(B)=cost(C)=cost(F)=cost(G)=\frac{1}{4}$. - -Then we select $S_1$, then $cost(A)=cost(E)=\frac{1}{2}$. - -Then we select $S_3$, then $cost(D)=1$. - -If element $x$ was covered by greedy set cover due to the addition of set $S^i$ at step $i$, then the cost of $x$ is $\frac{1}{\delta(S^i)}$. - -$$ -\textup{Total cost of GSC}=\sum_{x\in X}c(x)=\sum_{i=1}^{|C|}\sum_{X\textup{ covered at iteration }i}c(x)=\sum_{i=1}^{|C|}\delta(S^i)\frac{1}{\delta(S^i)}=|C| -$$ - -Claim: Consider any set $S$ that is a subset of $X$. The cost paid by the greedy set cover for $S$ is at most $H_{|S|}$. - -Suppose that the greedy set covers $S$ in order $x_1,x_2,\ldots,x_{|S|}$, where $\{x_1,x_2,\ldots,x_{|S|}\}=S$. - -When GSC covers $x_j$, $\{x_j,x_{j+1},\ldots,x_{|S|}\}$ are not covered. - -At this point, the GSC has the option of picking $S$ - -This implies that the $\delta(S)$ is at least $|S|-j+1$. - -Assume that $S$ is picked $\hat{S}$ for which $\delta(\hat{S})$ is maximized ($\hat{S}$ may be $S$ or other sets that have not covered $x_j$). - -So, $\delta(\hat{S})\geq \delta(S)\geq |S|-j+1$. - -So the cost of $x_j$ is $\delta(\hat{S})\leq \frac{1}{\delta(S)}\leq \frac{1}{|S|-j+1}$. - -Summing over all $j$, the cost of $S$ is at most $\sum_{j=1}^{|S|}\frac{1}{|S|-j+1}=H_{|S|}$. - -Back to the proof of approximation ratio: - -Let $C^*$ be optimal set cover. - -$$ -|C|=\sum_{x\in X}c(x)\leq \sum_{S_j\in C^*}\sum_{x\in S_j}c(x) -$$ - -This inequality holds because of counting element that is covered by more than one set. - -Since $\sum_{x\in S_j}c(x)\leq H_{|S_j|}$, by our claim. - -Let $d$ be the largest cardinality of any set in $C^*$. - -$$ -|C|\leq \sum_{S_j\in C^*}H_{|S_j|}\leq \sum_{S_j\in C^*}H_d=H_d|C^*| -$$ - -So the approximation ratio for greedy set cover is $H_d$. - -QED +# Lecture 8 + +## NP-optimization problem + +Cannot be solved in polynomial time. + +Example: + +- Maximum independent set +- Minimum vertex cover + +What can we do? + +- solve small instances +- hard instances are rare - average case analysis +- solve special cases +- find an approximate solution + +## Approximation algorithms + +We find a "good" solution in polynomial time, but may not be optimal. + +Example: + +- Minimum vertex cover: we will find a small vertex cover, but not necessarily the smallest one. +- Maximum independent set: we will find a large independent set, but not necessarily the largest one. + +Question: How do we quantify the quality of the solution? + +### Approximation ratio + +Intuition: + +How good is an algorithm $A$ compared to an optimal solution in the worst case? + +Definition: + +Consider algorithm $A$ for an NP-optimization problem $L$. Say for **any** instance $l$, $A$ finds a solution output $c_A(l)$ and the optimal solution is $c^*(l)$. + +Approximation ratio is either: + +$$ +\max_{l \in L} \frac{c_A(l)}{c^*(l)}=\alpha +$$ + +for maximization problems, or + +$$ +\min_{l \in L} \frac{c^A(l)}{c_*(l)}=\alpha +$$ + +for minimization problems. + +Example: + +Alice's Algorithm, $A$, finds a vertex cover of size $c_A(l)$ for instance $l(G)$. The optimal vertex cover has size $c^*(l)$. + +We want approximation ratio to be as close to 1 as possible. + +> Vertex cover: +> +> A vertex cover is a set of vertices that touches all edges. + +Let's try an approximation algorithm for the vertex cover problem, called Greedy cover. + +#### Greedy cover + +Pick any uncovered edge, both its endpoints are added to the cover $C$, until all edges are covered. + +Runtime: $O(m)$ + +Claim: Greedy cover is correct, and it finds a vertex cover. + +Proof: + +Algorithm only terminates when all edges are covered. + +Claim: Greedy cover is a 2-approximation algorithm. + +Proof: + +Look at the two edges we picked. + +Either it is covered by Greedy cover, or it is not. + +If it is not covered by Greedy cover, then we will add both endpoints to the cover. + +In worst case, Greedy cover will add both endpoints of each edge to the cover. (Consider the graph with disjoint edges.) + +Thus, the size of the vertex cover found by Greedy cover is at most twice the size of the optimal vertex cover. + +Thus, Greedy cover is a 2-approximation algorithm. + +> Min-cut: +> +> Given a graph $G$ and two vertices $s$ and $t$, find the minimum cut between $s$ and $t$. +> +> Max-cut: +> +> Given a graph $G$, find the maximum cut. + +#### Local cut + +Algorithm: + +- start with an arbitrary cut of $G$. +- While you can move a vertex from one side to the other side while increasing the size of the cut, do so. +- Return the cut found. + +We will prove its: + +- Runtime +- Feasibility +- Approximation ratio + +##### Runtime for local cut + +Since size of cut is at most $|E|$, the runtime is $O(m)$. + +When we move a vertex from one side to the other side, the size of the cut increases by at least 1. + +Thus, the algorithm terminates in at most $|V|$ steps. + +So the runtime is $O(|E||V|^2)$. + +##### Feasibility for local cut + +The algorithm only terminates when no more vertices can be moved. + +Thus, the cut found is a feasible solution. + +##### Approximation ratio for local cut + +This is a half-approximation algorithm. + +We need to show that the size of the cut found is at least half of the size of the optimal cut. + +We could first upper bound the size of the optimal cut is at most $|E|$. + +We will then prove that solution we found is at least half of the optimal cut $\frac{|E|}{2}$ for any graph $G$. + +Proof: + +When we terminate, no vertex could be moved + +Therefore, **The number of crossing edges is at least the number of non-crossing edges**. + +Let $d(u)$ be the degree of vertex $u\in V$. + +The total number of crossing edges for vertex $u$ is at least $\frac{1}{2}d(u)$. + +Summing over all vertices, the total number of crossing edges is at least $\frac{1}{2}\sum_{u\in V}d(u)=\frac{1}{2}|E|$. + +So the total number of non-crossing edges is at most $\frac{|E|}{2}$. + +QED + +#### Set cover + +Problem: + +You are collecting a set of magic cards. + +$X$ is the set of all possible cards. You want at least one of each card. + +Each dealer $j$ has a pack $S_j\subseteq X$ of cards. You have to buy entire pack or none from dealer $j$. + +Goal: What is the least number of packs you need to buy to get all cards? + +Formally: + +Input $X$ is a universe of $n$ elements, and a collection of subsets of $X$, $Y=\{S_1, S_2, \ldots, S_m\}\subseteq X$. + +Goal: Pick $C\subseteq Y$ such that $\bigcup_{S_i\in C}S_i=X$, and $|C|$ is minimized. + +Set cover is an NP-optimization problem. It is a generalization of the vertex cover problem. + +#### Greedy set cover + +Algorithm: + +- Start with empty set $C$. +- While there is an element $x$ in $X$ that is not covered, pick one such element $x\in S_i$ where $S_i$ is the set that has not been picked before. +- Add $S_i$ to $C$. +- Return $C$. + +```python +def greedy_set_cover(X, Y): + # X is the set of elements + # Y is the collection of sets, hashset by default + C = [] + def non_covered_elements(X, C): + # return the elements in X that are not covered by C + # O(|X|) + return [x for x in X if not any(x in c for c in C)] + non_covered = non_covered_elements(X, C) + # O(|X|) every loop reduce the size of non_covered by 1 + while non_covered: + max_cover,max_set = 0,None + # O(|Y|) + for S in Y: + # Intersection of two sets is O(min(|X|,|S|)) + cur_cover = len(set(non_covered) & set(S)) + if cur_cover > max_cover: + max_cover,max_set = cur_cover,S + C.append(max_set) + non_covered = non_covered_elements(X, C) + return C +``` + +It is not optimal. + +Need to prove its: + +- Correctness: + Keep picking until all elements are covered. +- Runtime: + $O(|X||Y|^2)$ +- Approximation ratio: + +##### Approximation ratio for greedy set cover + +> Harmonic number: +> +> $H_n=\sum_{i=1}^n\frac{1}{i}=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}=\Theta(\log n)$ + +We claim that the size of the set cover found is at most $H_n\log n$ times the size of the optimal set cover. + +###### First bound: + +Proof: + +If the optimal picks $k$ sets, then the size of the set cover found is at most $(1+\log n)k$ sets. + +Let $n=|X|$. + +Observe that + +For the first round, the elements that we not covered is $n$. +$$ +|U_0|=n +$$ + +In the second round, the elements that we not covered is at most $|U_0|-x$ where $x=|S_1|$ is the number of elements in the set picked in the first round. + +$$ +|U_1|=|U_0|-|S_1| +$$ + +... + +So $x_i\geq \frac{|U_{i-1}|}{k}$. + +We proceed by contradiction. + +Suppose all sets in the optimal solution are $< \frac{|U_0|}{k}$. Then the sum of the sizes of the sets in the optimal solution is $< |U_0|=n$. + +_There exists a least ratio of selection of sets determined by $k_i$. Otherwise the function (selecting the set cover) will not terminate (no such sets exists)_ + +> Some math magics: +> $$(1-\frac{1}{k})^k\leq \frac{1}{e}$$ + +So $n(1-\frac{1}{k})^{|C|-1}=1$, $|C|\leq 1+k\ln n$. + +So the size of the set cover found is at most $(1+\ln n)k$. + +QED + +So the greedy set cover is not too bad... + +###### Second bound: + +Greedy set cover is a $H_d$-approximation algorithm of set cover. + +Proof: + +Assign a cost to the elements of $X$ according to the decisions of the greedy set cover. + +Let $\delta(S^i)$ be the new number of elements covered by set $S^i$. + +$$ +\delta(S^i)=|S_i\cap U_{i-1}| +$$ + +If the element $x$ is added by step $i$, when set $S_i$ is picked, then the cost of $x$ to + +$$ +\frac{1}{\delta(S^i)}=\frac{1}{x_i} +$$ + +Example: + +$$ +\begin{aligned} +X&=\{A,B,C,D,E,F,G\}\\ +S_1&=\{A,C,E\}\\ +S_2&=\{B,C,F,G\}\\ +S_3&=\{B,D,F,G\}\\ +S_4&=\{D,G\} +\end{aligned} +$$ + +First we select $S_2$, then $cost(B)=cost(C)=cost(F)=cost(G)=\frac{1}{4}$. + +Then we select $S_1$, then $cost(A)=cost(E)=\frac{1}{2}$. + +Then we select $S_3$, then $cost(D)=1$. + +If element $x$ was covered by greedy set cover due to the addition of set $S^i$ at step $i$, then the cost of $x$ is $\frac{1}{\delta(S^i)}$. + +$$ +\textup{Total cost of GSC}=\sum_{x\in X}c(x)=\sum_{i=1}^{|C|}\sum_{X\textup{ covered at iteration }i}c(x)=\sum_{i=1}^{|C|}\delta(S^i)\frac{1}{\delta(S^i)}=|C| +$$ + +Claim: Consider any set $S$ that is a subset of $X$. The cost paid by the greedy set cover for $S$ is at most $H_{|S|}$. + +Suppose that the greedy set covers $S$ in order $x_1,x_2,\ldots,x_{|S|}$, where $\{x_1,x_2,\ldots,x_{|S|}\}=S$. + +When GSC covers $x_j$, $\{x_j,x_{j+1},\ldots,x_{|S|}\}$ are not covered. + +At this point, the GSC has the option of picking $S$ + +This implies that the $\delta(S)$ is at least $|S|-j+1$. + +Assume that $S$ is picked $\hat{S}$ for which $\delta(\hat{S})$ is maximized ($\hat{S}$ may be $S$ or other sets that have not covered $x_j$). + +So, $\delta(\hat{S})\geq \delta(S)\geq |S|-j+1$. + +So the cost of $x_j$ is $\delta(\hat{S})\leq \frac{1}{\delta(S)}\leq \frac{1}{|S|-j+1}$. + +Summing over all $j$, the cost of $S$ is at most $\sum_{j=1}^{|S|}\frac{1}{|S|-j+1}=H_{|S|}$. + +Back to the proof of approximation ratio: + +Let $C^*$ be optimal set cover. + +$$ +|C|=\sum_{x\in X}c(x)\leq \sum_{S_j\in C^*}\sum_{x\in S_j}c(x) +$$ + +This inequality holds because of counting element that is covered by more than one set. + +Since $\sum_{x\in S_j}c(x)\leq H_{|S_j|}$, by our claim. + +Let $d$ be the largest cardinality of any set in $C^*$. + +$$ +|C|\leq \sum_{S_j\in C^*}H_{|S_j|}\leq \sum_{S_j\in C^*}H_d=H_d|C^*| +$$ + +So the approximation ratio for greedy set cover is $H_d$. + +QED diff --git a/pages/CSE347/CSE347_L9.md b/content/CSE347/CSE347_L9.md similarity index 100% rename from pages/CSE347/CSE347_L9.md rename to content/CSE347/CSE347_L9.md diff --git a/pages/CSE347/Exam_reviews/CSE347_E1.md b/content/CSE347/Exam_reviews/CSE347_E1.md similarity index 98% rename from pages/CSE347/Exam_reviews/CSE347_E1.md rename to content/CSE347/Exam_reviews/CSE347_E1.md index 6a5629a..78c1443 100644 --- a/pages/CSE347/Exam_reviews/CSE347_E1.md +++ b/content/CSE347/Exam_reviews/CSE347_E1.md @@ -1,34 +1,34 @@ -# Exam 1 review - -## Greedy - -A Greedy Algorithm is an algorithm whose solution applies the same choice rule at each step over and over until no more choices can be made. - -- Stating and Proving a Greedy Algorithm -- State your algorithm (“at this step, make this choice”) -- Greedy Choice Property (Exchange Argument) -- Inductive Structure -- Optimal Substructure -- "Simple Induction" -- Asymptotic Runtime - -## Divide and conquer - -Stating and Proving a Dividing and Conquer Algorithm - -- Describe the divide, conquer, and combine steps of your algorithm. -- The combine step is the most important part of a divide and conquer algorithm, and in your recurrence this step is the "f (n)", or work done at each subproblem level. You need to show that you can combine the results of your subproblems somehow to get the solution for the entire problem. -- Provide and prove a base case (when you can divide no longer) -- Prove your induction step: suppose subproblems (two problems of size n/2, usually) of the same kind are solved optimally. Then, because of the combine step, the overall problem (of size n) will be solved optimally. -- Provide recurrence and solve for its runtime (Master Method) - -## Maximum Flow -Given a weighted directed acyclic graph with a source and a sink node, the goal is to see how much "flow" you can push from the source to the sink simultaneously. - -Finding the maximum flow can be solved by the Ford-Fulkerson Algorithm. Runtime (from lecture slides): $O(F (|V | + |E |))$. - -Fattest Path improvement: $O(log |V |(|V | + |E |))$ - -Min Cut-Max Flow: the maximum flow from source $s$ to sink $t$ is equal to the minimum sum of an $s-t$ cut. - +# Exam 1 review + +## Greedy + +A Greedy Algorithm is an algorithm whose solution applies the same choice rule at each step over and over until no more choices can be made. + +- Stating and Proving a Greedy Algorithm +- State your algorithm (“at this step, make this choice”) +- Greedy Choice Property (Exchange Argument) +- Inductive Structure +- Optimal Substructure +- "Simple Induction" +- Asymptotic Runtime + +## Divide and conquer + +Stating and Proving a Dividing and Conquer Algorithm + +- Describe the divide, conquer, and combine steps of your algorithm. +- The combine step is the most important part of a divide and conquer algorithm, and in your recurrence this step is the "f (n)", or work done at each subproblem level. You need to show that you can combine the results of your subproblems somehow to get the solution for the entire problem. +- Provide and prove a base case (when you can divide no longer) +- Prove your induction step: suppose subproblems (two problems of size n/2, usually) of the same kind are solved optimally. Then, because of the combine step, the overall problem (of size n) will be solved optimally. +- Provide recurrence and solve for its runtime (Master Method) + +## Maximum Flow +Given a weighted directed acyclic graph with a source and a sink node, the goal is to see how much "flow" you can push from the source to the sink simultaneously. + +Finding the maximum flow can be solved by the Ford-Fulkerson Algorithm. Runtime (from lecture slides): $O(F (|V | + |E |))$. + +Fattest Path improvement: $O(log |V |(|V | + |E |))$ + +Min Cut-Max Flow: the maximum flow from source $s$ to sink $t$ is equal to the minimum sum of an $s-t$ cut. + A cut is a partition of a graph into two disjoint sets by removing edges connecting the two parts. An $s-t$ cut will put $s$ and $t$ into the different sets. \ No newline at end of file diff --git a/pages/CSE347/Exam_reviews/CSE347_E2.md b/content/CSE347/Exam_reviews/CSE347_E2.md similarity index 100% rename from pages/CSE347/Exam_reviews/CSE347_E2.md rename to content/CSE347/Exam_reviews/CSE347_E2.md diff --git a/pages/CSE347/_meta.js b/content/CSE347/_meta.js similarity index 95% rename from pages/CSE347/_meta.js rename to content/CSE347/_meta.js index 73da9a7..8ae27a0 100644 --- a/pages/CSE347/_meta.js +++ b/content/CSE347/_meta.js @@ -1,5 +1,5 @@ export default { - index: "Course Description", + //index: "Course Description", "---":{ type: 'separator' }, diff --git a/pages/CSE347/index.md b/content/CSE347/index.md similarity index 100% rename from pages/CSE347/index.md rename to content/CSE347/index.md diff --git a/pages/CSE442T/CSE442T_L1.md b/content/CSE442T/CSE442T_L1.md similarity index 95% rename from pages/CSE442T/CSE442T_L1.md rename to content/CSE442T/CSE442T_L1.md index 47f7f38..8db27c2 100644 --- a/pages/CSE442T/CSE442T_L1.md +++ b/content/CSE442T/CSE442T_L1.md @@ -1,127 +1,127 @@ -# Lecture 1 - -## Chapter 1: Introduction - -### Alice sending information to Bob - -Assuming _Eve_ can always listen - -Rule 1. Message, Encryption to Code and Decryption to original Message. - -### Kerckhoffs' principle - -It states that the security of a cryptographic system shouldn't rely on the secrecy of the algorithm (Assuming Eve knows how everything works.) - -**Security is due to the security of the key.** - -### Private key encryption scheme - -Let $M$ be the set of message that Alice will send to Bob. (The message space) "plaintext" - -Let $K$ be the set of key that will ever be used. (The key space) - -$Gen$ be the key generation algorithm. - -$k\gets Gen(K)$ - -$c\gets Enc_k(m)$ denotes cipher encryption. - -$m'\gets Dec_k(c')$ $m'$ might be null for incorrect $c'$. - -$P[k\gets K:Dec_k(Enc_k(M))=m]=1$ The probability of decryption of encrypted message is original message is 1. - -*_in some cases we can allow the probability not be 1_ - -### Some examples of crypto system - -Let $M=\text{all five letter strings}$. - -And $K=[1,10^{10}]$ - -Example: - -$P[k=k']=\frac{1}{10^{10}}$ - -$Enc_{1234567890}("brion")="brion1234567890"$ - -$Dec_{1234567890}(brion1234567890)="brion"$ - -Seems not very secure but valid crypto system. - -### Early attempts for crypto system - -#### Caesar cipher - -$M=\text{finite string of texts}$ - -$K=[1,26]$ - -$Enc_k=[(i+K)\% 26\ for\ i \in m]=c$ - -$Dec_k=[(i+26-K)\% 26\ for\ i \in c]$ - -```python -def caesar_cipher_enc(s: str, k:int): - return ''.join([chr((ord(i)-ord('a')+k)%26+ord('a')) for i in s]) - -def caesar_cipher_dec(s: str, k:int): - return ''.join([chr((ord(i)-ord('a')+26-k)%26+ord('a')) for i in s]) -``` - -#### Substitution cipher - -$M=\text{finite string of texts}$ - -$K=\text{set of all bijective linear transformations (for English alphabet},|K|=26!\text{)}$ - -$Enc_k=[iK\ for\ i \in m]=c$ - -$Dec_k=[iK^{-1}\ for\ i \in c]$ - -Fails to frequency analysis - -#### Vigenere Cipher - -$M=\text{finite string of texts with length }m$ - -$K=\text{[0,26]}^n$ (assuming English alphabet) - -```python -def viginere_cipher_enc(s: str, k: List[int]): - res='' - n,m=len(s),len(k) - j=0 - for i in s: - res+=caesar_cipher_enc(i,k[j]) - j=(j+1)%m - return res - -def viginere_cipher_dec(s: str, k: List[int]): - res='' - n,m=len(s),len(k) - j=0 - for i in s: - res+=caesar_cipher_dec(i,k[j]) - j=(j+1)%m - return res -``` - -#### One time pad - -Completely random string, sufficiently long. - -$M=\text{finite string of texts with length }n$ - -$K=\text{[0,26]}^n$ (assuming English alphabet)$ - -$Enc_k=m\oplus k$ - -$Dec_k=c\oplus k$ - -```python -def one_time_pad_enc(s: str, k: List[int]): - return ''.join([chr((ord(i)-ord('a')+k[j])%26+ord('a')) for j,i in enumerate(s)]) - -def one_time_pad_dec(s: str, k: List[int]): - return ''.join([chr((ord(i)-ord('a')+26-k[j])%26+ord('a')) for j,i in enumerate(s)]) -``` +# Lecture 1 + +## Chapter 1: Introduction + +### Alice sending information to Bob + +Assuming _Eve_ can always listen + +Rule 1. Message, Encryption to Code and Decryption to original Message. + +### Kerckhoffs' principle + +It states that the security of a cryptographic system shouldn't rely on the secrecy of the algorithm (Assuming Eve knows how everything works.) + +**Security is due to the security of the key.** + +### Private key encryption scheme + +Let $M$ be the set of message that Alice will send to Bob. (The message space) "plaintext" + +Let $K$ be the set of key that will ever be used. (The key space) + +$Gen$ be the key generation algorithm. + +$k\gets Gen(K)$ + +$c\gets Enc_k(m)$ denotes cipher encryption. + +$m'\gets Dec_k(c')$ $m'$ might be null for incorrect $c'$. + +$P[k\gets K:Dec_k(Enc_k(M))=m]=1$ The probability of decryption of encrypted message is original message is 1. + +*_in some cases we can allow the probability not be 1_ + +### Some examples of crypto system + +Let $M=\text{all five letter strings}$. + +And $K=[1,10^{10}]$ + +Example: + +$P[k=k']=\frac{1}{10^{10}}$ + +$Enc_{1234567890}("brion")="brion1234567890"$ + +$Dec_{1234567890}(brion1234567890)="brion"$ + +Seems not very secure but valid crypto system. + +### Early attempts for crypto system + +#### Caesar cipher + +$M=\text{finite string of texts}$ + +$K=[1,26]$ + +$Enc_k=[(i+K)\% 26\ for\ i \in m]=c$ + +$Dec_k=[(i+26-K)\% 26\ for\ i \in c]$ + +```python +def caesar_cipher_enc(s: str, k:int): + return ''.join([chr((ord(i)-ord('a')+k)%26+ord('a')) for i in s]) + +def caesar_cipher_dec(s: str, k:int): + return ''.join([chr((ord(i)-ord('a')+26-k)%26+ord('a')) for i in s]) +``` + +#### Substitution cipher + +$M=\text{finite string of texts}$ + +$K=\text{set of all bijective linear transformations (for English alphabet},|K|=26!\text{)}$ + +$Enc_k=[iK\ for\ i \in m]=c$ + +$Dec_k=[iK^{-1}\ for\ i \in c]$ + +Fails to frequency analysis + +#### Vigenere Cipher + +$M=\text{finite string of texts with length }m$ + +$K=\text{[0,26]}^n$ (assuming English alphabet) + +```python +def viginere_cipher_enc(s: str, k: List[int]): + res='' + n,m=len(s),len(k) + j=0 + for i in s: + res+=caesar_cipher_enc(i,k[j]) + j=(j+1)%m + return res + +def viginere_cipher_dec(s: str, k: List[int]): + res='' + n,m=len(s),len(k) + j=0 + for i in s: + res+=caesar_cipher_dec(i,k[j]) + j=(j+1)%m + return res +``` + +#### One time pad + +Completely random string, sufficiently long. + +$M=\text{finite string of texts with length }n$ + +$K=\text{[0,26]}^n$ (assuming English alphabet)$ + +$Enc_k=m\oplus k$ + +$Dec_k=c\oplus k$ + +```python +def one_time_pad_enc(s: str, k: List[int]): + return ''.join([chr((ord(i)-ord('a')+k[j])%26+ord('a')) for j,i in enumerate(s)]) + +def one_time_pad_dec(s: str, k: List[int]): + return ''.join([chr((ord(i)-ord('a')+26-k[j])%26+ord('a')) for j,i in enumerate(s)]) +``` diff --git a/pages/CSE442T/CSE442T_L10.md b/content/CSE442T/CSE442T_L10.md similarity index 96% rename from pages/CSE442T/CSE442T_L10.md rename to content/CSE442T/CSE442T_L10.md index 8e87355..b913980 100644 --- a/pages/CSE442T/CSE442T_L10.md +++ b/content/CSE442T/CSE442T_L10.md @@ -1,210 +1,210 @@ -# Lecture 10 - -## Chapter 2: Computational Hardness - -### Discrete Log Assumption (Assumption 52.2) - -This is collection of one-way functions - -$$ -p\gets \tilde\Pi_n(\textup{ safe primes }), p=2q+1 -$$ - -$$ -a\gets \mathbb{Z}*_{p};g=a^2(\textup{ make sure }g\neq 1) -$$ - -$$ -f_{g,p}(x)=g^x\mod p -$$ - -$$ -f:\mathbb{Z}_q\to \mathbb{Z}^*_p -$$ - -#### Evidence for Discrete Log Assumption - -Best known algorithm to always solve discrete log mod p, $p\in \Pi_n$ - -$$ -O(2^{\sqrt{2}\sqrt{\log(n)}}) -$$ - -### RSA Assumption - -Let $e$ be the exponents - -$$ -P[p,q\gets \Pi_n;N\gets p\cdot q;e\gets \mathbb{Z}_{\phi(N)}^*;y\gets \mathbb{N}_n;x\gets \mathcal{A}(N,e,y);x^e=y\mod N]<\epsilon(n) -$$ - -#### Theorem 53.2 (RSA Algorithm) - -This is a collection of one-way functions - -$I=\{(N,e):N=p\cdot q,p,q\in \Pi_n \textup{ and } e\in \mathbb{Z}_{\phi(N)}^*\}$ - -$D_{(N,e)}=\mathbb{Z}_N^*$ - -$R_{(N,e)}=\mathbb{Z}_N^*$ - -$f_{(N,e)}(x)=x^e\mod N$ - -Example: - -On encryption side - -$p=5,q=11,N=5\times 11=55$, $\phi(N)=4*10=40$ - -pick $e\in \mathbb{Z}_{40}^*$. say $e=3$, and $f(x)=x^3\mod 55$ - -pick $y\in \mathbb{Z}_{55}^*$. say $y=17$. We have $(55,3,17)$ - -$x^{40}\equiv 1\mod 55$ - -$x^{41}\equiv x\mod 55$ - -$x^{40k+1}\equiv x \mod 55$ - -Since $x^a\equiv x^{a\mod 40}\mod 55$ (by corollary of Fermat's little Theorem: $a^x\mod N=a^{x\mod \Phi(N)}\mod N$ -s ) - -The problem is, what can we multiply by $3$ to get $1\mod \phi(N)=1\mod 40$. - -by computing the multiplicative inverse using extended Euclidean algorithm we have $3\cdot 27\equiv 1\mod 40$. - -$x^3\equiv 17\mod 55$ - -$x\equiv 17^{27}\mod 55$ - -On adversary side. - -they don't know $\phi(N)=40$ - -$$ -f(N,e):\mathbb{Z}_N^*\to \mathbb{Z}_N^* -$$ -is a bijection. - -Proof: Suppose $x_1^e\equiv x_2^e\mod n$ - -Then let $d=e^{-1}\mod \phi(N)$ (exists b/c $e\in\phi(N)^*$) - -So $(x_1^e)^d\equiv (x_2^e)^d\mod N$ - -So $x_1^{e\cdot d\mod \phi(N)}\equiv x_2^{e\cdot d\mod \phi(N)}\mod N$ (Euler's Theorem) - -$x_1\equiv x_2\mod N$ - -So it's one-to-one. - -QED - -Let $y\in \mathbb{Z}_N^*$, letting $x=y^d\mod N$, where $d\equiv e^{-1}\mod \phi(N)$ - -$x^e\equiv (y^d)^e \equiv y\mod n$ - -Proof: - -It's easy to sample from $I$: - -* pick $p,q\in \Pi_n$. $N=p\cdot q$ -* compute $\phi(N)=(p-1)(q-1)$ -* pick $e\gets \mathbb{Z}^*_N$. If $gcd(e,\phi(N))\neq 1$, pick again ($\mathbb{Z}_{\phi_(N)}^*$ has plenty of elements.) - -Easy to sample $\mathbb{\mathbb{Z}_N^*}$ (domain). - -Easy to compute $x^e\mod N$. - -Hard to invert: - -$$ -\begin{aligned} -&~~~~P[(N,e)\in I;x\gets \mathbb{Z}_N^*;y=x^e\mod N:f(\mathcal{A}((N,e),y))=y]\\ -&=P[(N,e)\in I;x\gets \mathbb{Z}_N^*;y=x^e\mod N:x\gets \mathcal{A}((N,e),y)]\\ -&=P[(N,e)\in I;y\gets \mathbb{Z}_N^*;y=x^e\mod N:x\gets \mathcal{A}((N,e),y),x^e\equiv y\mod N]\\ -\end{aligned} -$$ - -By RSA assumption - -The second equality follows because for any finite $D$ and bijection $f:D\to D$, sampling $y\in D$ directly is equivalent to sampling $x\gets D$, then computing $y=f(x)$. - -QED - -#### Theorem If inverting RSA is hard, then factoring is hard. - -$$ -\textup{ RSA assumption }\implies \textup{ Factoring assumption} -$$ - -If inverting RSA is hard, then factoring is hard. - -i.e If factoring is easy, then inverting RSA is easy. - -Proof: - -Suppose $\mathcal{A}$ is an adversary that breaks the factoring assumption, then - -$$ -P[p\gets \Pi_n;q\gets \Pi_n;N=p\cdot q;\mathcal{A}(N)=(p,q)]>\frac{1}{p(n)} -$$ - -infinitely often.for a polynomial $p$. - -Then we designing $B$ to invert RSA. - -Suppose - -$p,q\gets \Pi_n;N=p\cdot q;e\gets \mathbb{Z}_{\phi(N)}^*;x\gets \mathbb{Z}^n;y=x^e\mod N$ - -``` python -def B(N,e,y): - """ - Goal: find x - """ - p,q=A(N) - if n!=p*q: - return None - phiN=(p-1)*(q-1) - # find modular inverse of e \mod N - d=extended_euclidean_algorithm(e,phiN) - # returns (y**d)%N - x=fast_modular_exponent(y,d,N) - return x -``` - -So the probability of B succeeds is equal to A succeeds, which $>\frac{1}{p(n)}$ infinitely often, breaking RSA assumption. - -Remaining question: Can $x$ be found without factoring $N$? $y=x^e\mod N$ - -### One-way permutation (Definition 55.1) - -A collection function $\mathcal{F}=\{f_i:D_i\to R_i\}_{i\in I}$ is a one-way permutation if - -1. $\forall i,f_i$ is a permutation -2. $\mathcal{F}$ is a collection of one-way functions - -_basically, a one-way permutation is a collection of one-way functions that maps $\{0,1\}^n$ to $\{0,1\}^n$ in a bijection way._ - -### Trapdoor permutations - -Idea: $f:D\to R$ is a one-way permutation. - -$y\gets R$. - -* Finding $x$ such that $f(x)=y$ is hard. -* With some secret info about $f$, finding $x$ is easy. - -$\mathcal{F}=\{f_i:D_i\to R_i\}_{i\in I}$ - -1. $\forall i,f_i$ is a permutation -2. $(i,t)\gets Gen(1^n)$ efficient. ($i\in I$ paired with $t$), $t$ is the "trapdoor info" -3. $\forall i,D_i$ can be sampled efficiently. -4. $\forall i,\forall x,f_i(x)$ can be computed in polynomial time. -5. $P[(i,t)\gets Gen(1^n);y\gets R_i:f_i(\mathcal{A}(1^n,i,y))=y]<\epsilon(n)$ (note: $\mathcal{A}$ is not given $t$) -6. (trapdoor) There is a p.p.t. $B$ such that given $i,y,t$, B always finds x such that $f_i(x)=y$. $t$ is the "trapdoor info" - -#### Theorem RSA is a trapdoor - -RSA collection of trapdoor permutation with factorization $(p,q)$ of $N$, or $\phi(N)$, as trapdoor info $f$. +# Lecture 10 + +## Chapter 2: Computational Hardness + +### Discrete Log Assumption (Assumption 52.2) + +This is collection of one-way functions + +$$ +p\gets \tilde\Pi_n(\textup{ safe primes }), p=2q+1 +$$ + +$$ +a\gets \mathbb{Z}*_{p};g=a^2(\textup{ make sure }g\neq 1) +$$ + +$$ +f_{g,p}(x)=g^x\mod p +$$ + +$$ +f:\mathbb{Z}_q\to \mathbb{Z}^*_p +$$ + +#### Evidence for Discrete Log Assumption + +Best known algorithm to always solve discrete log mod p, $p\in \Pi_n$ + +$$ +O(2^{\sqrt{2}\sqrt{\log(n)}}) +$$ + +### RSA Assumption + +Let $e$ be the exponents + +$$ +P[p,q\gets \Pi_n;N\gets p\cdot q;e\gets \mathbb{Z}_{\phi(N)}^*;y\gets \mathbb{N}_n;x\gets \mathcal{A}(N,e,y);x^e=y\mod N]<\epsilon(n) +$$ + +#### Theorem 53.2 (RSA Algorithm) + +This is a collection of one-way functions + +$I=\{(N,e):N=p\cdot q,p,q\in \Pi_n \textup{ and } e\in \mathbb{Z}_{\phi(N)}^*\}$ + +$D_{(N,e)}=\mathbb{Z}_N^*$ + +$R_{(N,e)}=\mathbb{Z}_N^*$ + +$f_{(N,e)}(x)=x^e\mod N$ + +Example: + +On encryption side + +$p=5,q=11,N=5\times 11=55$, $\phi(N)=4*10=40$ + +pick $e\in \mathbb{Z}_{40}^*$. say $e=3$, and $f(x)=x^3\mod 55$ + +pick $y\in \mathbb{Z}_{55}^*$. say $y=17$. We have $(55,3,17)$ + +$x^{40}\equiv 1\mod 55$ + +$x^{41}\equiv x\mod 55$ + +$x^{40k+1}\equiv x \mod 55$ + +Since $x^a\equiv x^{a\mod 40}\mod 55$ (by corollary of Fermat's little Theorem: $a^x\mod N=a^{x\mod \Phi(N)}\mod N$ +s ) + +The problem is, what can we multiply by $3$ to get $1\mod \phi(N)=1\mod 40$. + +by computing the multiplicative inverse using extended Euclidean algorithm we have $3\cdot 27\equiv 1\mod 40$. + +$x^3\equiv 17\mod 55$ + +$x\equiv 17^{27}\mod 55$ + +On adversary side. + +they don't know $\phi(N)=40$ + +$$ +f(N,e):\mathbb{Z}_N^*\to \mathbb{Z}_N^* +$$ +is a bijection. + +Proof: Suppose $x_1^e\equiv x_2^e\mod n$ + +Then let $d=e^{-1}\mod \phi(N)$ (exists b/c $e\in\phi(N)^*$) + +So $(x_1^e)^d\equiv (x_2^e)^d\mod N$ + +So $x_1^{e\cdot d\mod \phi(N)}\equiv x_2^{e\cdot d\mod \phi(N)}\mod N$ (Euler's Theorem) + +$x_1\equiv x_2\mod N$ + +So it's one-to-one. + +QED + +Let $y\in \mathbb{Z}_N^*$, letting $x=y^d\mod N$, where $d\equiv e^{-1}\mod \phi(N)$ + +$x^e\equiv (y^d)^e \equiv y\mod n$ + +Proof: + +It's easy to sample from $I$: + +* pick $p,q\in \Pi_n$. $N=p\cdot q$ +* compute $\phi(N)=(p-1)(q-1)$ +* pick $e\gets \mathbb{Z}^*_N$. If $gcd(e,\phi(N))\neq 1$, pick again ($\mathbb{Z}_{\phi_(N)}^*$ has plenty of elements.) + +Easy to sample $\mathbb{\mathbb{Z}_N^*}$ (domain). + +Easy to compute $x^e\mod N$. + +Hard to invert: + +$$ +\begin{aligned} +&~~~~P[(N,e)\in I;x\gets \mathbb{Z}_N^*;y=x^e\mod N:f(\mathcal{A}((N,e),y))=y]\\ +&=P[(N,e)\in I;x\gets \mathbb{Z}_N^*;y=x^e\mod N:x\gets \mathcal{A}((N,e),y)]\\ +&=P[(N,e)\in I;y\gets \mathbb{Z}_N^*;y=x^e\mod N:x\gets \mathcal{A}((N,e),y),x^e\equiv y\mod N]\\ +\end{aligned} +$$ + +By RSA assumption + +The second equality follows because for any finite $D$ and bijection $f:D\to D$, sampling $y\in D$ directly is equivalent to sampling $x\gets D$, then computing $y=f(x)$. + +QED + +#### Theorem If inverting RSA is hard, then factoring is hard. + +$$ +\textup{ RSA assumption }\implies \textup{ Factoring assumption} +$$ + +If inverting RSA is hard, then factoring is hard. + +i.e If factoring is easy, then inverting RSA is easy. + +Proof: + +Suppose $\mathcal{A}$ is an adversary that breaks the factoring assumption, then + +$$ +P[p\gets \Pi_n;q\gets \Pi_n;N=p\cdot q;\mathcal{A}(N)=(p,q)]>\frac{1}{p(n)} +$$ + +infinitely often.for a polynomial $p$. + +Then we designing $B$ to invert RSA. + +Suppose + +$p,q\gets \Pi_n;N=p\cdot q;e\gets \mathbb{Z}_{\phi(N)}^*;x\gets \mathbb{Z}^n;y=x^e\mod N$ + +``` python +def B(N,e,y): + """ + Goal: find x + """ + p,q=A(N) + if n!=p*q: + return None + phiN=(p-1)*(q-1) + # find modular inverse of e \mod N + d=extended_euclidean_algorithm(e,phiN) + # returns (y**d)%N + x=fast_modular_exponent(y,d,N) + return x +``` + +So the probability of B succeeds is equal to A succeeds, which $>\frac{1}{p(n)}$ infinitely often, breaking RSA assumption. + +Remaining question: Can $x$ be found without factoring $N$? $y=x^e\mod N$ + +### One-way permutation (Definition 55.1) + +A collection function $\mathcal{F}=\{f_i:D_i\to R_i\}_{i\in I}$ is a one-way permutation if + +1. $\forall i,f_i$ is a permutation +2. $\mathcal{F}$ is a collection of one-way functions + +_basically, a one-way permutation is a collection of one-way functions that maps $\{0,1\}^n$ to $\{0,1\}^n$ in a bijection way._ + +### Trapdoor permutations + +Idea: $f:D\to R$ is a one-way permutation. + +$y\gets R$. + +* Finding $x$ such that $f(x)=y$ is hard. +* With some secret info about $f$, finding $x$ is easy. + +$\mathcal{F}=\{f_i:D_i\to R_i\}_{i\in I}$ + +1. $\forall i,f_i$ is a permutation +2. $(i,t)\gets Gen(1^n)$ efficient. ($i\in I$ paired with $t$), $t$ is the "trapdoor info" +3. $\forall i,D_i$ can be sampled efficiently. +4. $\forall i,\forall x,f_i(x)$ can be computed in polynomial time. +5. $P[(i,t)\gets Gen(1^n);y\gets R_i:f_i(\mathcal{A}(1^n,i,y))=y]<\epsilon(n)$ (note: $\mathcal{A}$ is not given $t$) +6. (trapdoor) There is a p.p.t. $B$ such that given $i,y,t$, B always finds x such that $f_i(x)=y$. $t$ is the "trapdoor info" + +#### Theorem RSA is a trapdoor + +RSA collection of trapdoor permutation with factorization $(p,q)$ of $N$, or $\phi(N)$, as trapdoor info $f$. diff --git a/pages/CSE442T/CSE442T_L11.md b/content/CSE442T/CSE442T_L11.md similarity index 96% rename from pages/CSE442T/CSE442T_L11.md rename to content/CSE442T/CSE442T_L11.md index cfae233..6045bea 100644 --- a/pages/CSE442T/CSE442T_L11.md +++ b/content/CSE442T/CSE442T_L11.md @@ -1,114 +1,114 @@ -# Lecture 11 - -Exam info posted tonight. - -## Chapter 3: Indistinguishability and pseudo-randomness - -### Pseudo-randomness - -Idea: **Efficiently** produce many bits - -which "appear" truly random. - -#### One-time pad - -$m\in\{0,1\}^n$ - -$Gen(1^n):k\gets \{0,1\}^N$ - -$Enc_k(m)=m\oplus k$ - -$Dec_k(c)=c\oplus k$ - -Advantage: Perfectly secret - -Disadvantage: Impractical - -The goal of pseudo-randomness is to make the algorithm, computationally secure, and practical. - -Let $\{X_n\}$ be a sequence of distributions over $\{0,1\}^{l(n)}$, where $l(n)$ is a polynomial of $n$. - -"Probability ensemble" - -Example: - -Let $U_n$ be the uniform distribution over $\{0,1\}^n$ - -For all $x\in \{0,1\}^n$ - -$P[x\gets U_n]=\frac{1}{2^n}$ - -For $1\leq i\leq n$, $P[x_i=1]=\frac{1}{2}$ - -For $1\leq i n$ (expansion) -3. $\{x\gets \{0,1\}^n:G(x)\}_n\approx \{u\gets \{0,1\}^{l(n)}\}$ - -#### Definition 78.3 (Hard-core bit (predicate) (HCB)) - -Hard-core bit (predicate) (HCB): $h:\{0,1\}^n\to \{0,1\}$ is a hard-core bit of $f:\{0,1\}^n\to \{0,1\}^*$ if for every adversary $A$, - -$$ -Pr[x\gets \{0,1\}^n;y=f(x);A(1^n,y)=h(x)]\leq \frac{1}{2}+\epsilon(n) -$$ - -Ideas: $f:\{0,1\}^n\to \{0,1\}^*$ is a one-way function. - -Given $y=f(x)$, it is hard to recover $x$. A cannot produce all of $x$ but can know some bits of $x$. - -$h(x)$ is just a yes/no question regarding $x$. - -Example: - -In RSA function, we pick $p,q\in \Pi^n$ as primes and $N=pq$. $e\gets \mathbb{Z}_N^*$ and $f(x)=x^e\mod N$. - -$h(x)=x_n$ is a HCB of $f$. Given RSA assumption. - -**h(x) is not necessarily one of the bits of $x=x_1x_2\cdots x_n$.** - -#### Theorem Any one-way function has a HCB. - -A HCB can be produced for any one-way function. - -Let $f:\{0,1\}^n\to \{0,1\}^*$ be a strong one-way function. - -Define $g:\{0,1\}^{2n}\to \{0,1\}^*$ as $g(x,r)=(f(x), r),x\in \{0,1\}^n,r\in \{0,1\}^n$. $g$ is a strong one-way function. (proved in homework) - -$$ -h(x,r)=\langle x,r\rangle=x_1r_1+ x_2r_2+\cdots + x_nr_n\mod 2 -$$ - -$\langle x,1^n\rangle=x_1+x_2+\cdots +x_n\mod 2$ - -$\langle x,0^{n-1}1\rangle=x_ n$ - -Ideas of proof: - -If A could reliably find $\langle x,1^n\rangle$, with $r$ being completely random, then it could find $x$ too often. - -### Pseudorandom Generator from HCB - -1. $G(x)=\{0,1\}^n\to \{0,1\}^{n+1}$ -2. $G(x)=\{0,1\}^n\to \{0,1\}^{l(n)}$ - -For (1), - -#### Theorem HCB generates PRG - -Let $f:\{0,1\}^n\to \{0,1\}^n$ be a one-way permutation (bijective) with a HCB $h$. Then $G(x)=f(x)|| h(x)$ is a PRG. - -Proof: - -Efficiently computable: $f$ is one-way so $h$ is efficiently computable. - -Expansion: $n\frac{1}{2}+\epsilon(n) -$$ - -This contradicts the HCB definition of $h$. - -### Construction of PRG - -$G'=\{0,1\}^n\to \{0,1\}^{l(n)}$ - -using PRG $G:\{0,1\}^n\to \{0,1\}^{n+1}$ - -Let $s\gets \{0,1\}^n$ be a random string. - -We proceed by the following construction: - -$G(s)=X_1||b_1$ - -$G(X_1)=X_2||b_2$ - -$G(X_2)=X_3||b_3$ - -$\cdots$ - -$G(X_{l(n)-1})=X_{l(n)}||b_{l(n)}$ - -$G'(s)=b_1b_2b_3\cdots b_{l(n)}$ - -We claim $G':\{0,1\}^n\to \{0,1\}^{l(n)}$ is a PRG. - -#### Corollary: Combining constructions - -$f:\{0,1\}^n\to \{0,1\}^n$ is a one-way permutation with a HCB $h: \{0,1\}^n\to \{0,1\}$. - -$G(s)=h(x)||h(f(x))||h(f^2(x))\cdots h(f^{l(n)-1}(x))$ is a PRG. Where $f^a(x)=f(f^{a-1}(x))$. - -Proof: - -$G'$ is a PRG: - -1. Efficiently computable: since we are computing $G'$ by applying $G$ multiple times (polynomial of $l(n)$ times). -2. Expansion: $n n$ (expansion) +3. $\{x\gets \{0,1\}^n:G(x)\}_n\approx \{u\gets \{0,1\}^{l(n)}\}$ + +#### Definition 78.3 (Hard-core bit (predicate) (HCB)) + +Hard-core bit (predicate) (HCB): $h:\{0,1\}^n\to \{0,1\}$ is a hard-core bit of $f:\{0,1\}^n\to \{0,1\}^*$ if for every adversary $A$, + +$$ +Pr[x\gets \{0,1\}^n;y=f(x);A(1^n,y)=h(x)]\leq \frac{1}{2}+\epsilon(n) +$$ + +Ideas: $f:\{0,1\}^n\to \{0,1\}^*$ is a one-way function. + +Given $y=f(x)$, it is hard to recover $x$. A cannot produce all of $x$ but can know some bits of $x$. + +$h(x)$ is just a yes/no question regarding $x$. + +Example: + +In RSA function, we pick $p,q\in \Pi^n$ as primes and $N=pq$. $e\gets \mathbb{Z}_N^*$ and $f(x)=x^e\mod N$. + +$h(x)=x_n$ is a HCB of $f$. Given RSA assumption. + +**h(x) is not necessarily one of the bits of $x=x_1x_2\cdots x_n$.** + +#### Theorem Any one-way function has a HCB. + +A HCB can be produced for any one-way function. + +Let $f:\{0,1\}^n\to \{0,1\}^*$ be a strong one-way function. + +Define $g:\{0,1\}^{2n}\to \{0,1\}^*$ as $g(x,r)=(f(x), r),x\in \{0,1\}^n,r\in \{0,1\}^n$. $g$ is a strong one-way function. (proved in homework) + +$$ +h(x,r)=\langle x,r\rangle=x_1r_1+ x_2r_2+\cdots + x_nr_n\mod 2 +$$ + +$\langle x,1^n\rangle=x_1+x_2+\cdots +x_n\mod 2$ + +$\langle x,0^{n-1}1\rangle=x_ n$ + +Ideas of proof: + +If A could reliably find $\langle x,1^n\rangle$, with $r$ being completely random, then it could find $x$ too often. + +### Pseudorandom Generator from HCB + +1. $G(x)=\{0,1\}^n\to \{0,1\}^{n+1}$ +2. $G(x)=\{0,1\}^n\to \{0,1\}^{l(n)}$ + +For (1), + +#### Theorem HCB generates PRG + +Let $f:\{0,1\}^n\to \{0,1\}^n$ be a one-way permutation (bijective) with a HCB $h$. Then $G(x)=f(x)|| h(x)$ is a PRG. + +Proof: + +Efficiently computable: $f$ is one-way so $h$ is efficiently computable. + +Expansion: $n\frac{1}{2}+\epsilon(n) +$$ + +This contradicts the HCB definition of $h$. + +### Construction of PRG + +$G'=\{0,1\}^n\to \{0,1\}^{l(n)}$ + +using PRG $G:\{0,1\}^n\to \{0,1\}^{n+1}$ + +Let $s\gets \{0,1\}^n$ be a random string. + +We proceed by the following construction: + +$G(s)=X_1||b_1$ + +$G(X_1)=X_2||b_2$ + +$G(X_2)=X_3||b_3$ + +$\cdots$ + +$G(X_{l(n)-1})=X_{l(n)}||b_{l(n)}$ + +$G'(s)=b_1b_2b_3\cdots b_{l(n)}$ + +We claim $G':\{0,1\}^n\to \{0,1\}^{l(n)}$ is a PRG. + +#### Corollary: Combining constructions + +$f:\{0,1\}^n\to \{0,1\}^n$ is a one-way permutation with a HCB $h: \{0,1\}^n\to \{0,1\}$. + +$G(s)=h(x)||h(f(x))||h(f^2(x))\cdots h(f^{l(n)-1}(x))$ is a PRG. Where $f^a(x)=f(f^{a-1}(x))$. + +Proof: + +$G'$ is a PRG: + +1. Efficiently computable: since we are computing $G'$ by applying $G$ multiple times (polynomial of $l(n)$ times). +2. Expansion: $n n$, and $G$ is pseudorandom - -$$ -\{G(U_n)\}\approx \{U_{l(n)}\} -$$ - -Back to the experiment we did long time ago: - -||Group 1|Group 2| -|---|---|---| -|$00000$ or $11111$|3|16| -|4 of 1's|42|56| -|balanced|too often|usual| -|consecutive repeats|0|4| - -So Group 1 is human, Group 2 is computer. - -## Chapter 3: Indistinguishability and Pseudorandomness - -### Computationally secure encryption - -Recall with perfect security, - -$$ -P[k\gets Gen(1^n):Enc_k(m_1)=c] = P[k\gets Gen(1^n):Enc_k(m_2)=c] -$$ - -for all $m_1,m_2\in M$ and $c\in C$. - -$(Gen,Enc,Dec)$ is **single message secure** if $\forall n.u.p.p.t \mathcal{D}$ and for all $n\in \mathbb{N}$, $\forall m_1,m_2\gets \{0,1\}^n \in M^n$, $\mathcal{D}$ distinguishes $Enc_k(m_1)$ and $Enc_k(m_2)$ with at most negligble probability. - -$$ -P[k\gets Gen(1^n):\mathcal{D}(Enc_k(m_1),Enc_k(m_2))=1] \leq \epsilon(n) -$$ - -By the prediction lemma, ($\mathcal{A}$ is a ppt, you can also name it as $\mathcal{D}$) - -$$ -P[b\gets \{0,1\}:k\gets Gen(1^n):\mathcal{A}(Enc_k(m_b)) = b] \leq \frac{1}{2} + \frac{\epsilon(n)}{2} -$$ - -and the above equation is $\frac{1}{2}$ for perfect secrecy. - -### Construction of single message secure cryptosystem - -cryptosystem with shorter keys. Mimic OTP(one time pad) with shorter keys with pseudorandom randomness. - -$K=\{0,1\}^n$, $\mathcal{M}=\{0,1\}^{l(n)}$, $G:K \to \mathcal{M}$ is a PRG. - -$Gen(1^n)$: $k\gets \{0,1\}^n$; output $k$. - -$Enc_k(m)$: $r\gets \{0,1\}^{l(n)}$; output $G(k)\oplus m$. - -$Dec_k(c)$: output $G(k)\oplus c$. - -Proof of security: - -Let $m_0,m_1\in \mathcal{M}$ be two messages, and $\mathcal{D}$ is a n.u.p.p.t distinguisher. - -Suppose $\{K\gets Gen(1^n):Enc_k(m_i)\}$ is distinguished for $i=0,1$ by $\mathcal{D}$ and by $\mu(n)\geq\frac{1}{poly(n)}$. - -Strategy: Move to OTP, then flip message. - -$$ -H_0(Enc_k(m_0)) = \{k\gets \{0,1\}^n: m_0\oplus G(k)\} -$$ -$$ -H_1(OTP(m_1)) = \{u\gets U_{l(n)}: m_o\oplus u\} -$$ -$$ -H_2(OTP(m_1)) = \{u\gets U_{l(n)}: m_1\oplus u\} -$$ -$$ -H_3(Enc_k(m_0)) = \{k\gets \{0,1\}^n: m_1\oplus G(k)\} -$$ - -By hybrid argument, 2 neighboring messages are indistinguishable. - -However, $H_0$ and $H_1$ are indistinguishable since $G(U_n)$ and $U_{l(n)}$ are indistinguishable. - -$H_1$ and $H_2$ are indistinguishable by perfect secrecy of OTP. - -$H_2$ and $H_3$ are indistinguishable since $G(U_n)$ and $U_{l(n)}$ are indistinguishable. - -Which leads to a contradiction. - -### Multi-message secure encryption - -$(Gen,Enc,Dec)$ is multi-message secure if $\forall n.u.p.p.t \mathcal{D}$ and for all $n\in \mathbb{N}$, and $q(n)\in poly(n)$. - -$$ -\overline{m}=(m_1,\dots,m_{q(n)}) -$$ -$$ -\overline{m}'=(m_1',\dots,m_{q(n)}') -$$ - -are list of $q(n)$ messages in $\{0,1\}^n$. - -$\mathcal{D}$ distinguishes $Enc_k(\overline{m})$ and $Enc_k(\overline{m}')$ with at most negligble probability. - -$$ -P[k\gets Gen(1^n):\mathcal{D}(Enc_k(\overline{m}),Enc_k(\overline{m}'))=1] \leq \frac{1}{2} + \epsilon(n) -$$ - -**THIS IS NOT MULTI-MESSAGE SECURE.** - -We can take $\overline{m}=(0^n,0^n)\to (G(k),G(k))$ and $\overline{m}'=(0^n,1^n)\to (G(k),G(k)+1^n)$ the distinguisher can easily distinguish if some message was sent twice. - -What we need is that the distinguisher cannot distinguish if some message was sent twice. To achieve multi-message security, we need our encryption function to use randomness (or change states) for each message, otherwise $Enc_k(0^n)$ will return the same on consecutive messages. - -Our fix is, if we can agree on a random function $F:\{0,1\}^n\to \{0,1\}^n$ satisfied that: for each input $x\in\{0,1\}^n$, $F(x)$ is chosen uniformly at random. - -$Gen(1^n):$ Choose random function $F:\{0,1\}^n\to \{0,1\}^n$. - -$Enc_F(m):$ let $r\gets U_n$; output $(r,F(r)\oplus m)$. - -$Dec_F(m):$ Given $(r,c)$, output $m=F(r)\oplus c$. - -Ideas: Adversary sees $r$ but has no Ideas about $F(r)$. (we choose all outputs at random) - -If we could do this, this is MMS (multi-message secure). - -Proof: - -Suppose $m_1,m_2,\dots,m_{q(n)}$, $m_1',\dots,m_{q(n)}'$ are sent to the encryption oracle. - -Suppose the encryption are distinguished by $\mathcal{D}$ with probability $\frac{1}{2}+\epsilon(n)$. - -Strategy: move to OTP with hybrid argument. - -Suppose we choose a random function - -$$ -H_0:\{F\gets RF_n:((r_1,m_1\oplus F(r_1)),(r_2,m_2\oplus F(r_2)),\dots,(r_{q(n)},m_{q(n)}\oplus F(r_{q(n)})))\} -$$ - -and - -$$ -H_1:\{OTP:(r_1,m_1\oplus u_1),(r_2,m_2\oplus u_2),\dots,(r_{q(n)},m_{q(n)}\oplus u_{q(n)})\} -$$ - -$r_i,u_i\in U_n$. - -By hybrid argument, $H_0$ and $H_1$ are indistinguishable if $r_1,\dots,r_{q(n)}$ are different, these are the same. - -$F(r_1),\dots,F(r_{q(n)})$ are chosen uniformly and independently at random. - -only possible problem is $r_i=r_j$ for some $i\neq j$, and $P[r_i=r_j]=\frac{1}{2^n}$. - -And the probability that at least one pair are equal - -$$ -P[\text{at least one pair are equal}] =P[\bigcup_{i\neq j}\{r_i=r_j\}] \leq \sum_{i\neq j}P[r_i=r_j]=\binom{n}{2}\frac{1}{2^n} < \frac{n^2}{2^{n+1}} -$$ - -which is negligible. - -Unfortunately, we cannot do this in practice. - -How many random functions are there? - -The length of description of $F$ is $n 2^n$. - -For each $x\in \{0,1\}^n$, there are $2^n$ possible values for $F(x)$. - -So the total number of random functions is $(2^n)^{2^n}=2^{n2^n}$. - - +# Lecture 14 + +## Recap + +$\exists$ one-way functions $\implies$ $\exists$ PRG expand by any polynomial amount + +$\exists G:\{0,1\}^n \to \{0,1\}^{l(n)}$ s.t. $G$ is efficiently computable, $l(n) > n$, and $G$ is pseudorandom + +$$ +\{G(U_n)\}\approx \{U_{l(n)}\} +$$ + +Back to the experiment we did long time ago: + +||Group 1|Group 2| +|---|---|---| +|$00000$ or $11111$|3|16| +|4 of 1's|42|56| +|balanced|too often|usual| +|consecutive repeats|0|4| + +So Group 1 is human, Group 2 is computer. + +## Chapter 3: Indistinguishability and Pseudorandomness + +### Computationally secure encryption + +Recall with perfect security, + +$$ +P[k\gets Gen(1^n):Enc_k(m_1)=c] = P[k\gets Gen(1^n):Enc_k(m_2)=c] +$$ + +for all $m_1,m_2\in M$ and $c\in C$. + +$(Gen,Enc,Dec)$ is **single message secure** if $\forall n.u.p.p.t \mathcal{D}$ and for all $n\in \mathbb{N}$, $\forall m_1,m_2\gets \{0,1\}^n \in M^n$, $\mathcal{D}$ distinguishes $Enc_k(m_1)$ and $Enc_k(m_2)$ with at most negligble probability. + +$$ +P[k\gets Gen(1^n):\mathcal{D}(Enc_k(m_1),Enc_k(m_2))=1] \leq \epsilon(n) +$$ + +By the prediction lemma, ($\mathcal{A}$ is a ppt, you can also name it as $\mathcal{D}$) + +$$ +P[b\gets \{0,1\}:k\gets Gen(1^n):\mathcal{A}(Enc_k(m_b)) = b] \leq \frac{1}{2} + \frac{\epsilon(n)}{2} +$$ + +and the above equation is $\frac{1}{2}$ for perfect secrecy. + +### Construction of single message secure cryptosystem + +cryptosystem with shorter keys. Mimic OTP(one time pad) with shorter keys with pseudorandom randomness. + +$K=\{0,1\}^n$, $\mathcal{M}=\{0,1\}^{l(n)}$, $G:K \to \mathcal{M}$ is a PRG. + +$Gen(1^n)$: $k\gets \{0,1\}^n$; output $k$. + +$Enc_k(m)$: $r\gets \{0,1\}^{l(n)}$; output $G(k)\oplus m$. + +$Dec_k(c)$: output $G(k)\oplus c$. + +Proof of security: + +Let $m_0,m_1\in \mathcal{M}$ be two messages, and $\mathcal{D}$ is a n.u.p.p.t distinguisher. + +Suppose $\{K\gets Gen(1^n):Enc_k(m_i)\}$ is distinguished for $i=0,1$ by $\mathcal{D}$ and by $\mu(n)\geq\frac{1}{poly(n)}$. + +Strategy: Move to OTP, then flip message. + +$$ +H_0(Enc_k(m_0)) = \{k\gets \{0,1\}^n: m_0\oplus G(k)\} +$$ +$$ +H_1(OTP(m_1)) = \{u\gets U_{l(n)}: m_o\oplus u\} +$$ +$$ +H_2(OTP(m_1)) = \{u\gets U_{l(n)}: m_1\oplus u\} +$$ +$$ +H_3(Enc_k(m_0)) = \{k\gets \{0,1\}^n: m_1\oplus G(k)\} +$$ + +By hybrid argument, 2 neighboring messages are indistinguishable. + +However, $H_0$ and $H_1$ are indistinguishable since $G(U_n)$ and $U_{l(n)}$ are indistinguishable. + +$H_1$ and $H_2$ are indistinguishable by perfect secrecy of OTP. + +$H_2$ and $H_3$ are indistinguishable since $G(U_n)$ and $U_{l(n)}$ are indistinguishable. + +Which leads to a contradiction. + +### Multi-message secure encryption + +$(Gen,Enc,Dec)$ is multi-message secure if $\forall n.u.p.p.t \mathcal{D}$ and for all $n\in \mathbb{N}$, and $q(n)\in poly(n)$. + +$$ +\overline{m}=(m_1,\dots,m_{q(n)}) +$$ +$$ +\overline{m}'=(m_1',\dots,m_{q(n)}') +$$ + +are list of $q(n)$ messages in $\{0,1\}^n$. + +$\mathcal{D}$ distinguishes $Enc_k(\overline{m})$ and $Enc_k(\overline{m}')$ with at most negligble probability. + +$$ +P[k\gets Gen(1^n):\mathcal{D}(Enc_k(\overline{m}),Enc_k(\overline{m}'))=1] \leq \frac{1}{2} + \epsilon(n) +$$ + +**THIS IS NOT MULTI-MESSAGE SECURE.** + +We can take $\overline{m}=(0^n,0^n)\to (G(k),G(k))$ and $\overline{m}'=(0^n,1^n)\to (G(k),G(k)+1^n)$ the distinguisher can easily distinguish if some message was sent twice. + +What we need is that the distinguisher cannot distinguish if some message was sent twice. To achieve multi-message security, we need our encryption function to use randomness (or change states) for each message, otherwise $Enc_k(0^n)$ will return the same on consecutive messages. + +Our fix is, if we can agree on a random function $F:\{0,1\}^n\to \{0,1\}^n$ satisfied that: for each input $x\in\{0,1\}^n$, $F(x)$ is chosen uniformly at random. + +$Gen(1^n):$ Choose random function $F:\{0,1\}^n\to \{0,1\}^n$. + +$Enc_F(m):$ let $r\gets U_n$; output $(r,F(r)\oplus m)$. + +$Dec_F(m):$ Given $(r,c)$, output $m=F(r)\oplus c$. + +Ideas: Adversary sees $r$ but has no Ideas about $F(r)$. (we choose all outputs at random) + +If we could do this, this is MMS (multi-message secure). + +Proof: + +Suppose $m_1,m_2,\dots,m_{q(n)}$, $m_1',\dots,m_{q(n)}'$ are sent to the encryption oracle. + +Suppose the encryption are distinguished by $\mathcal{D}$ with probability $\frac{1}{2}+\epsilon(n)$. + +Strategy: move to OTP with hybrid argument. + +Suppose we choose a random function + +$$ +H_0:\{F\gets RF_n:((r_1,m_1\oplus F(r_1)),(r_2,m_2\oplus F(r_2)),\dots,(r_{q(n)},m_{q(n)}\oplus F(r_{q(n)})))\} +$$ + +and + +$$ +H_1:\{OTP:(r_1,m_1\oplus u_1),(r_2,m_2\oplus u_2),\dots,(r_{q(n)},m_{q(n)}\oplus u_{q(n)})\} +$$ + +$r_i,u_i\in U_n$. + +By hybrid argument, $H_0$ and $H_1$ are indistinguishable if $r_1,\dots,r_{q(n)}$ are different, these are the same. + +$F(r_1),\dots,F(r_{q(n)})$ are chosen uniformly and independently at random. + +only possible problem is $r_i=r_j$ for some $i\neq j$, and $P[r_i=r_j]=\frac{1}{2^n}$. + +And the probability that at least one pair are equal + +$$ +P[\text{at least one pair are equal}] =P[\bigcup_{i\neq j}\{r_i=r_j\}] \leq \sum_{i\neq j}P[r_i=r_j]=\binom{n}{2}\frac{1}{2^n} < \frac{n^2}{2^{n+1}} +$$ + +which is negligible. + +Unfortunately, we cannot do this in practice. + +How many random functions are there? + +The length of description of $F$ is $n 2^n$. + +For each $x\in \{0,1\}^n$, there are $2^n$ possible values for $F(x)$. + +So the total number of random functions is $(2^n)^{2^n}=2^{n2^n}$. + + diff --git a/pages/CSE442T/CSE442T_L15.md b/content/CSE442T/CSE442T_L15.md similarity index 95% rename from pages/CSE442T/CSE442T_L15.md rename to content/CSE442T/CSE442T_L15.md index 9e57d2f..d5673bc 100644 --- a/pages/CSE442T/CSE442T_L15.md +++ b/content/CSE442T/CSE442T_L15.md @@ -1,189 +1,189 @@ -# Lecture 15 - -## Chapter 3: Indistinguishability and Pseudorandomness - -### Random Function - -$F:\{0,1\}^n\to \{0,1\}^n$ - -For each $x\in \{0,1\}^n$, there are $2^n$ possible values for $F(x)$. - -pick $y=F(x)\gets \{0,1\}^n$ independently at random. ($n$ bits) - -This generates $n\cdot 2^n$ random bits to specify $F$. - -### Equivalent description of $F$ - -```python -# initialized empty list L -L=collections.defaultdict(int) -# initialize n bits constant -n=10 -def F(x): - """ simulation of random function - param: - x: n bits - return: - y: n bits - """ - if L[x] is not None: - return L[x] - else: - # y is a random n-bit string - y=random.randbits(n) - L[x]=y - return y -``` - -However, this is not a good random function since two communicator may not agree on the same $F$. - -### Pseudorandom Function - -$f:\{0,1\}^n\to \{0,1\}^n$ - -#### Oracle Access (for function $g$) - -$O_g$ is a p.p.t. that given $x\in \{0,1\}^n$ outputs $g(x)$. - -The distinguisher $D$ is given oracle access to $O_g$ and outputs $1$ if $g$ is random and $0$ otherwise. It can make polynomially many queries. - -### Oracle indistinguishability - -$\{F_n\}$ and $\{G_n\}$ are sequence of distribution on functions - -$$ -f:\{0,1\}^{l_1(n)}\to \{0,1\}^{l_2(n)} -$$ - -that are computationally indistinguishable - -$$ -\{f_n\}\sim \{g_n\} -$$ - -if for all p.p.t. $D$ (with oracle access to $F_n$ and $G_n$), - -$$ -\left|P[f\gets F_n:D^f(1^n)=1]-P[g\gets G_n:D^g(1^n)=1]\right|< \epsilon(n) -$$ - -where $\epsilon(n)$ is negligible. - -Under this property, we still have: - -- Closure properties. under efficient procedures. -- Prediction lemma. -- Hybrid lemma. - -### Pseudorandom Function Family - -Definition: $\{f_s:\{0,1\}^\{0.1\}^{|S|}\to \{0,1\}^P$ $t_0s\in \{0,1\}^n\}$ is a pseudorandom function family if $\{f_s\}_{s\in \{0,1\}^n}$ are oracle indistinguishable. - -- It is easy to compute for every $x\in \{0,1\}^{|S|}$. -- $\{s \gets\{0,1\}^n\}_n\approx \{F\gets RF_n,F\}$ is indistinguishable from the uniform distribution over $\{0,1\}^P$. - - $R$ is truly random function. - -Example: - -For $s\in \{0,1\}^n$, define $f_s:\overline{x}\mapsto s\cdot \overline{s}$. - -$\mathcal{D}$ gives oracle access to $g(0^n)=\overline{y_0}$, $g(1^n)=\overline{y_1}$. If $\overline{y_0}+\overline{y_1}=1^n$, then $\mathcal{D}$ outputs $1$ otherwise $0$. - -```python -def O_g(x): - pass - -def D(): - # bit_stream(0,n) is a n-bit string of 0s - y0=O_g(bit_stream(0,n)) - y1=O_g(bit_stream(1,n)) - if y0+y1==bit_stream(1,n): - return 1 - else: - return 0 -``` - -If $g=f_s$, then $D$ returns $\overline{s}+\overline{s}+1^n =1^n$. - -$$ -P[f_s\gets D^{f_s}(1^n)=1]=1 -$$ - -$$ -P[F\gets RF^n,D^F(1^n)=1]=\frac{1}{2^n} -$$ - -#### Theorem PRG exists then PRF family exists. - -Proof: - -Let $g:\{0,1\}^n\to \{0,1\}^{2n}$ be a PRG. - -$$ -g(\overline{x})=[g_0(\overline{x})] [g_1(\overline{x})] -$$ - -Then we choose a random $s\in \{0,1\}^n$ (initial seed) and define $\overline{x}\gets \{0,1\}^n$, $\overline{x}=x_1\cdots x_n$. - -$$ -f_s(\overline{x})=f_s(x_1\cdots x_n)=g_{x_n}(\dots (g_{x_2}(g_{x_1}(s)))) -$$ - -```python -s=random.randbits(n) - -#???? - -def g(x): - if x[0]==0: - return g(f_s(x[1:])) - else: - return g(f_s(x[1:])) - -def f_s(x): - return g(x) - -``` - -Suppose $g:\{0,1\}^3\to \{0,1\}^6$ is a PRG. - -| $x$ | $f_s(x)$ | -| --- | -------- | -| 000 | 110011 | -| 001 | 010010 | -| 010 | 001001 | -| 011 | 000110 | -| 100 | 100000 | -| 101 | 110110 | -| 110 | 000111 | -| 111 | 001110 | - -Suppose the initial seed is $011$, then the constructed function tree goes as follows: - -Example: - -$$ -\begin{aligned} -f_s(110)&=g_0(g_1(g_1(s)))\\ -&=g_0(g_1(110))\\ -&=g_0(111)\\ -&=001 -\end{aligned} -$$ - -$$ -\begin{aligned} -f_s(010)&=g_0(g_1(g_0(s)))\\ -&=g_0(g_1(000))\\ -&=g_0(001)\\ -&=010 -\end{aligned} -$$ - -Assume that $D$ distinguishes $f_s$ and $F\gets RF_n$ with non-negligible probability. - -By hybrid argument, there exists a hybrid $H_i$ such that $D$ distinguishes $H_i$ and $H_{i+1}$ with non-negligible probability. - -For $H_0$, - -QED +# Lecture 15 + +## Chapter 3: Indistinguishability and Pseudorandomness + +### Random Function + +$F:\{0,1\}^n\to \{0,1\}^n$ + +For each $x\in \{0,1\}^n$, there are $2^n$ possible values for $F(x)$. + +pick $y=F(x)\gets \{0,1\}^n$ independently at random. ($n$ bits) + +This generates $n\cdot 2^n$ random bits to specify $F$. + +### Equivalent description of $F$ + +```python +# initialized empty list L +L=collections.defaultdict(int) +# initialize n bits constant +n=10 +def F(x): + """ simulation of random function + param: + x: n bits + return: + y: n bits + """ + if L[x] is not None: + return L[x] + else: + # y is a random n-bit string + y=random.randbits(n) + L[x]=y + return y +``` + +However, this is not a good random function since two communicator may not agree on the same $F$. + +### Pseudorandom Function + +$f:\{0,1\}^n\to \{0,1\}^n$ + +#### Oracle Access (for function $g$) + +$O_g$ is a p.p.t. that given $x\in \{0,1\}^n$ outputs $g(x)$. + +The distinguisher $D$ is given oracle access to $O_g$ and outputs $1$ if $g$ is random and $0$ otherwise. It can make polynomially many queries. + +### Oracle indistinguishability + +$\{F_n\}$ and $\{G_n\}$ are sequence of distribution on functions + +$$ +f:\{0,1\}^{l_1(n)}\to \{0,1\}^{l_2(n)} +$$ + +that are computationally indistinguishable + +$$ +\{f_n\}\sim \{g_n\} +$$ + +if for all p.p.t. $D$ (with oracle access to $F_n$ and $G_n$), + +$$ +\left|P[f\gets F_n:D^f(1^n)=1]-P[g\gets G_n:D^g(1^n)=1]\right|< \epsilon(n) +$$ + +where $\epsilon(n)$ is negligible. + +Under this property, we still have: + +- Closure properties. under efficient procedures. +- Prediction lemma. +- Hybrid lemma. + +### Pseudorandom Function Family + +Definition: $\{f_s:\{0,1\}^\{0.1\}^{|S|}\to \{0,1\}^P$ $t_0s\in \{0,1\}^n\}$ is a pseudorandom function family if $\{f_s\}_{s\in \{0,1\}^n}$ are oracle indistinguishable. + +- It is easy to compute for every $x\in \{0,1\}^{|S|}$. +- $\{s \gets\{0,1\}^n\}_n\approx \{F\gets RF_n,F\}$ is indistinguishable from the uniform distribution over $\{0,1\}^P$. + - $R$ is truly random function. + +Example: + +For $s\in \{0,1\}^n$, define $f_s:\overline{x}\mapsto s\cdot \overline{s}$. + +$\mathcal{D}$ gives oracle access to $g(0^n)=\overline{y_0}$, $g(1^n)=\overline{y_1}$. If $\overline{y_0}+\overline{y_1}=1^n$, then $\mathcal{D}$ outputs $1$ otherwise $0$. + +```python +def O_g(x): + pass + +def D(): + # bit_stream(0,n) is a n-bit string of 0s + y0=O_g(bit_stream(0,n)) + y1=O_g(bit_stream(1,n)) + if y0+y1==bit_stream(1,n): + return 1 + else: + return 0 +``` + +If $g=f_s$, then $D$ returns $\overline{s}+\overline{s}+1^n =1^n$. + +$$ +P[f_s\gets D^{f_s}(1^n)=1]=1 +$$ + +$$ +P[F\gets RF^n,D^F(1^n)=1]=\frac{1}{2^n} +$$ + +#### Theorem PRG exists then PRF family exists. + +Proof: + +Let $g:\{0,1\}^n\to \{0,1\}^{2n}$ be a PRG. + +$$ +g(\overline{x})=[g_0(\overline{x})] [g_1(\overline{x})] +$$ + +Then we choose a random $s\in \{0,1\}^n$ (initial seed) and define $\overline{x}\gets \{0,1\}^n$, $\overline{x}=x_1\cdots x_n$. + +$$ +f_s(\overline{x})=f_s(x_1\cdots x_n)=g_{x_n}(\dots (g_{x_2}(g_{x_1}(s)))) +$$ + +```python +s=random.randbits(n) + +#???? + +def g(x): + if x[0]==0: + return g(f_s(x[1:])) + else: + return g(f_s(x[1:])) + +def f_s(x): + return g(x) + +``` + +Suppose $g:\{0,1\}^3\to \{0,1\}^6$ is a PRG. + +| $x$ | $f_s(x)$ | +| --- | -------- | +| 000 | 110011 | +| 001 | 010010 | +| 010 | 001001 | +| 011 | 000110 | +| 100 | 100000 | +| 101 | 110110 | +| 110 | 000111 | +| 111 | 001110 | + +Suppose the initial seed is $011$, then the constructed function tree goes as follows: + +Example: + +$$ +\begin{aligned} +f_s(110)&=g_0(g_1(g_1(s)))\\ +&=g_0(g_1(110))\\ +&=g_0(111)\\ +&=001 +\end{aligned} +$$ + +$$ +\begin{aligned} +f_s(010)&=g_0(g_1(g_0(s)))\\ +&=g_0(g_1(000))\\ +&=g_0(001)\\ +&=010 +\end{aligned} +$$ + +Assume that $D$ distinguishes $f_s$ and $F\gets RF_n$ with non-negligible probability. + +By hybrid argument, there exists a hybrid $H_i$ such that $D$ distinguishes $H_i$ and $H_{i+1}$ with non-negligible probability. + +For $H_0$, + +QED diff --git a/pages/CSE442T/CSE442T_L16.md b/content/CSE442T/CSE442T_L16.md similarity index 95% rename from pages/CSE442T/CSE442T_L16.md rename to content/CSE442T/CSE442T_L16.md index 6502bf9..3930b30 100644 --- a/pages/CSE442T/CSE442T_L16.md +++ b/content/CSE442T/CSE442T_L16.md @@ -1,134 +1,134 @@ -# Lecture 16 - -## Chapter 3: Indistinguishability and Pseudorandomness - -PRG exists $\implies$ Pseudorandom function family exists. - -### Multi-message secure encryption - -$Gen(1^n):$ Output $f_i:\{0,1\}^n\to \{0,1\}^n$ from PRF family - -$Enc_i(m):$ Random $r\gets \{0,1\}^n$ -Ouput $(r,m\oplus f_i(r))$ - -$Dec_i(r,c):$ Output $c\oplus f_i(r)$ - -Proof of security: - -Suppose $D$ distinguishes, for infinitly many $n$. - -The encryption of $a$ pair of lists - -(1) $\{i\gets Gen(1^n):(r_1,m_1\oplus f_i(r_1)),(r_2,m_2\oplus f_i(r_2)),(r_3,m_3\oplus f_i(r_3)),\ldots,(r_q,m_q\oplus f_i(r_q)), \}$ - -(2) $\{F\gets RF_n: (r_1,m_1\oplus F(r_1))\ldots\}$ - -(3) One-time pad $\{(r_1,m_1\oplus s_1)\}$ - -(4) One-time pad $\{(r_1,m_1'\oplus s_1)\}$ - -If (1) (2) distinguished, - -$(r_1,f_i(r_1)),\ldots,(r_q,f_i(r_q))$ is distinguished from - -$(r_1,F(r_1)),\ldots, (r_q,F(r_q))$ - -So $D$ distinguishing output of $r_1,\ldots, r_q$ of PRF from the RF, this contradicts with definition of PRF. - -QED - -Noe we have - -(RSA assumption and Discrete log assumption for one-way function exists.) - -One-way function exists $\implies$ - -Pseudo random generator exists $\implies$ - -Pseudo random function familiy exists $\implies$ - -Mult-message secure encryption exists. - -### Public key cryptography - -1970s. - -The goal was to agree/share a key without meeting in advance - -#### Diffie-Helmann Key exchange - -A and B create a secret key together without meeting. - -Rely on discrete log assumption. - -They pulicly agree on modulus $p$ and generator $g$. - -Alice picks random exponent $a$ and computes $g^a\mod p$ - -Bob picks random exponent $b$ and computes $g^b\mod p$ - -and they send result to each other. - -And Alice do $(g^b)^a$ where Bob do $(g^a)^b$. - -#### Diffie-Helmann assumption - -With $g^a,g^b$ no one can compute $g^{ab}$. - -#### Public key encryption scheme - -Ideas: The recipient Bob distributes opened Bob-locks - -- Once closed, only Bob can open it. - -Public-key encryption scheme: - -1. $Gen(1^n):$ Outputs $(pk,sk)$ -2. $Enc_{pk}(m):$ Efficient for all $m,pk$ -3. $Dec_{sk}(c):$ Efficient for all $c,sk$ -4. $P[(pk,sk)\gets Gen(1^n):Dec_{sk}(Enc_{pk}(m))=m]=1$ - -Let $A, E$ knows $pk$ not $sk$ and $B$ knows $pk,sk$. - -Adversary can now encrypt any message $m$ with the public key. - -- Perfect secrecy impossible -- Randomness necessary - -#### Security of public key - -$\forall n.u.p.p.t D,\exists \epsilon(n)$ such that $\forall n,m_0,m_1\in \{0,1\}^n$ - -$$ -\{(pk,sk)\gets Gen(1^n):(pk,Enc_{pk}(m_0))\} \{(pk,sk)\gets Gen(1^n):(pk,Enc_{pk}(m_1))\} -$$ - -are distinguished by at most $\epsilon (n)$ - -This "single" message security implies multi-message security! - -_Left as exercise_ - -We will achieve security in sending a single bit $0,1$ - -Time for trapdoor permutation. (EX. RSA) - -#### Encryption Scheme via Trapdoor Permutation - -Given family of trapdoor permutation $\{f_i\}$ with hardcore bit $h(i)$ - -$Gen(1^n):(f_i,f_i^{-1})$, where $f_i^{-1}$ uses trapdoor permutation of $t$ - -$Output ((f_i,h_i),f_i^{-1})$ - -$m=0$ or $1$. - -$Enc_{pk}(m):r\gets\{0,1\}^n$ - -$Output (f_i(r),h_i(r)+m)$ - -$Dec_{sk}(c_1,c_2)$ - -$r=f_i^{-1}(c_1)$ - +# Lecture 16 + +## Chapter 3: Indistinguishability and Pseudorandomness + +PRG exists $\implies$ Pseudorandom function family exists. + +### Multi-message secure encryption + +$Gen(1^n):$ Output $f_i:\{0,1\}^n\to \{0,1\}^n$ from PRF family + +$Enc_i(m):$ Random $r\gets \{0,1\}^n$ +Ouput $(r,m\oplus f_i(r))$ + +$Dec_i(r,c):$ Output $c\oplus f_i(r)$ + +Proof of security: + +Suppose $D$ distinguishes, for infinitly many $n$. + +The encryption of $a$ pair of lists + +(1) $\{i\gets Gen(1^n):(r_1,m_1\oplus f_i(r_1)),(r_2,m_2\oplus f_i(r_2)),(r_3,m_3\oplus f_i(r_3)),\ldots,(r_q,m_q\oplus f_i(r_q)), \}$ + +(2) $\{F\gets RF_n: (r_1,m_1\oplus F(r_1))\ldots\}$ + +(3) One-time pad $\{(r_1,m_1\oplus s_1)\}$ + +(4) One-time pad $\{(r_1,m_1'\oplus s_1)\}$ + +If (1) (2) distinguished, + +$(r_1,f_i(r_1)),\ldots,(r_q,f_i(r_q))$ is distinguished from + +$(r_1,F(r_1)),\ldots, (r_q,F(r_q))$ + +So $D$ distinguishing output of $r_1,\ldots, r_q$ of PRF from the RF, this contradicts with definition of PRF. + +QED + +Noe we have + +(RSA assumption and Discrete log assumption for one-way function exists.) + +One-way function exists $\implies$ + +Pseudo random generator exists $\implies$ + +Pseudo random function familiy exists $\implies$ + +Mult-message secure encryption exists. + +### Public key cryptography + +1970s. + +The goal was to agree/share a key without meeting in advance + +#### Diffie-Helmann Key exchange + +A and B create a secret key together without meeting. + +Rely on discrete log assumption. + +They pulicly agree on modulus $p$ and generator $g$. + +Alice picks random exponent $a$ and computes $g^a\mod p$ + +Bob picks random exponent $b$ and computes $g^b\mod p$ + +and they send result to each other. + +And Alice do $(g^b)^a$ where Bob do $(g^a)^b$. + +#### Diffie-Helmann assumption + +With $g^a,g^b$ no one can compute $g^{ab}$. + +#### Public key encryption scheme + +Ideas: The recipient Bob distributes opened Bob-locks + +- Once closed, only Bob can open it. + +Public-key encryption scheme: + +1. $Gen(1^n):$ Outputs $(pk,sk)$ +2. $Enc_{pk}(m):$ Efficient for all $m,pk$ +3. $Dec_{sk}(c):$ Efficient for all $c,sk$ +4. $P[(pk,sk)\gets Gen(1^n):Dec_{sk}(Enc_{pk}(m))=m]=1$ + +Let $A, E$ knows $pk$ not $sk$ and $B$ knows $pk,sk$. + +Adversary can now encrypt any message $m$ with the public key. + +- Perfect secrecy impossible +- Randomness necessary + +#### Security of public key + +$\forall n.u.p.p.t D,\exists \epsilon(n)$ such that $\forall n,m_0,m_1\in \{0,1\}^n$ + +$$ +\{(pk,sk)\gets Gen(1^n):(pk,Enc_{pk}(m_0))\} \{(pk,sk)\gets Gen(1^n):(pk,Enc_{pk}(m_1))\} +$$ + +are distinguished by at most $\epsilon (n)$ + +This "single" message security implies multi-message security! + +_Left as exercise_ + +We will achieve security in sending a single bit $0,1$ + +Time for trapdoor permutation. (EX. RSA) + +#### Encryption Scheme via Trapdoor Permutation + +Given family of trapdoor permutation $\{f_i\}$ with hardcore bit $h(i)$ + +$Gen(1^n):(f_i,f_i^{-1})$, where $f_i^{-1}$ uses trapdoor permutation of $t$ + +$Output ((f_i,h_i),f_i^{-1})$ + +$m=0$ or $1$. + +$Enc_{pk}(m):r\gets\{0,1\}^n$ + +$Output (f_i(r),h_i(r)+m)$ + +$Dec_{sk}(c_1,c_2)$ + +$r=f_i^{-1}(c_1)$ + $m=c_2+h_1(r)$ \ No newline at end of file diff --git a/pages/CSE442T/CSE442T_L17.md b/content/CSE442T/CSE442T_L17.md similarity index 96% rename from pages/CSE442T/CSE442T_L17.md rename to content/CSE442T/CSE442T_L17.md index dd6ac4c..b84e8b3 100644 --- a/pages/CSE442T/CSE442T_L17.md +++ b/content/CSE442T/CSE442T_L17.md @@ -1,159 +1,159 @@ -# Lecture 17 - -## Chapter 3: Indistinguishability and Pseudorandomness - -### Public key encryption scheme (1-bit) - -$Gen(1^n):(f_i, f_i^{-1})$ - -$f_i$ is the trapdoor permutation. (eg. RSA) - -$Output((f_i, h_i), f_i^{-1})$, where $(f_i, h_i)$ is the public key and $f_i^{-1}$ is the secret key. - -$Enc_{pk}(m):r\gets \{0, 1\}^n$ - -$Output(f_i(r), h_i(r)\oplus m)$ - -where $f_i(r)$ is denoted as $c_1$ and $h_i(r)\oplus m$ is the tag $c_2$. - -The decryption function is: - -$Dec_{sk}(c_1, c_2)$: - -$r=f_i^{-1}(c_1)$ - -$m=c_2\oplus h_i(r)$ - -#### Validity of the decryption - -Proof of the validity of the decryption: Exercise. - -#### Security of the encryption scheme - -The encryption scheme is secure under this construction (Trapdoor permutation (TDP), Hardcore bit (HCB)). - -Proof: - -We proceed by contradiction. (Constructing contradiction with definition of hardcore bit.) - -Assume that there exists a distinguisher $\mathcal{D}$ that can distinguish the encryption of $0$ and $1$ with non-negligible probability $\mu(n)$. - -$$ -\{(pk,sk)\gets Gen(1^n):(pk,Enc_{pk}(0))\} v.s.\{(pk,sk)\gets Gen(1^n):(pk,Enc_{pk}(1))\} \geq \mu(n) -$$ - -By prediction lemma (the distinguisher can be used to create and adversary that can break the security of the encryption scheme with non-negligible probability $\mu(n)$). - -$$ -P[m\gets \{0,1\}; (pk,sk)\gets Gen(1^n):\mathcal{A}(pk,Enc_{pk}(m))=m]\geq \frac{1}{2}+\mu(n) -$$ - -We will use this to construct an agent $B$ which can determine the hardcore bit $h_i(r)$ of the trapdoor permutation $f_i(r)$ with non-negligible probability. - -$f_i,h_i$ are determined. - -$B$ is given $f_i(r)$ and $h_i(r)$ and outputs $b\in \{0,1\}$. - -- $r\gets \{0,1\}^n$ is chosen uniformly at random. -- $y=f_i(r)$ is given to $B$. -- $b=h_i(r)$ is given to $B$. -- Choose $c_2\gets \{0,1\}= h_i(r)\oplus m$ uniformly at random. -- Then use $\mathcal{A}$ with $pk=(f_i, h_i),Enc_{pk}(m)=(f_i(r), h_i(r)\oplus m)$ to determine whether $r$ is $0$ or $1$. -- Let $m'\gets \mathcal{A}(pk,(y,c_2))$. -- Since $c_2=h_i(r)\oplus m$, we have $m=b\oplus c_2$, $b=m'\oplus c_2$. -- Output $b=m'\oplus c_2$. - -The probability that $B$ correctly guesses $b$ given $f_i,h_i$ is: - -$$ -\begin{aligned} -&~~~~~P[r\gets \{0,1\}^n: y=f_i(r), b=h_i(r): B(f_i,h_i,y)=b]\\ -&=P[r\gets \{0,1\}^n,c_2\gets \{0,1\}: y=f_i(r), b=h_i(r):\mathcal{A}((f_i,h_i),(y,c_2))=(c_2+b)]\\ -&=P[r\gets \{0,1\}^n,m\gets \{0,1\}: y=f_i(r), b=h_i(r):\mathcal{A}((f_i,h_i),(y,b\oplus m))=m]\\ -&>\frac{1}{2}+\mu(n) -\end{aligned} -$$ - -This contradicts the definition of hardcore bit. - -QED - -### Public key encryption scheme (multi-bit) - -Let $m\in \{0,1\}^k$. - -We can choose random $r_i\in \{0,1\}^n$, $y_i=f_i(r_i)$, $b_i=h_i(r_i),c_i=m_i\oplus b_i$. - -$Enc_{pk}(m)=((y_1,c_1),\cdots,(y_k,c_k)),c\in \{0,1\}^k$ - -$Dec_{sk}:r_k=f_i^{-1}(y_k),h_i(r_k)\oplus c_k=m_k$ - -### Special public key cryptosystem: El-Gamal (based on Diffie-Hellman Assumption) - -#### Definition 105.1 Decisional Diffie-Hellman Assumption (DDH) - -> Define the group of squares mod $p$ as follows: -> -> $p=2q+1$, $q\in \Pi_{n-1}$, $g\gets \mathbb{Z}_p^*/\{1\}$, $y=g^2$ -> -> $G=\{y,y^2,\cdots,y^q=1\}\mod p$ - -These two listed below are indistinguishable. - -$\{p\gets \tilde{\Pi_n};y\gets Gen_q;a,b\gets \mathbb{Z}_q:(p,y,y^a,y^b,y^{ab})\}_n$ - -$\{p\gets \tilde{\Pi_n};y\gets Gen_q;a,b,\bold{z}\gets \mathbb{Z}_q:(p,y,y^a,y^b,y^\bold{z})\}_n$ - -> (Computational) Diffie-Hellman Assumption: -> -> Hard to compute $y^{ab}$ given $p,y,y^a,y^b$. - -So DDH assumption implies discrete logarithm assumption. - -Ideas: - -If one can find $a,b$ from $y^a,y^b$, then one can find $ab$ from $y^{ab}$ and compare to $\bold{z}$ to check whether $y^\bold{z}$ is a valid DDH tuple. - -#### El-Gamal encryption scheme (public key cryptosystem) - -$Gen(1^n)$: - -$p\gets \tilde{\Pi_n},g\gets \mathbb{Z}_p^*/\{1\},y\gets Gen_q,a\gets \mathbb{Z}_q$ - -Output: - -$pk=(p,y,y^a\mod p)$ (public key) - -$sk=(p,y,a)$ (secret key) - -**Message space:** $G_q=\{y,y^2,\cdots,y^q=1\}$ - -$Enc_{pk}(m)$: - -$b\gets \mathbb{Z}_q$ - -$c_1=y^b\mod p,c_2=(y^{ab}\cdot m)\mod p$ - -Output: $(c_1,c_2)$ - -$Dec_{sk}(c_1,c_2)$: - -Since $c_2=(y^{ab}\cdot m)\mod p$, we have $m=\frac{c_2}{c_1^a}\mod p$ - -Output: $m$ - -#### Security of El-Gamal encryption scheme - -Proof: - -If not secure, then there exists a distinguisher $\mathcal{D}$ that can distinguish the encryption of $m_1,m_2\in G_q$ with non-negligible probability $\mu(n)$. - -$$ -\{(pk,sk)\gets Gen(1^n):D(pk,Enc_{pk}(m_1))\}\text{ vs. }\\ -\{(pk,sk)\gets Gen(1^n):D(pk,Enc_{pk}(m_2))\}\geq \mu(n) -$$ - -And proceed by contradiction. This contradicts the DDH assumption. - -QED - +# Lecture 17 + +## Chapter 3: Indistinguishability and Pseudorandomness + +### Public key encryption scheme (1-bit) + +$Gen(1^n):(f_i, f_i^{-1})$ + +$f_i$ is the trapdoor permutation. (eg. RSA) + +$Output((f_i, h_i), f_i^{-1})$, where $(f_i, h_i)$ is the public key and $f_i^{-1}$ is the secret key. + +$Enc_{pk}(m):r\gets \{0, 1\}^n$ + +$Output(f_i(r), h_i(r)\oplus m)$ + +where $f_i(r)$ is denoted as $c_1$ and $h_i(r)\oplus m$ is the tag $c_2$. + +The decryption function is: + +$Dec_{sk}(c_1, c_2)$: + +$r=f_i^{-1}(c_1)$ + +$m=c_2\oplus h_i(r)$ + +#### Validity of the decryption + +Proof of the validity of the decryption: Exercise. + +#### Security of the encryption scheme + +The encryption scheme is secure under this construction (Trapdoor permutation (TDP), Hardcore bit (HCB)). + +Proof: + +We proceed by contradiction. (Constructing contradiction with definition of hardcore bit.) + +Assume that there exists a distinguisher $\mathcal{D}$ that can distinguish the encryption of $0$ and $1$ with non-negligible probability $\mu(n)$. + +$$ +\{(pk,sk)\gets Gen(1^n):(pk,Enc_{pk}(0))\} v.s.\{(pk,sk)\gets Gen(1^n):(pk,Enc_{pk}(1))\} \geq \mu(n) +$$ + +By prediction lemma (the distinguisher can be used to create and adversary that can break the security of the encryption scheme with non-negligible probability $\mu(n)$). + +$$ +P[m\gets \{0,1\}; (pk,sk)\gets Gen(1^n):\mathcal{A}(pk,Enc_{pk}(m))=m]\geq \frac{1}{2}+\mu(n) +$$ + +We will use this to construct an agent $B$ which can determine the hardcore bit $h_i(r)$ of the trapdoor permutation $f_i(r)$ with non-negligible probability. + +$f_i,h_i$ are determined. + +$B$ is given $f_i(r)$ and $h_i(r)$ and outputs $b\in \{0,1\}$. + +- $r\gets \{0,1\}^n$ is chosen uniformly at random. +- $y=f_i(r)$ is given to $B$. +- $b=h_i(r)$ is given to $B$. +- Choose $c_2\gets \{0,1\}= h_i(r)\oplus m$ uniformly at random. +- Then use $\mathcal{A}$ with $pk=(f_i, h_i),Enc_{pk}(m)=(f_i(r), h_i(r)\oplus m)$ to determine whether $r$ is $0$ or $1$. +- Let $m'\gets \mathcal{A}(pk,(y,c_2))$. +- Since $c_2=h_i(r)\oplus m$, we have $m=b\oplus c_2$, $b=m'\oplus c_2$. +- Output $b=m'\oplus c_2$. + +The probability that $B$ correctly guesses $b$ given $f_i,h_i$ is: + +$$ +\begin{aligned} +&~~~~~P[r\gets \{0,1\}^n: y=f_i(r), b=h_i(r): B(f_i,h_i,y)=b]\\ +&=P[r\gets \{0,1\}^n,c_2\gets \{0,1\}: y=f_i(r), b=h_i(r):\mathcal{A}((f_i,h_i),(y,c_2))=(c_2+b)]\\ +&=P[r\gets \{0,1\}^n,m\gets \{0,1\}: y=f_i(r), b=h_i(r):\mathcal{A}((f_i,h_i),(y,b\oplus m))=m]\\ +&>\frac{1}{2}+\mu(n) +\end{aligned} +$$ + +This contradicts the definition of hardcore bit. + +QED + +### Public key encryption scheme (multi-bit) + +Let $m\in \{0,1\}^k$. + +We can choose random $r_i\in \{0,1\}^n$, $y_i=f_i(r_i)$, $b_i=h_i(r_i),c_i=m_i\oplus b_i$. + +$Enc_{pk}(m)=((y_1,c_1),\cdots,(y_k,c_k)),c\in \{0,1\}^k$ + +$Dec_{sk}:r_k=f_i^{-1}(y_k),h_i(r_k)\oplus c_k=m_k$ + +### Special public key cryptosystem: El-Gamal (based on Diffie-Hellman Assumption) + +#### Definition 105.1 Decisional Diffie-Hellman Assumption (DDH) + +> Define the group of squares mod $p$ as follows: +> +> $p=2q+1$, $q\in \Pi_{n-1}$, $g\gets \mathbb{Z}_p^*/\{1\}$, $y=g^2$ +> +> $G=\{y,y^2,\cdots,y^q=1\}\mod p$ + +These two listed below are indistinguishable. + +$\{p\gets \tilde{\Pi_n};y\gets Gen_q;a,b\gets \mathbb{Z}_q:(p,y,y^a,y^b,y^{ab})\}_n$ + +$\{p\gets \tilde{\Pi_n};y\gets Gen_q;a,b,\bold{z}\gets \mathbb{Z}_q:(p,y,y^a,y^b,y^\bold{z})\}_n$ + +> (Computational) Diffie-Hellman Assumption: +> +> Hard to compute $y^{ab}$ given $p,y,y^a,y^b$. + +So DDH assumption implies discrete logarithm assumption. + +Ideas: + +If one can find $a,b$ from $y^a,y^b$, then one can find $ab$ from $y^{ab}$ and compare to $\bold{z}$ to check whether $y^\bold{z}$ is a valid DDH tuple. + +#### El-Gamal encryption scheme (public key cryptosystem) + +$Gen(1^n)$: + +$p\gets \tilde{\Pi_n},g\gets \mathbb{Z}_p^*/\{1\},y\gets Gen_q,a\gets \mathbb{Z}_q$ + +Output: + +$pk=(p,y,y^a\mod p)$ (public key) + +$sk=(p,y,a)$ (secret key) + +**Message space:** $G_q=\{y,y^2,\cdots,y^q=1\}$ + +$Enc_{pk}(m)$: + +$b\gets \mathbb{Z}_q$ + +$c_1=y^b\mod p,c_2=(y^{ab}\cdot m)\mod p$ + +Output: $(c_1,c_2)$ + +$Dec_{sk}(c_1,c_2)$: + +Since $c_2=(y^{ab}\cdot m)\mod p$, we have $m=\frac{c_2}{c_1^a}\mod p$ + +Output: $m$ + +#### Security of El-Gamal encryption scheme + +Proof: + +If not secure, then there exists a distinguisher $\mathcal{D}$ that can distinguish the encryption of $m_1,m_2\in G_q$ with non-negligible probability $\mu(n)$. + +$$ +\{(pk,sk)\gets Gen(1^n):D(pk,Enc_{pk}(m_1))\}\text{ vs. }\\ +\{(pk,sk)\gets Gen(1^n):D(pk,Enc_{pk}(m_2))\}\geq \mu(n) +$$ + +And proceed by contradiction. This contradicts the DDH assumption. + +QED + diff --git a/pages/CSE442T/CSE442T_L18.md b/content/CSE442T/CSE442T_L18.md similarity index 97% rename from pages/CSE442T/CSE442T_L18.md rename to content/CSE442T/CSE442T_L18.md index 4a66426..93c4555 100644 --- a/pages/CSE442T/CSE442T_L18.md +++ b/content/CSE442T/CSE442T_L18.md @@ -1,148 +1,148 @@ -# Lecture 18 - -## Chapter 5: Authentication - -### 5.1 Introduction - -Signatures - -**private key** - -Alice and Bob share a secret key $k$. - -Message Authentication Codes (MACs) - -**public key** - -Any one can verify the signature. - -Digital Signatures - -#### Definitions 134.1 - -A message authentication codes (MACs) is a triple $(Gen, Tag, Ver)$ where - -- $k\gets Gen(1^k)$ is a p.p.t. algorithm that takes as input a security parameter $k$ and outputs a key $k$. -- $\sigma\gets Tag_k(m)$ is a p.p.t. algorithm that takes as input a key $k$ and a message $m$ and outputs a tag $\sigma$. -- $Ver_k(m, \sigma)$ is a deterministic algorithm that takes as input a key $k$, a message $m$, and a tag $\sigma$ and outputs "Accept" if $\sigma$ is a valid tag for $m$ under $k$ and "Reject" otherwise. - -For all $n\in\mathbb{N}$, all $m\in\mathcal{M}_n$. - -$$ -P[k\gets Gen(1^k):Ver_k(m, Tag_k(m))=\textup {``Accept''}]=1 -$$ - -#### Definition 134.2 (Security of MACs) - -Security: Prevent an adversary from producing any accepted $(m, \sigma)$ pair that they haven't seen before. - -- Assume they have seen some history of signed messages. $(m_1, \sigma_1), (m_2, \sigma_2), \ldots, (m_q, \sigma_q)$. -- Adversary $\mathcal{A}$ has oracle access to $Tag_k$. Goal is to produce a new $(m, \sigma)$ pair that is accepted but none of $(m_1, \sigma_1), (m_2, \sigma_2), \ldots, (m_q, \sigma_q)$. - -$\forall$ n.u.p.p.t. adversary $\mathcal{A}$ with oracle access to $Tag_k(\cdot)$, - -$$ -\Pr[k\gets Gen(1^k);(m, \sigma)\gets\mathcal{A}^{Tag_k(\cdot)}(1^k);\mathcal{A}\textup{ did not query }m \textup{ and } Ver_k(m, \sigma)=\textup{``Accept''}]<\epsilon(n) -$$ - -#### MACs scheme - -$F=\{f_s\}$ is a PRF family. - -$f_s:\{0,1\}^{|S|}\to\{0,1\}^{|S|}$ - -$Gen(1^k): s\gets \{0,1\}^n$ - -$Tag_k(m)$ outputs $f_s(m)$. - -$Ver_s(m, \sigma)$ outputs "Accept" if $f_s(m)=\sigma$ and "Reject" otherwise. - -Proof of security (Outline): - -Suppose we used $F\gets RF_n$ (true random function). - -If $\mathcal{A}$ wants $F(m)$ for $m\in \{m_1, \ldots, m_q\}$. $F(m)\gets U_n$. - -$$ -\begin{aligned} -&P[F\gets RF_n; (m, \sigma)\gets\mathcal{A}^{F(\cdot)}(1^k);\mathcal{A}\textup{ did not query }m \textup{ and } Ver_k(m, \sigma)=\textup{``Accept''}]\\ -&=P[F\gets RF_n; (m, \sigma)\gets F(m)]\\ -&=\frac{1}{2^n}<\epsilon(n) -\end{aligned} -$$ - -Suppose an adversary $\mathcal{A}$ has $\frac{1}{p(n)}$ chance of success with our PRF-based scheme... - -This could be used to distinguish PRF $f_s$ from a random function. - -The distinguisher runs as follows: - -- Runs $\mathcal{A}(1^n)$ -- Whenever $\mathcal{A}$ asks for $Tag_k(m)$, we ask our oracle for $f(m)$ -- $(m, \sigma)\gets\mathcal{A}^{F(\cdot)}(1^n)$ -- Query oracle for $f(m)$ -- If $\sigma=f(m)$, output 1 -- Otherwise, output 0 - -$D$ will output 1 for PRF with probability $\frac{1}{p(n)}$ and for RF with probability $\frac{1}{2^n}$. - -#### Definition 135.1(Digital Signature D.S. over $\{M_n\}_n$) - -A digital signature scheme is a triple $(Gen, Sign, Ver)$ where - -- $(pk,sk)\gets Gen(1^k)$ is a p.p.t. algorithm that takes as input a security parameter $k$ and outputs a public key $pk$ and a secret key $sk$. -- $\sigma\gets Sign_{sk}(m)$ is a p.p.t. algorithm that takes as input a secret key $sk$ and a message $m$ and outputs a signature $\sigma$. -- $Ver_{pk}(m, \sigma)$ is a deterministic algorithm that takes as input a public key $pk$, a message $m$, and a signature $\sigma$ and outputs "Accept" if $\sigma$ is a valid signature for $m$ under $pk$ and "Reject" otherwise. - -For all $n\in\mathbb{N}$, all $m\in\mathcal{M}_n$. - -$$ -P[(pk,sk)\gets Gen(1^k); \sigma\gets Sign_{sk}(m); Ver_{pk}(m, \sigma)=\textup{``Accept''}]=1 -$$ - -#### Security of Digital Signature - -$$ -P[(pk,sk)\gets Gen(1^k); (m, \sigma)\gets\mathcal{A}^{Sign_{sk}(\cdot)}(1^k);\mathcal{A}\textup{ did not query }m \textup{ and } Ver_{pk}(m, \sigma)=\textup{``Accept''}]<\epsilon(n) -$$ - -For all n.u.p.p.t. adversary $\mathcal{A}$ with oracle access to $Sign_{sk}(\cdot)$. - -### 5.4 One time security: $\mathcal{A}$ can only use oracle once. - -Output $(m, \sigma)$ if $m\neq m$ - -Security parameter $n$ - -One time security on $\{0,1\}^n$ - -One time security on $\{0,1\}^*$ - -Regular security on $\{0,1\}^*$ - -Note: the adversary automatically has access to $Ver_{pk}(\cdot)$ - -#### One time security scheme (Lamport Scheme on $\{0,1\}^n$) - -$Gen(1^k)$: $\mathbb{Z}_n$ random n-bit string - -$sk$: List 0: $\bar{x_1}^0, \bar{x_2}^0, \ldots, \bar{x_n}^0$ - -List 1: $\bar{x_1}^1, \bar{x_2}^1, \ldots, \bar{x_n}^1$ - -All $\bar{x_i}^j\in\{0,1\}^n$ - -$pk$: For a strong one-way function $f$ - -List 0: $f(\bar{x_1}^0), f(\bar{x_2}^0), \ldots, f(\bar{x_n}^0)$ - -List 1: $f(\bar{x_1}^1), f(\bar{x_2}^1), \ldots, f(\bar{x_n}^1)$ - -$Sign_{sk}(m):(m_1, m_2, \ldots, m_n)\mapsto(\bar{x_1}^{m_1}, \bar{x_2}^{m_2}, \ldots, \bar{x_n}^{m_n})$ - -$Ver_{pk}(m, \sigma)$: output "Accept" if $\sigma$ is a prefix of $f(m)$ and "Reject" otherwise. - -> Example: When we sign a message $01100$, $$Sign_{sk}(01100)=(\bar{x_1}^0, \bar{x_2}^1, \bar{x_3}^1, \bar{x_4}^0, \bar{x_5}^0)$$ -> We only reveal the $x_1^0, x_2^1, x_3^1, x_4^0, x_5^0$ -> For the second signature, we need to reveal exactly different bits. -> The adversary can query the oracle for $f(0^n)$ (reveals list0) and $f(1^n)$ (reveals list1) to produce any valid signature they want. +# Lecture 18 + +## Chapter 5: Authentication + +### 5.1 Introduction + +Signatures + +**private key** + +Alice and Bob share a secret key $k$. + +Message Authentication Codes (MACs) + +**public key** + +Any one can verify the signature. + +Digital Signatures + +#### Definitions 134.1 + +A message authentication codes (MACs) is a triple $(Gen, Tag, Ver)$ where + +- $k\gets Gen(1^k)$ is a p.p.t. algorithm that takes as input a security parameter $k$ and outputs a key $k$. +- $\sigma\gets Tag_k(m)$ is a p.p.t. algorithm that takes as input a key $k$ and a message $m$ and outputs a tag $\sigma$. +- $Ver_k(m, \sigma)$ is a deterministic algorithm that takes as input a key $k$, a message $m$, and a tag $\sigma$ and outputs "Accept" if $\sigma$ is a valid tag for $m$ under $k$ and "Reject" otherwise. + +For all $n\in\mathbb{N}$, all $m\in\mathcal{M}_n$. + +$$ +P[k\gets Gen(1^k):Ver_k(m, Tag_k(m))=\textup {``Accept''}]=1 +$$ + +#### Definition 134.2 (Security of MACs) + +Security: Prevent an adversary from producing any accepted $(m, \sigma)$ pair that they haven't seen before. + +- Assume they have seen some history of signed messages. $(m_1, \sigma_1), (m_2, \sigma_2), \ldots, (m_q, \sigma_q)$. +- Adversary $\mathcal{A}$ has oracle access to $Tag_k$. Goal is to produce a new $(m, \sigma)$ pair that is accepted but none of $(m_1, \sigma_1), (m_2, \sigma_2), \ldots, (m_q, \sigma_q)$. + +$\forall$ n.u.p.p.t. adversary $\mathcal{A}$ with oracle access to $Tag_k(\cdot)$, + +$$ +\Pr[k\gets Gen(1^k);(m, \sigma)\gets\mathcal{A}^{Tag_k(\cdot)}(1^k);\mathcal{A}\textup{ did not query }m \textup{ and } Ver_k(m, \sigma)=\textup{``Accept''}]<\epsilon(n) +$$ + +#### MACs scheme + +$F=\{f_s\}$ is a PRF family. + +$f_s:\{0,1\}^{|S|}\to\{0,1\}^{|S|}$ + +$Gen(1^k): s\gets \{0,1\}^n$ + +$Tag_k(m)$ outputs $f_s(m)$. + +$Ver_s(m, \sigma)$ outputs "Accept" if $f_s(m)=\sigma$ and "Reject" otherwise. + +Proof of security (Outline): + +Suppose we used $F\gets RF_n$ (true random function). + +If $\mathcal{A}$ wants $F(m)$ for $m\in \{m_1, \ldots, m_q\}$. $F(m)\gets U_n$. + +$$ +\begin{aligned} +&P[F\gets RF_n; (m, \sigma)\gets\mathcal{A}^{F(\cdot)}(1^k);\mathcal{A}\textup{ did not query }m \textup{ and } Ver_k(m, \sigma)=\textup{``Accept''}]\\ +&=P[F\gets RF_n; (m, \sigma)\gets F(m)]\\ +&=\frac{1}{2^n}<\epsilon(n) +\end{aligned} +$$ + +Suppose an adversary $\mathcal{A}$ has $\frac{1}{p(n)}$ chance of success with our PRF-based scheme... + +This could be used to distinguish PRF $f_s$ from a random function. + +The distinguisher runs as follows: + +- Runs $\mathcal{A}(1^n)$ +- Whenever $\mathcal{A}$ asks for $Tag_k(m)$, we ask our oracle for $f(m)$ +- $(m, \sigma)\gets\mathcal{A}^{F(\cdot)}(1^n)$ +- Query oracle for $f(m)$ +- If $\sigma=f(m)$, output 1 +- Otherwise, output 0 + +$D$ will output 1 for PRF with probability $\frac{1}{p(n)}$ and for RF with probability $\frac{1}{2^n}$. + +#### Definition 135.1(Digital Signature D.S. over $\{M_n\}_n$) + +A digital signature scheme is a triple $(Gen, Sign, Ver)$ where + +- $(pk,sk)\gets Gen(1^k)$ is a p.p.t. algorithm that takes as input a security parameter $k$ and outputs a public key $pk$ and a secret key $sk$. +- $\sigma\gets Sign_{sk}(m)$ is a p.p.t. algorithm that takes as input a secret key $sk$ and a message $m$ and outputs a signature $\sigma$. +- $Ver_{pk}(m, \sigma)$ is a deterministic algorithm that takes as input a public key $pk$, a message $m$, and a signature $\sigma$ and outputs "Accept" if $\sigma$ is a valid signature for $m$ under $pk$ and "Reject" otherwise. + +For all $n\in\mathbb{N}$, all $m\in\mathcal{M}_n$. + +$$ +P[(pk,sk)\gets Gen(1^k); \sigma\gets Sign_{sk}(m); Ver_{pk}(m, \sigma)=\textup{``Accept''}]=1 +$$ + +#### Security of Digital Signature + +$$ +P[(pk,sk)\gets Gen(1^k); (m, \sigma)\gets\mathcal{A}^{Sign_{sk}(\cdot)}(1^k);\mathcal{A}\textup{ did not query }m \textup{ and } Ver_{pk}(m, \sigma)=\textup{``Accept''}]<\epsilon(n) +$$ + +For all n.u.p.p.t. adversary $\mathcal{A}$ with oracle access to $Sign_{sk}(\cdot)$. + +### 5.4 One time security: $\mathcal{A}$ can only use oracle once. + +Output $(m, \sigma)$ if $m\neq m$ + +Security parameter $n$ + +One time security on $\{0,1\}^n$ + +One time security on $\{0,1\}^*$ + +Regular security on $\{0,1\}^*$ + +Note: the adversary automatically has access to $Ver_{pk}(\cdot)$ + +#### One time security scheme (Lamport Scheme on $\{0,1\}^n$) + +$Gen(1^k)$: $\mathbb{Z}_n$ random n-bit string + +$sk$: List 0: $\bar{x_1}^0, \bar{x_2}^0, \ldots, \bar{x_n}^0$ + +List 1: $\bar{x_1}^1, \bar{x_2}^1, \ldots, \bar{x_n}^1$ + +All $\bar{x_i}^j\in\{0,1\}^n$ + +$pk$: For a strong one-way function $f$ + +List 0: $f(\bar{x_1}^0), f(\bar{x_2}^0), \ldots, f(\bar{x_n}^0)$ + +List 1: $f(\bar{x_1}^1), f(\bar{x_2}^1), \ldots, f(\bar{x_n}^1)$ + +$Sign_{sk}(m):(m_1, m_2, \ldots, m_n)\mapsto(\bar{x_1}^{m_1}, \bar{x_2}^{m_2}, \ldots, \bar{x_n}^{m_n})$ + +$Ver_{pk}(m, \sigma)$: output "Accept" if $\sigma$ is a prefix of $f(m)$ and "Reject" otherwise. + +> Example: When we sign a message $01100$, $$Sign_{sk}(01100)=(\bar{x_1}^0, \bar{x_2}^1, \bar{x_3}^1, \bar{x_4}^0, \bar{x_5}^0)$$ +> We only reveal the $x_1^0, x_2^1, x_3^1, x_4^0, x_5^0$ +> For the second signature, we need to reveal exactly different bits. +> The adversary can query the oracle for $f(0^n)$ (reveals list0) and $f(1^n)$ (reveals list1) to produce any valid signature they want. diff --git a/pages/CSE442T/CSE442T_L19.md b/content/CSE442T/CSE442T_L19.md similarity index 100% rename from pages/CSE442T/CSE442T_L19.md rename to content/CSE442T/CSE442T_L19.md diff --git a/pages/CSE442T/CSE442T_L2.md b/content/CSE442T/CSE442T_L2.md similarity index 96% rename from pages/CSE442T/CSE442T_L2.md rename to content/CSE442T/CSE442T_L2.md index 0bc0789..52fa65f 100644 --- a/pages/CSE442T/CSE442T_L2.md +++ b/content/CSE442T/CSE442T_L2.md @@ -1,97 +1,97 @@ -# Lecture 2 - -## Probability review - -Sample space $S=\text{set of outcomes (possible results of experiments)}$ - -Event $A\subseteq S$ - -$P[A]=P[$ outcome $x\in A]$ - -$P[\{x\}]=P[x]$ - -Conditional probability: - -$P[A|B]={P[A\cap B]\over P[B]}$ - -Assuming $B$ is the known information. Moreover, $P[B]>0$ - -Probability that $A$ and $B$ occurring: $P[A\cap B]=P[A|B]\cdot P[B]$ - -$P[B\cap A]=P[B|A]\cdot P[A]$ - -So $P[A|B]={P[B|A]\cdot P[A]\over P[B]}$ (Bayes Theorem) - -**There is always a chance that random guess would be the password... Although really, really, low...** - -### Law of total probability - -Let $S=\bigcup_{i=1}^n B_i$. and $B_i$ are disjoint events. - -$A=\bigcup_{i=1}^n A\cap B_i$ ($A\cap B_i$ are all disjoint) - -$P[A]=\sum^n_{i=1} P[A|B_i]\cdot P[B_i]$ - -## Chapter 1: Introduction - -### Defining security - -#### Perfect Secrecy (Shannon Secrecy) - -$k\gets Gen()$ $k\in K$ - -$c\gets Enc_k(m)$ or we can also write as $c\gets Enc(k,m)$ for $m\in M$ - -And the decryption procedure: - -$m'\gets Dec_k(c')$, $m'$ might be null. - -$P[k\gets Gen(): Dec_k(Enc_k(m))=m]=1$ - -#### Definition 11.1 (Shannon Secrecy) - -Distribution $D$ over the message space $M$ - -$P[k\gets Gen;m\gets D: m=m'|c\gets Enc_k(m)]=P[m\gets D: m=m']$ - -Basically, we cannot gain any information from the encoded message. - -Code shall not contain any information changing the distribution of expectation of message after viewing the code. - -**NO INFO GAINED** - -#### Definition 11.2 (Perfect Secrecy) - -For any 2 messages, say $m_1,m_2\in M$ and for any possible cipher $c$, - -$P[k\gets Gen:c\gets Enc_k(m_1)]=P[k\gets Gen():c\gets Enc_k(m_2)]$ - -For a fixed $c$, any message (have a equal probability) could be encrypted to that... - -#### Theorem 12.3 - -Shannon secrecy is equivalent to perfect secrecy. - -Proof: - -If a crypto-system satisfy perfect secrecy, then it also satisfy Shannon secrecy. - -Let $(Gen,Enc,Dec)$ be a perfectly secret crypto-system with $K$ and $M$. - -Let $D$ be any distribution over messages. - -Let $m'\in M$. - -$$ -={P_k[c\gets Enc_k(m')]\cdot P[m=m']\over P_{k,m}[c\gets Enc_k(m)]}\\ -$$ - -$$ -P[k\gets Gen();m\gets D:m=m'|c\gets Enc_k(m)]={P_{k,m}[c\gets Enc_k(m)\vert m=m']\cdot P[m=m']\over P_{k,m}[c\gets Enc_k(m)]}\\ -P_{k,m}[c\gets Enc_k(m)]=\sum^n_{i=1}P_{k,m}[c\gets Enc_k(m)|m=m_i]\cdot P[m=m_i]\\ -=\sum^n_{i=1}P_{K,m_i}[c\gets Enc_k(m_i)]\cdot P[m=m_i] -$$ - -and $P_{k,m_i}[c\gets Enc_k(m_i)]$ is constant due to perfect secrecy - +# Lecture 2 + +## Probability review + +Sample space $S=\text{set of outcomes (possible results of experiments)}$ + +Event $A\subseteq S$ + +$P[A]=P[$ outcome $x\in A]$ + +$P[\{x\}]=P[x]$ + +Conditional probability: + +$P[A|B]={P[A\cap B]\over P[B]}$ + +Assuming $B$ is the known information. Moreover, $P[B]>0$ + +Probability that $A$ and $B$ occurring: $P[A\cap B]=P[A|B]\cdot P[B]$ + +$P[B\cap A]=P[B|A]\cdot P[A]$ + +So $P[A|B]={P[B|A]\cdot P[A]\over P[B]}$ (Bayes Theorem) + +**There is always a chance that random guess would be the password... Although really, really, low...** + +### Law of total probability + +Let $S=\bigcup_{i=1}^n B_i$. and $B_i$ are disjoint events. + +$A=\bigcup_{i=1}^n A\cap B_i$ ($A\cap B_i$ are all disjoint) + +$P[A]=\sum^n_{i=1} P[A|B_i]\cdot P[B_i]$ + +## Chapter 1: Introduction + +### Defining security + +#### Perfect Secrecy (Shannon Secrecy) + +$k\gets Gen()$ $k\in K$ + +$c\gets Enc_k(m)$ or we can also write as $c\gets Enc(k,m)$ for $m\in M$ + +And the decryption procedure: + +$m'\gets Dec_k(c')$, $m'$ might be null. + +$P[k\gets Gen(): Dec_k(Enc_k(m))=m]=1$ + +#### Definition 11.1 (Shannon Secrecy) + +Distribution $D$ over the message space $M$ + +$P[k\gets Gen;m\gets D: m=m'|c\gets Enc_k(m)]=P[m\gets D: m=m']$ + +Basically, we cannot gain any information from the encoded message. + +Code shall not contain any information changing the distribution of expectation of message after viewing the code. + +**NO INFO GAINED** + +#### Definition 11.2 (Perfect Secrecy) + +For any 2 messages, say $m_1,m_2\in M$ and for any possible cipher $c$, + +$P[k\gets Gen:c\gets Enc_k(m_1)]=P[k\gets Gen():c\gets Enc_k(m_2)]$ + +For a fixed $c$, any message (have a equal probability) could be encrypted to that... + +#### Theorem 12.3 + +Shannon secrecy is equivalent to perfect secrecy. + +Proof: + +If a crypto-system satisfy perfect secrecy, then it also satisfy Shannon secrecy. + +Let $(Gen,Enc,Dec)$ be a perfectly secret crypto-system with $K$ and $M$. + +Let $D$ be any distribution over messages. + +Let $m'\in M$. + +$$ +={P_k[c\gets Enc_k(m')]\cdot P[m=m']\over P_{k,m}[c\gets Enc_k(m)]}\\ +$$ + +$$ +P[k\gets Gen();m\gets D:m=m'|c\gets Enc_k(m)]={P_{k,m}[c\gets Enc_k(m)\vert m=m']\cdot P[m=m']\over P_{k,m}[c\gets Enc_k(m)]}\\ +P_{k,m}[c\gets Enc_k(m)]=\sum^n_{i=1}P_{k,m}[c\gets Enc_k(m)|m=m_i]\cdot P[m=m_i]\\ +=\sum^n_{i=1}P_{K,m_i}[c\gets Enc_k(m_i)]\cdot P[m=m_i] +$$ + +and $P_{k,m_i}[c\gets Enc_k(m_i)]$ is constant due to perfect secrecy + $\sum^n_{i=1}P_{k,m_i}[c\gets Enc_k(m_i)]\cdot P[m=m_i]=\sum^n_{i=1} P[m=m_i]=1$ \ No newline at end of file diff --git a/pages/CSE442T/CSE442T_L20.md b/content/CSE442T/CSE442T_L20.md similarity index 100% rename from pages/CSE442T/CSE442T_L20.md rename to content/CSE442T/CSE442T_L20.md diff --git a/pages/CSE442T/CSE442T_L21.md b/content/CSE442T/CSE442T_L21.md similarity index 100% rename from pages/CSE442T/CSE442T_L21.md rename to content/CSE442T/CSE442T_L21.md diff --git a/pages/CSE442T/CSE442T_L22.md b/content/CSE442T/CSE442T_L22.md similarity index 100% rename from pages/CSE442T/CSE442T_L22.md rename to content/CSE442T/CSE442T_L22.md diff --git a/pages/CSE442T/CSE442T_L23.md b/content/CSE442T/CSE442T_L23.md similarity index 100% rename from pages/CSE442T/CSE442T_L23.md rename to content/CSE442T/CSE442T_L23.md diff --git a/pages/CSE442T/CSE442T_L24.md b/content/CSE442T/CSE442T_L24.md similarity index 100% rename from pages/CSE442T/CSE442T_L24.md rename to content/CSE442T/CSE442T_L24.md diff --git a/pages/CSE442T/CSE442T_L3.md b/content/CSE442T/CSE442T_L3.md similarity index 96% rename from pages/CSE442T/CSE442T_L3.md rename to content/CSE442T/CSE442T_L3.md index 5c9f0cb..8c00221 100644 --- a/pages/CSE442T/CSE442T_L3.md +++ b/content/CSE442T/CSE442T_L3.md @@ -1,115 +1,115 @@ -# Lecture 3 - -All algorithms $C(x)\to y$, $x,y\in \{0,1\}^*$ - -P.P.T= Probabilistic Polynomial-time Turing Machine. - -## Chapter 2: Computational Hardness - -### Turing Machine: Mathematical model for a computer program - -A machine that can: - -1. Read in put -2. Read/Write working tape move left/right -3. Can change state - -### Assumptions - -Anything can be accomplished by a real computer program can be accomplished by a "sufficiently complicated" Turing Machine (TM). - -### Polynomial time - -We say $C(x),|x|=n,n\to \infty$ runs in polynomial time if it uses at most $T(n)$ operations bounded by some polynomials. $\exist c>0$ such that $T(n)=O(n^c)$ - -If we can argue that algorithm runs in polynomially-many constant-time operations, then this is true for the T.M. - -$p,q$ are polynomials in $n$, - -$p(n)+q(n),p(n)q(n),p(q(n))$ are polynomial of $n$. - -Polynomial-time $\approx$ "efficient" for this course. - -### Probabilistic - -Our algorithm's have access to random "coin-flips" we can produce poly(n) random bits. - -$P[C(x)\text{ takes at most }T(n)\text{ steps }]=1$ - -Our adversary $a(x)$ will be a P.P.T which is non-uniform (n.u.) (programs description size can grow polynomially in n) - -### Efficient private key encryption scheme - -#### Definition 3.2 (Efficient private key encryption scheme) - -The triple $(Gen,Enc,Dec)$ is an efficient private key encryption scheme over the message space $M$ and key space $K$ if: - -1. $Gen(1^n)$ is a randomized p.p.t that outputs $k\in K$ -2. $Enc_k(m)$ is a potentially randomized p.p.t that outputs $c$ given $m\in M$ -3. $Dec_k(c')$ is a deterministic p.p.t that outputs $m$ or "null" -4. $P_k[Dec_k(Enc_k(m))=m]=1,\forall m\in M$ - -### Negligible function - -$\epsilon:\mathbb{N}\to \mathbb{R}$ is a negligible function if $\forall c>0$, $\exists N\in\mathbb{N}$ such that $\forall n\geq N, \epsilon(n)<\frac{1}{n^c}$ (looks like definition of limits huh) (Definition 27.2) - -Idea: for any polynomial, even $n^{100}$, in the long run $\epsilon(n)\leq \frac{1}{n^{100}}$ - -Example: $\epsilon (n)=\frac{1}{2^n}$, $\epsilon (n)=\frac{1}{n^{\log (n)}}$ - -Non-example: $\epsilon (n)=O(\frac{1}{n^c})\forall c$ - -### One-way function - -Idea: We are always okay with our chance of failure being negligible. - -Foundational concept of cryptography - -Goal: making $Enc_k(m),Dec_k(c')$ easy and $Dec^{-1}(c')$ hard. - -#### Definition 27.3 (Strong one-way function) - -$$ -f:\{0,1\}^n\to \{0,1\}^*(n\to \infty) -$$ - -There is a negligible function $\epsilon (n)$ such that for any adversary $\mathcal{A}$ (n.u.p.p.t) - -$$ -P[x\gets\{0,1\}^n;y=f(x):f(\mathcal{A}(y))=y]\leq\epsilon(n) -$$ - -_Probability of guessing a message $x'$ with the same output as the correct message $x$ is negligible_ - -and - -there is a p.p.t which computes $f(x)$ for any $x$. - -- Hard to go back from output -- Easy to find output - -$a$ sees output y, they wan to find some $x'$ such that $f(x')=y$. - -Example: Suppose $f$ is one-to-one, then $a$ must find our $x$, $P[x'=x]=\frac{1}{2^n}$, which is negligible. - -Why do we allow $a$ to get a different $x'$? - -> Suppose the definition is $P[x\gets\{0,1\}^n;y=f(x):\mathcal{A}(y)=x]\neq\epsilon(n)$, then a trivial function $f(x)=x$ would also satisfy the definition. - -To be technically fair, $\mathcal{A}(y)=\mathcal{A}(y,1^n)$, size of input $\approx n$, let them use $poly(n)$ operations. (we also tells the input size is $n$ to $\mathcal{A}$) - -#### Do one-way function exists? - -Unknown, actually... - -But we think so! - -We will need to use various assumptions. one that we believe very strongly based on evidence/experience - -Example: - -$p,q$ are large random primes - -$N=p\cdot q$ - -Factoring $N$ is hard. (without knowing $p,q$) +# Lecture 3 + +All algorithms $C(x)\to y$, $x,y\in \{0,1\}^*$ + +P.P.T= Probabilistic Polynomial-time Turing Machine. + +## Chapter 2: Computational Hardness + +### Turing Machine: Mathematical model for a computer program + +A machine that can: + +1. Read in put +2. Read/Write working tape move left/right +3. Can change state + +### Assumptions + +Anything can be accomplished by a real computer program can be accomplished by a "sufficiently complicated" Turing Machine (TM). + +### Polynomial time + +We say $C(x),|x|=n,n\to \infty$ runs in polynomial time if it uses at most $T(n)$ operations bounded by some polynomials. $\exist c>0$ such that $T(n)=O(n^c)$ + +If we can argue that algorithm runs in polynomially-many constant-time operations, then this is true for the T.M. + +$p,q$ are polynomials in $n$, + +$p(n)+q(n),p(n)q(n),p(q(n))$ are polynomial of $n$. + +Polynomial-time $\approx$ "efficient" for this course. + +### Probabilistic + +Our algorithm's have access to random "coin-flips" we can produce poly(n) random bits. + +$P[C(x)\text{ takes at most }T(n)\text{ steps }]=1$ + +Our adversary $a(x)$ will be a P.P.T which is non-uniform (n.u.) (programs description size can grow polynomially in n) + +### Efficient private key encryption scheme + +#### Definition 3.2 (Efficient private key encryption scheme) + +The triple $(Gen,Enc,Dec)$ is an efficient private key encryption scheme over the message space $M$ and key space $K$ if: + +1. $Gen(1^n)$ is a randomized p.p.t that outputs $k\in K$ +2. $Enc_k(m)$ is a potentially randomized p.p.t that outputs $c$ given $m\in M$ +3. $Dec_k(c')$ is a deterministic p.p.t that outputs $m$ or "null" +4. $P_k[Dec_k(Enc_k(m))=m]=1,\forall m\in M$ + +### Negligible function + +$\epsilon:\mathbb{N}\to \mathbb{R}$ is a negligible function if $\forall c>0$, $\exists N\in\mathbb{N}$ such that $\forall n\geq N, \epsilon(n)<\frac{1}{n^c}$ (looks like definition of limits huh) (Definition 27.2) + +Idea: for any polynomial, even $n^{100}$, in the long run $\epsilon(n)\leq \frac{1}{n^{100}}$ + +Example: $\epsilon (n)=\frac{1}{2^n}$, $\epsilon (n)=\frac{1}{n^{\log (n)}}$ + +Non-example: $\epsilon (n)=O(\frac{1}{n^c})\forall c$ + +### One-way function + +Idea: We are always okay with our chance of failure being negligible. + +Foundational concept of cryptography + +Goal: making $Enc_k(m),Dec_k(c')$ easy and $Dec^{-1}(c')$ hard. + +#### Definition 27.3 (Strong one-way function) + +$$ +f:\{0,1\}^n\to \{0,1\}^*(n\to \infty) +$$ + +There is a negligible function $\epsilon (n)$ such that for any adversary $\mathcal{A}$ (n.u.p.p.t) + +$$ +P[x\gets\{0,1\}^n;y=f(x):f(\mathcal{A}(y))=y]\leq\epsilon(n) +$$ + +_Probability of guessing a message $x'$ with the same output as the correct message $x$ is negligible_ + +and + +there is a p.p.t which computes $f(x)$ for any $x$. + +- Hard to go back from output +- Easy to find output + +$a$ sees output y, they wan to find some $x'$ such that $f(x')=y$. + +Example: Suppose $f$ is one-to-one, then $a$ must find our $x$, $P[x'=x]=\frac{1}{2^n}$, which is negligible. + +Why do we allow $a$ to get a different $x'$? + +> Suppose the definition is $P[x\gets\{0,1\}^n;y=f(x):\mathcal{A}(y)=x]\neq\epsilon(n)$, then a trivial function $f(x)=x$ would also satisfy the definition. + +To be technically fair, $\mathcal{A}(y)=\mathcal{A}(y,1^n)$, size of input $\approx n$, let them use $poly(n)$ operations. (we also tells the input size is $n$ to $\mathcal{A}$) + +#### Do one-way function exists? + +Unknown, actually... + +But we think so! + +We will need to use various assumptions. one that we believe very strongly based on evidence/experience + +Example: + +$p,q$ are large random primes + +$N=p\cdot q$ + +Factoring $N$ is hard. (without knowing $p,q$) diff --git a/pages/CSE442T/CSE442T_L4.md b/content/CSE442T/CSE442T_L4.md similarity index 96% rename from pages/CSE442T/CSE442T_L4.md rename to content/CSE442T/CSE442T_L4.md index 056690e..d904f0b 100644 --- a/pages/CSE442T/CSE442T_L4.md +++ b/content/CSE442T/CSE442T_L4.md @@ -1,140 +1,140 @@ -# Lecture 4 - -## Recap - -Negligible function $\epsilon(n)$ if $\forall c>0,\exist N$ such that $n>N$, $\epsilon (n)<\frac{1}{n^c}$ - -Example: - -$\epsilon(n)=2^{-n},\epsilon(n)=\frac{1}{n^{\log (\log n)}}$ - -## Chapter 2: Computational Hardness - -### One-way function - -#### Strong One-Way Function - -1. $\exists$ a P.P.T. that computes $f(x),\forall x\in\{0,1\}^n$ -2. $\forall \mathcal{A}$ adversaries, $\exists \epsilon(n),\forall n$. - -$$ -P[x\gets \{0,1\}^n;y=f(x):f(\mathcal{A}(y,1^n))=y]<\epsilon(n) -$$ - -_That is, the probability of success guessing should decreasing (exponentially) as encrypted message increase (linearly)..._ - -To negate statement 2: - -$$ -P[x\gets \{0,1\}^n;y=f(x):f(\mathcal{A}(y,1^n))=y]=\mu(n) -$$ - -is a negligible function. - -Negation: - -$\exists \mathcal{A}$, $P[x\gets \{0,1\}^n;y=f(x):f(\mathcal{A}(y,1^n))=y]=\mu(n)$ is not a negligible function. - -That is, $\exists c>0,\forall N \exists n>N \epsilon(n)>\frac{1}{n^c}$ - -$\mu(n)>\frac{1}{n^c}$ for infinitely many $n$. or infinitely often. - -> Keep in mind: $P[success]=\frac{1}{n^c}$, it can try $O(n^c)$ times and have a good chance of succeeding at least once. - -#### Definition 28.4 (Weak one-way function) - -$f:\{0,1\}^n\to \{0,1\}^*$ - -1. $\exists$ a P.P.T. that computes $f(x),\forall x\in\{0,1\}^n$ -2. $\forall \mathcal{A}$ adversaries, $\exists \epsilon(n),\forall n$. - -$$ -P[x\gets \{0,1\}^n;y=f(x):f(\mathcal{A}(y,1^n))=y]<1-\frac{1}{p(n)} -$$ - -_The probability of success should not be too close to 1_ - -### Probability - -#### Useful bound $00,\exist N$ such that $n>N$, $\epsilon (n)<\frac{1}{n^c}$ + +Example: + +$\epsilon(n)=2^{-n},\epsilon(n)=\frac{1}{n^{\log (\log n)}}$ + +## Chapter 2: Computational Hardness + +### One-way function + +#### Strong One-Way Function + +1. $\exists$ a P.P.T. that computes $f(x),\forall x\in\{0,1\}^n$ +2. $\forall \mathcal{A}$ adversaries, $\exists \epsilon(n),\forall n$. + +$$ +P[x\gets \{0,1\}^n;y=f(x):f(\mathcal{A}(y,1^n))=y]<\epsilon(n) +$$ + +_That is, the probability of success guessing should decreasing (exponentially) as encrypted message increase (linearly)..._ + +To negate statement 2: + +$$ +P[x\gets \{0,1\}^n;y=f(x):f(\mathcal{A}(y,1^n))=y]=\mu(n) +$$ + +is a negligible function. + +Negation: + +$\exists \mathcal{A}$, $P[x\gets \{0,1\}^n;y=f(x):f(\mathcal{A}(y,1^n))=y]=\mu(n)$ is not a negligible function. + +That is, $\exists c>0,\forall N \exists n>N \epsilon(n)>\frac{1}{n^c}$ + +$\mu(n)>\frac{1}{n^c}$ for infinitely many $n$. or infinitely often. + +> Keep in mind: $P[success]=\frac{1}{n^c}$, it can try $O(n^c)$ times and have a good chance of succeeding at least once. + +#### Definition 28.4 (Weak one-way function) + +$f:\{0,1\}^n\to \{0,1\}^*$ + +1. $\exists$ a P.P.T. that computes $f(x),\forall x\in\{0,1\}^n$ +2. $\forall \mathcal{A}$ adversaries, $\exists \epsilon(n),\forall n$. + +$$ +P[x\gets \{0,1\}^n;y=f(x):f(\mathcal{A}(y,1^n))=y]<1-\frac{1}{p(n)} +$$ + +_The probability of success should not be too close to 1_ + +### Probability + +#### Useful bound $0\frac{2^n}{2n}$ - -e.g. -$$ -P[x\gets \{0,1\}^n:x\in prime]\geq {\frac{2^n}{2n}\over 2^n}=\frac{1}{2n} -$$ - -Theorem: - -$$ -f_{mult}:\{0,1\}^{2n}\to \{0,1\}^{2n},f_{mult}(x_1,x_2)=x_1\cdot x_2 -$$ - -Idea: There are enough pairs of primes to make this difficult. - -> Reminder: Weak on-way if easy to compute and $\exist p(n)$, -> $P[\mathcal{A}\ \text{inverts=success}]<1-\frac{1}{p(n)}$ -> $P[\mathcal{A}\ \text{inverts=failure}]>\frac{1}{p(n)}$ high enough - -### Prove one-way function (under assumptions) - -To prove $f$ is on-way (under assumption) - -1. Show $\exists p.p.t$ solves $f(x),\forall x$. -2. Proof by contradiction. - - For weak: Provide $p(n)$ that we know works. - - Assume $\exists \mathcal{A}$ such that $P[\mathcal{A}\ \text{inverts}]>1-\frac{1}{p(n)}$ - - For strong: Provide $p(n)$ that we know works. - - Assume $\exists \mathcal{A}$ such that $P[\mathcal{A}\ \text{inverts}]>\frac{1}{p(n)}$ - -Construct p.p.t $\mathcal{B}$ -which uses $\mathcal{A}$ to solve a problem, which contradicts assumption or known fact. - -Back to Theorem: - -We will show that $p(n)=8n^2$ works. - -We claim $\forall \mathcal{A}$, - -$$ -P[(x_1,x_2)\gets \{0,1\}^{2n};y=f_{mult}(x_1,x_2):f(\mathcal{A}(y))=y]<1-\frac{1}{8n^2} -$$ - -For the sake of contradiction, suppose - -$$ -\exists \mathcal{A} \textup{ such that} P[\mathcal{A}\ \text{inverts}]>1-\frac{1}{8n^2} -$$ - -We will use this $\mathcal{A}$ to design p.p.t $B$ which can factor 2 random primes with non-negligible prob. - -```python -def A(y): - # the adversary algorithm - # expecting N to be product of random integer, don't need to be prime - -def is_prime(x): - # test if x is a prime - -def gen(n): - # generate number up to n bits - -def B(y): - # N is the input cipher - x1,x2=gen(n),gen(n) - p=x1*x2 - if is_prime(x1) and is_prime(x2): - return A(p) - return A(y) -``` - -How often does $\mathcal{B}$ succeed/fail? - -$\mathcal{B}$ fails to factor $N=p\dot q$, if: - -- $x$ and $y$ are not both prime - - $P_e=1-P(x\in \Pi_n)P(y\in \Pi_n)\leq 1-(\frac{1}{2n})^2=1-\frac{1}{4n^2}$ -- if $\mathcal{A}$ fails to factor - - $P_f<\frac{1}{8n^2}$ - -So - -$$ -P[\mathcal{B} \text{ fails}]\leq P[E\cup F]\leq P[E]+P[F]\leq (1-\frac{1}{4n^2}+\frac{1}{8n^2})=1-\frac{1}{8n^2} -$$ - -So - -$$ -P[\mathcal{B} \text{ succeed}]\geq \frac{1}{8n^2} (\text{non-negligible}) -$$ - -This contradicting factoring assumption. Therefore, our assumption that $\mathcal{A}$ exists was wrong. - -Therefore $\forall \mathcal{A}$, $P[(x_1,x_2)\gets \{0,1\}^{2n};y=f_{mult}(x_1,x_2):f(\mathcal{A}(y))=y]<1-\frac{1}{8n^2}$ is wrong. +# Lecture 5 + +## Chapter 2: Computational Hardness + +Proving that there are one-way functions relies on assumptions. + +Factoring Assumption: $\forall \mathcal{A}, \exist \epsilon (n)$, let $p,q\in \Pi_n,p,q<2^n$ + +$$ +P[p\gets \Pi_n;q\gets \Pi_n;N=p\cdot q:\mathcal{A}(N)\in \{p,q\}]<\epsilon(n) +$$ + +Evidence: To this point, best known procedure to always factor has run time $O(2^{\sqrt{n}\sqrt{log(n)}})$ + +Distribution of prime numbers: + +- We have infinitely many prime +- Prime Number Theorem $\pi(n)\approx\frac{n}{\ln(n)}$, that means, $\frac{1}{\ln n}$ of all integers are prime. + +We want to (guaranteed to) find prime: + +$\pi(n)>\frac{2^n}{2n}$ + +e.g. +$$ +P[x\gets \{0,1\}^n:x\in prime]\geq {\frac{2^n}{2n}\over 2^n}=\frac{1}{2n} +$$ + +Theorem: + +$$ +f_{mult}:\{0,1\}^{2n}\to \{0,1\}^{2n},f_{mult}(x_1,x_2)=x_1\cdot x_2 +$$ + +Idea: There are enough pairs of primes to make this difficult. + +> Reminder: Weak on-way if easy to compute and $\exist p(n)$, +> $P[\mathcal{A}\ \text{inverts=success}]<1-\frac{1}{p(n)}$ +> $P[\mathcal{A}\ \text{inverts=failure}]>\frac{1}{p(n)}$ high enough + +### Prove one-way function (under assumptions) + +To prove $f$ is on-way (under assumption) + +1. Show $\exists p.p.t$ solves $f(x),\forall x$. +2. Proof by contradiction. + - For weak: Provide $p(n)$ that we know works. + - Assume $\exists \mathcal{A}$ such that $P[\mathcal{A}\ \text{inverts}]>1-\frac{1}{p(n)}$ + - For strong: Provide $p(n)$ that we know works. + - Assume $\exists \mathcal{A}$ such that $P[\mathcal{A}\ \text{inverts}]>\frac{1}{p(n)}$ + +Construct p.p.t $\mathcal{B}$ +which uses $\mathcal{A}$ to solve a problem, which contradicts assumption or known fact. + +Back to Theorem: + +We will show that $p(n)=8n^2$ works. + +We claim $\forall \mathcal{A}$, + +$$ +P[(x_1,x_2)\gets \{0,1\}^{2n};y=f_{mult}(x_1,x_2):f(\mathcal{A}(y))=y]<1-\frac{1}{8n^2} +$$ + +For the sake of contradiction, suppose + +$$ +\exists \mathcal{A} \textup{ such that} P[\mathcal{A}\ \text{inverts}]>1-\frac{1}{8n^2} +$$ + +We will use this $\mathcal{A}$ to design p.p.t $B$ which can factor 2 random primes with non-negligible prob. + +```python +def A(y): + # the adversary algorithm + # expecting N to be product of random integer, don't need to be prime + +def is_prime(x): + # test if x is a prime + +def gen(n): + # generate number up to n bits + +def B(y): + # N is the input cipher + x1,x2=gen(n),gen(n) + p=x1*x2 + if is_prime(x1) and is_prime(x2): + return A(p) + return A(y) +``` + +How often does $\mathcal{B}$ succeed/fail? + +$\mathcal{B}$ fails to factor $N=p\dot q$, if: + +- $x$ and $y$ are not both prime + - $P_e=1-P(x\in \Pi_n)P(y\in \Pi_n)\leq 1-(\frac{1}{2n})^2=1-\frac{1}{4n^2}$ +- if $\mathcal{A}$ fails to factor + - $P_f<\frac{1}{8n^2}$ + +So + +$$ +P[\mathcal{B} \text{ fails}]\leq P[E\cup F]\leq P[E]+P[F]\leq (1-\frac{1}{4n^2}+\frac{1}{8n^2})=1-\frac{1}{8n^2} +$$ + +So + +$$ +P[\mathcal{B} \text{ succeed}]\geq \frac{1}{8n^2} (\text{non-negligible}) +$$ + +This contradicting factoring assumption. Therefore, our assumption that $\mathcal{A}$ exists was wrong. + +Therefore $\forall \mathcal{A}$, $P[(x_1,x_2)\gets \{0,1\}^{2n};y=f_{mult}(x_1,x_2):f(\mathcal{A}(y))=y]<1-\frac{1}{8n^2}$ is wrong. diff --git a/pages/CSE442T/CSE442T_L6.md b/content/CSE442T/CSE442T_L6.md similarity index 96% rename from pages/CSE442T/CSE442T_L6.md rename to content/CSE442T/CSE442T_L6.md index 6d5a123..09e3c1b 100644 --- a/pages/CSE442T/CSE442T_L6.md +++ b/content/CSE442T/CSE442T_L6.md @@ -1,114 +1,114 @@ -# Lecture 6 - -## Review - -$$ -f_{mult}:\{0,1\}^{2n}\to \{0,1\}^{2n} -$$ - -is a weak one-way. - -$P[\mathcal{A}\ \text{invert}]\leq 1-\frac{1}{8n^2}$ over $x,y\in$ random integers $\{0,1\}^n$ - -## Chapter 2: Computational Hardness - -### Converting weak one-way function to strong one-way function - -By factoring assumptions, $\exists$ strong one-way function - -$f:\{0,1\}^N\to \{0,1\}^N$ for infinitely many $N$. - -$f=\left(f_{mult}(x_1,y_1),f_{mult}(x_2,y_2),\dots,f_{mult}(x_q,y_q)\right)$, $x_i,y_i\in \{0,1\}^n$. - -$f:\{0,1\}^{8n^4}\to \{0,1\}^{8n^4}$ - -Idea: With high probability, at least one pair $(x_i,y_i)$ are both prime. - -Factoring assumption: $\mathcal{A}$ has low chance of factoring $f_{mult}(x_i,y_i)$ - -Use $P[x \textup{ is prime}]\geq\frac{1}{2n}$ - -$$ -P[\forall p,q \in x_i,y_i, p\textup{ and } q \textup{ is not prime }]=P[p,q \in x_i,y_i, p\textup{ and } q \textup{ is not prime }]^q -$$ - -$$ -P[\forall p,q \in x_i,y_i, p\textup{ and } q \textup{ is not prime }]\leq(1-\frac{1}{4n^2})^{4n^3}\leq (e^{-\frac{1}{4n^2}})^{4n^3}=e^{-n} -$$ - -### Proof of strong one-way function - -1. $f_{mult}$ is efficiently computable, and we compute it poly-many times. -2. Suppose it's not hard to invert. Then - $\exists \text{n.u.p.p.t.}\ \mathcal{A}$such that $P[w\gets \{0,1\}^{8n^4};z=f(w):f(\mathcal{A}(z))=0]=\mu (n)>\frac{1}{p(n)}$ - -We will use this to construct $\mathcal{B}$ that breaks factoring assumption. - -$p\gets \Pi_n,q\gets \Pi_n,N=p\cdot q$ - -```psudocode -function B: - Receives N - Sample (x,y) q times - Compute z_i = f_mult(x_i,y_i) for each i - From i=1 to q - check if both x_i y_i are prime - If yes, - z_i = N - break // replace first instance - Let z = (z_1,z_2,...,z_q) // z_k = N hopefully - ((x_1,y_1),...,(x_k,y_k),...,(x_q,y_q)) <- a(z) - if (x_k,y_k) was replaced - return x_k,y_k - else - return null -``` - -Let $E$ be the event that all pairs of sampled integers were not both prime. - -Let $F$ be the event that $\mathcal{A}$ failed to invert - -$P[\mathcal{B} \text{ fails}]\leq P[E\cup F]\leq P[E]+P[F]\leq e^{-n}+(1-\frac{1}{p(n)})=1-(\frac{1}{p(n)}-e^{-n})\leq 1-\frac{1}{2p(n)}$ - -$P[\mathcal{B} \text{ succeeds}]=P[p\gets \Pi_n,q\gets \Pi_n,N=p\cdot q:\mathcal{B}(N)\in \{p,q\}]\geq \frac{1}{2p(n)}$ - -Contradicting factoring assumption - -We've defined one-way functions to hae domain $\{0,1\}^n$ for some $n$. - -Our strong one-way function $f(n)$ - -- Takes $4n^3$ pairs of random integers -- Multiplies all pairs -- Hope at least pair are both prime $p,q$ b/c we know $N=p\cdot q$ is hard to factor - -### General collection of strong one-way functions - -$F=\{f_i:D_i\to R_i\},i\in I$, $I$ is the index set. - -1. We can effectively choose $i\gets I$ using $Gen$. -2. $\forall i$ we ca efficiently sample $x\gets D_i$. -3. $\forall i\forall x\in D_i,f_i(x)$ is efficiently computable -4. For any n.u.p.p.t $\mathcal{A}$, $\exists$ negligible function $\epsilon (n)$. - $P[i\gets Gen(1^n);x\gets D_i;y=f_i(x):f(\mathcal{A}(y,i,1^n))=y]\leq \epsilon(n)$ - -#### An instance of strong one-way function under factoring assumption - -$f_{mult,n}:(\Pi_n\times \Pi_n)\to \{0,1\}^{2n}$ is a collection of strong one way function. - -Ideas of proof: - -1. $n\gets Gen(1^n)$ -2. We can efficiently sample $p,q$ (with justifications) -3. Factoring assumption - -Algorithm for sampling a random prime $p\gets \Pi_n$ - -1. $x\gets \{0,1\}^n$ (n bit integer) -2. Check if $x$ is prime. - - Deterministic poly-time procedure - - In practice, a much faster randomized procedure (Miller-Rabin) used - - $P[x\cancel{\in} \text{prime}|\text{test said x prime}]<\epsilon(n)$ - -3. If not, repeat. Do this for polynomial number of times +# Lecture 6 + +## Review + +$$ +f_{mult}:\{0,1\}^{2n}\to \{0,1\}^{2n} +$$ + +is a weak one-way. + +$P[\mathcal{A}\ \text{invert}]\leq 1-\frac{1}{8n^2}$ over $x,y\in$ random integers $\{0,1\}^n$ + +## Chapter 2: Computational Hardness + +### Converting weak one-way function to strong one-way function + +By factoring assumptions, $\exists$ strong one-way function + +$f:\{0,1\}^N\to \{0,1\}^N$ for infinitely many $N$. + +$f=\left(f_{mult}(x_1,y_1),f_{mult}(x_2,y_2),\dots,f_{mult}(x_q,y_q)\right)$, $x_i,y_i\in \{0,1\}^n$. + +$f:\{0,1\}^{8n^4}\to \{0,1\}^{8n^4}$ + +Idea: With high probability, at least one pair $(x_i,y_i)$ are both prime. + +Factoring assumption: $\mathcal{A}$ has low chance of factoring $f_{mult}(x_i,y_i)$ + +Use $P[x \textup{ is prime}]\geq\frac{1}{2n}$ + +$$ +P[\forall p,q \in x_i,y_i, p\textup{ and } q \textup{ is not prime }]=P[p,q \in x_i,y_i, p\textup{ and } q \textup{ is not prime }]^q +$$ + +$$ +P[\forall p,q \in x_i,y_i, p\textup{ and } q \textup{ is not prime }]\leq(1-\frac{1}{4n^2})^{4n^3}\leq (e^{-\frac{1}{4n^2}})^{4n^3}=e^{-n} +$$ + +### Proof of strong one-way function + +1. $f_{mult}$ is efficiently computable, and we compute it poly-many times. +2. Suppose it's not hard to invert. Then + $\exists \text{n.u.p.p.t.}\ \mathcal{A}$such that $P[w\gets \{0,1\}^{8n^4};z=f(w):f(\mathcal{A}(z))=0]=\mu (n)>\frac{1}{p(n)}$ + +We will use this to construct $\mathcal{B}$ that breaks factoring assumption. + +$p\gets \Pi_n,q\gets \Pi_n,N=p\cdot q$ + +```psudocode +function B: + Receives N + Sample (x,y) q times + Compute z_i = f_mult(x_i,y_i) for each i + From i=1 to q + check if both x_i y_i are prime + If yes, + z_i = N + break // replace first instance + Let z = (z_1,z_2,...,z_q) // z_k = N hopefully + ((x_1,y_1),...,(x_k,y_k),...,(x_q,y_q)) <- a(z) + if (x_k,y_k) was replaced + return x_k,y_k + else + return null +``` + +Let $E$ be the event that all pairs of sampled integers were not both prime. + +Let $F$ be the event that $\mathcal{A}$ failed to invert + +$P[\mathcal{B} \text{ fails}]\leq P[E\cup F]\leq P[E]+P[F]\leq e^{-n}+(1-\frac{1}{p(n)})=1-(\frac{1}{p(n)}-e^{-n})\leq 1-\frac{1}{2p(n)}$ + +$P[\mathcal{B} \text{ succeeds}]=P[p\gets \Pi_n,q\gets \Pi_n,N=p\cdot q:\mathcal{B}(N)\in \{p,q\}]\geq \frac{1}{2p(n)}$ + +Contradicting factoring assumption + +We've defined one-way functions to hae domain $\{0,1\}^n$ for some $n$. + +Our strong one-way function $f(n)$ + +- Takes $4n^3$ pairs of random integers +- Multiplies all pairs +- Hope at least pair are both prime $p,q$ b/c we know $N=p\cdot q$ is hard to factor + +### General collection of strong one-way functions + +$F=\{f_i:D_i\to R_i\},i\in I$, $I$ is the index set. + +1. We can effectively choose $i\gets I$ using $Gen$. +2. $\forall i$ we ca efficiently sample $x\gets D_i$. +3. $\forall i\forall x\in D_i,f_i(x)$ is efficiently computable +4. For any n.u.p.p.t $\mathcal{A}$, $\exists$ negligible function $\epsilon (n)$. + $P[i\gets Gen(1^n);x\gets D_i;y=f_i(x):f(\mathcal{A}(y,i,1^n))=y]\leq \epsilon(n)$ + +#### An instance of strong one-way function under factoring assumption + +$f_{mult,n}:(\Pi_n\times \Pi_n)\to \{0,1\}^{2n}$ is a collection of strong one way function. + +Ideas of proof: + +1. $n\gets Gen(1^n)$ +2. We can efficiently sample $p,q$ (with justifications) +3. Factoring assumption + +Algorithm for sampling a random prime $p\gets \Pi_n$ + +1. $x\gets \{0,1\}^n$ (n bit integer) +2. Check if $x$ is prime. + - Deterministic poly-time procedure + - In practice, a much faster randomized procedure (Miller-Rabin) used + + $P[x\cancel{\in} \text{prime}|\text{test said x prime}]<\epsilon(n)$ + +3. If not, repeat. Do this for polynomial number of times diff --git a/pages/CSE442T/CSE442T_L7.md b/content/CSE442T/CSE442T_L7.md similarity index 96% rename from pages/CSE442T/CSE442T_L7.md rename to content/CSE442T/CSE442T_L7.md index 500223e..5f37c67 100644 --- a/pages/CSE442T/CSE442T_L7.md +++ b/content/CSE442T/CSE442T_L7.md @@ -1,120 +1,120 @@ -# Lecture 7 - -## Chapter 2: Computational Hardness - -### Letter choosing experiment - -For 100 letter tiles, - -$p_1,...,p_{27}$ (with one blank) - -$(p_1)^2+\dots +(p_{27})^2\geq\frac{1}{27}$ - -For any $p_1,...,p_n$, $0\leq p_i\leq 1$. - -$\sum p_i=1$ - -$P[\text{the same event twice in a row}]=p_1^2+p_2^2....+p_n^2$ - -By Cauchy-Schwarz: $|u\cdot v|^2 \leq ||u||\cdot ||v||^2$. - -let $\vec{u}=(p_1,...,p_n)$, $\vec{v}=(1,..,1)$, so $(p_1^2+p_2^2....+p_n)^2\leq (p_1^2+p_2^2....+p_n^2)\cdot n$. So $p_1^2+p_2^2....+p_n^2\geq \frac{1}{n}$ - -So for an adversary $\mathcal{A}$, who random choose $x'$ and output $f(x')=f(x)$ if matched. $P[f(x)=f(x')]\geq\frac{1}{|Y|}$ - -So $P[x\gets f(x);y=f(x):\mathcal{A}(y,1^n)=y]\geq \frac{1}{|Y|}$ - -### Modular arithmetic - -For $a,b\in \mathbb{Z}$, $N\in \mathbb{Z}^2$ - -$a\equiv b \mod N\iff N|(a-b)\iff \exists k\in \mathbb{Z}, a-b=kN,a=kN+b$ - -Ex: $N=23$, $-20\equiv 3\equiv 26\equiv 49\equiv 72\mod 23$. - -#### Equivalent relations for any $N$ on $\mathbb{Z}$ - -$a\equiv a\mod N$ - -$a\equiv b\mod N\iff b\equiv a\mod N$ - -$a\equiv b\mod N$ and $b\equiv c\mod N\implies a\equiv c\mod N$ - -#### Division Theorem - -For any $a\in \mathbb{Z}$, and $N\in\mathbb{Z}^+$, $\exists unique\ r,0\leq rb>0)$ - -$gcd(a,b)=gcd(b,a\mod b)$ - -```python -def euclidean_algorithm(a,b): - if ab_3$, and $q_2$ in worst case is $1$, so $b_3<\frac{b_1}{2}$ - -$T(n)=2\Theta(\log b)=O(\log n)$ (linear in size of bits input) - -##### Extended Euclidean algorithm - -Our goal is to find $x,y$ such that $ax+by=gcd(a,b)$ - -Given $a\cdot x\equiv b\mod N$, we do euclidean algorithm to find $gcd(a,b)=d$, then reverse the steps to find $x,y$ such that $ax+by=d$ - -```python -def extended_euclidean_algorithm(a,b): - if a%b==0: return (0,1) - x,y=extended_euclidean_algorithm(b,a%b) - return (y,x-y*(a//b)) -``` - -Example: $a=12,b=43$, $gcd(12,43)=1$ - -$$ -\begin{aligned} -43&=3\cdot 12+7\\ -12&=1\cdot 7+5\\ -7&=1\cdot 5+2\\ -5&=2\cdot 2+1\\ -2&=2\cdot 1+0\\ -1&=1\cdot 5-2\cdot 2\\ -1&=1\cdot 5-2\cdot (7-1\cdot 5)\\ -1&=3\cdot 5-2\cdot 7\\ -1&=3\cdot (12-1\cdot 7)-2\cdot 7\\ -1&=3\cdot 12-5\cdot 7\\ -1&=3\cdot 12-5\cdot (43-3\cdot 12)\\ -1&=-5\cdot 43+18\cdot 12\\ -\end{aligned} -$$ - -So $x=-5,y=18$ +# Lecture 7 + +## Chapter 2: Computational Hardness + +### Letter choosing experiment + +For 100 letter tiles, + +$p_1,...,p_{27}$ (with one blank) + +$(p_1)^2+\dots +(p_{27})^2\geq\frac{1}{27}$ + +For any $p_1,...,p_n$, $0\leq p_i\leq 1$. + +$\sum p_i=1$ + +$P[\text{the same event twice in a row}]=p_1^2+p_2^2....+p_n^2$ + +By Cauchy-Schwarz: $|u\cdot v|^2 \leq ||u||\cdot ||v||^2$. + +let $\vec{u}=(p_1,...,p_n)$, $\vec{v}=(1,..,1)$, so $(p_1^2+p_2^2....+p_n)^2\leq (p_1^2+p_2^2....+p_n^2)\cdot n$. So $p_1^2+p_2^2....+p_n^2\geq \frac{1}{n}$ + +So for an adversary $\mathcal{A}$, who random choose $x'$ and output $f(x')=f(x)$ if matched. $P[f(x)=f(x')]\geq\frac{1}{|Y|}$ + +So $P[x\gets f(x);y=f(x):\mathcal{A}(y,1^n)=y]\geq \frac{1}{|Y|}$ + +### Modular arithmetic + +For $a,b\in \mathbb{Z}$, $N\in \mathbb{Z}^2$ + +$a\equiv b \mod N\iff N|(a-b)\iff \exists k\in \mathbb{Z}, a-b=kN,a=kN+b$ + +Ex: $N=23$, $-20\equiv 3\equiv 26\equiv 49\equiv 72\mod 23$. + +#### Equivalent relations for any $N$ on $\mathbb{Z}$ + +$a\equiv a\mod N$ + +$a\equiv b\mod N\iff b\equiv a\mod N$ + +$a\equiv b\mod N$ and $b\equiv c\mod N\implies a\equiv c\mod N$ + +#### Division Theorem + +For any $a\in \mathbb{Z}$, and $N\in\mathbb{Z}^+$, $\exists unique\ r,0\leq rb>0)$ + +$gcd(a,b)=gcd(b,a\mod b)$ + +```python +def euclidean_algorithm(a,b): + if ab_3$, and $q_2$ in worst case is $1$, so $b_3<\frac{b_1}{2}$ + +$T(n)=2\Theta(\log b)=O(\log n)$ (linear in size of bits input) + +##### Extended Euclidean algorithm + +Our goal is to find $x,y$ such that $ax+by=gcd(a,b)$ + +Given $a\cdot x\equiv b\mod N$, we do euclidean algorithm to find $gcd(a,b)=d$, then reverse the steps to find $x,y$ such that $ax+by=d$ + +```python +def extended_euclidean_algorithm(a,b): + if a%b==0: return (0,1) + x,y=extended_euclidean_algorithm(b,a%b) + return (y,x-y*(a//b)) +``` + +Example: $a=12,b=43$, $gcd(12,43)=1$ + +$$ +\begin{aligned} +43&=3\cdot 12+7\\ +12&=1\cdot 7+5\\ +7&=1\cdot 5+2\\ +5&=2\cdot 2+1\\ +2&=2\cdot 1+0\\ +1&=1\cdot 5-2\cdot 2\\ +1&=1\cdot 5-2\cdot (7-1\cdot 5)\\ +1&=3\cdot 5-2\cdot 7\\ +1&=3\cdot (12-1\cdot 7)-2\cdot 7\\ +1&=3\cdot 12-5\cdot 7\\ +1&=3\cdot 12-5\cdot (43-3\cdot 12)\\ +1&=-5\cdot 43+18\cdot 12\\ +\end{aligned} +$$ + +So $x=-5,y=18$ diff --git a/pages/CSE442T/CSE442T_L8.md b/content/CSE442T/CSE442T_L8.md similarity index 97% rename from pages/CSE442T/CSE442T_L8.md rename to content/CSE442T/CSE442T_L8.md index c41ff30..e594dbd 100644 --- a/pages/CSE442T/CSE442T_L8.md +++ b/content/CSE442T/CSE442T_L8.md @@ -1,74 +1,74 @@ -# Lecture 8 - -## Chapter 2: Computational Hardness - -### Computational number theory/arithmetic - -We want to have a easy-to-use one-way functions for cryptography. - -How to find $a^x\mod N$ quickly. $a,x,N$ are positive integers. We want to reduce $[a\mod N]$ - -Example: $129^{39}\mod 41\equiv (129\mod 41)^{39}\mod 41=6^{39}\mod 41$ - -Find the binary representation of $x$. e.g. express as sums of powers of 2. - -`x=39=bin(1,0,0,1,1,1)` - -Repeatedly square $floor(\log_2(x))$ times. - -$$ -\begin{aligned} - 6^{39}\mod 41&=6^{32+4+2+1}\mod 41\\ - &=(6^{32}\mod 41)(6^{4}\mod 41)(6^{2}\mod 41)(6^{1}\mod 41)\mod 41\\ - &=(-4)(25)(-5)(6)\mod 41\\ - &=7 -\end{aligned} -$$ - -The total multiplication steps is $floor(\log_2(x))$ - -_looks like fast exponentiation right?_ - -Goal: $f_{g,p}(x)=g^x\mod p$ is a one-way function, for certain choice of $p,g$ (and assumptions) - -#### A group (Nice day one for MODERN ALGEBRA) - -A group $G$ is a set with, a binary operation $\oplus$. and $\forall a,b\in G$, $a \oplus b\to c$ - -1. $a,b\in G,a\oplus b\in G$ (closure) -2. $(a\oplus b)\oplus c=a\oplus(b\oplus c)$ (associativity) -3. $\exists e$ such that $\forall a\in G, e\oplus g=g=g\oplus e$ (identity element) -4. $\exists g^{-1}\in G$ such that $g\oplus g^{-1}=e$ (inverse element) - -Example: - -- $\mathbb{Z}_N=\{0,1,2,3,...,N-1\}$ with addition $\mod N$, with identity element $0$. $a\in \mathbb{Z}_N, a^{-1}=N-a$. -- A even simpler group is $\Z$ with addition. -- $\mathbb{Z}_N^*=\{x:x\in \mathbb{Z},1 \leq x\leq N: gcd(x,N)=1\}$ with multiplication $\mod N$ (we can do division here! yeah...). - - If $N=p$ is prime, then $\mathbb{Z}_p^*=\{1,2,3,...,p-1\}$ - - If $N=24$, then $\mathbb{Z}_{24}^*=\{1,5,7,11,13,17,19,23\}$ - - Identity is $1$. - - Let $a\in \mathbb{Z}_N^*$, by Euclidean algorithm, $gcd(a,N)=1$,$\exists x,y \in Z$ such that $ax+Ny=1,ax\equiv 1\mod N,x=a^{-1}$ - - $a,b\in \mathbb{Z}_N^*$. Want to show $gcd(ab,N)=1$. If $gcd(ab,N)=d>1$, then some prime $p|d$. so $p|(a,b)$, which means $p|a$ or $p|b$. In either case, $gcd(a,N)>d$ or $gcd(b,N)>d$, which contradicts that $a,b\in \mathbb{C}_N^*$ - -#### Euler's totient function - -$\phi:\mathbb{Z}^+\to \mathbb{Z}^+,\phi(N)=|\mathbb{Z}_N^*|=|\{1\leq x\leq N:gcd(x,N)=1\}|$ - -Example: $\phi(1)=1$, $\phi(24)=8$, $\phi (p)=p-1$, $\phi(p\cdot q)=(p-1)(q-1)$ - -#### Euler's Theorem - -For any $a\in \mathbb{Z}_N^*$, $a^{\phi(N)}\equiv 1\mod N$ - -Consequence: $a^x\mod N$, $x=K\cdot \phi(N)+r,0\leq r\leq \phi(N)$ - -$$ -a^x\equiv a^{K \cdot \phi (N) +r}\equiv ( a^{\phi(n)} )^K \cdot a^r \mod N$ -$$ - -So computing $a^x\mod N$ is polynomial in $\log (N)$ by reducing $a\mod N$ and $x\mod \phi(N)1$, then some prime $p|d$. so $p|(a,b)$, which means $p|a$ or $p|b$. In either case, $gcd(a,N)>d$ or $gcd(b,N)>d$, which contradicts that $a,b\in \mathbb{C}_N^*$ + +#### Euler's totient function + +$\phi:\mathbb{Z}^+\to \mathbb{Z}^+,\phi(N)=|\mathbb{Z}_N^*|=|\{1\leq x\leq N:gcd(x,N)=1\}|$ + +Example: $\phi(1)=1$, $\phi(24)=8$, $\phi (p)=p-1$, $\phi(p\cdot q)=(p-1)(q-1)$ + +#### Euler's Theorem + +For any $a\in \mathbb{Z}_N^*$, $a^{\phi(N)}\equiv 1\mod N$ + +Consequence: $a^x\mod N$, $x=K\cdot \phi(N)+r,0\leq r\leq \phi(N)$ + +$$ +a^x\equiv a^{K \cdot \phi (N) +r}\equiv ( a^{\phi(n)} )^K \cdot a^r \mod N$ +$$ + +So computing $a^x\mod N$ is polynomial in $\log (N)$ by reducing $a\mod N$ and $x\mod \phi(N)0: - if k==0: - raise ValueError(f"Damn, {i} generates 0 for group {p}") - sg.append(k) - k=(k*i)%p - step-=1 - sg.append(1) - # if len(sg)!=(p-1): continue - g.append((i,[j for j in sg])) - return g -``` - -### (Computational) Diffie-Hellman assumption - -If $p$ is a randomly sampled safe prime. - -Denote safe prime as $\tilde{\Pi}_n=\{p\in \Pi_n:q=\frac{p-1}{2}\in \Pi_{n-1}\}$ - -Then - -$$ -P\left[p\gets \tilde{\Pi_n};a\gets\mathbb{Z}_p^*;g=a^2\neq 1;x\gets \mathbb{Z}_q;y=g^x\mod p:\mathcal{A}(y)=x\right]\leq \epsilon(n) -$$ - -$p\gets \tilde{\Pi_n};a\gets\mathbb{Z}_p^*;g=a^2\neq 1$ is the function condition when we do the encryption on cyclic groups. - -Notes: $f:\Z_q\to \mathbb{Z}_p^*$ is one-to-one, so $f(\mathcal{A}(y))\iff \mathcal{A}(y)=x$ +# Lecture 9 + +## Chapter 2: Computational Hardness + +### Continue on Cyclic groups + +$$ +\begin{aligned} +107^{662}\mod 51&=(107\mod 51)^{662}\mod 51\\ +&=5^{662}\mod 51 +\end{aligned} +$$ + +Remind that $\phi(p),p\in\Pi,\phi(p)=p-1$. + +$51=3\times 17,\phi(51)=\phi(3)\times \phi(17)=2\times 16=32$, So $5^{32}\mod 1$ + +$5^2\equiv 25\mod 51=25$ +$5^4\equiv (5^2)^2\equiv(25)^2 \mod 51\equiv 625\mod 51=13$ +$5^8\equiv (5^4)^2\equiv(13)^2 \mod 51\equiv 169\mod 51=16$ +$5^16\equiv (5^8)^2\equiv(16)^2 \mod 51\equiv 256\mod 51=1$ + +$$ +\begin{aligned} +5^{662}\mod 51&=107^{662\mod 32}\mod 51\\ +&=5^{22}\mod 51\\ +&=5^{16}\cdot 5^4\cdot 5^2\mod 51\\ +&=19 +\end{aligned} +$$ + +For $a\in \mathbb{Z}_N^*$, the order of $a$, $o(a)$ is the smallest positive $k$ such that $a^k\equiv 1\mod N$. $o(a)\leq \phi(N),o(a)|\phi (N)$ + +In a general finite group + +$g^{|G|}=e$ (identity) + +$o(g)\vert |G|$ + +If a group $G=\{a,a^2,a^3,...,e\}$ $G$ is cyclic + +In a cyclic group, if $o(a)=|G|$, then a is a generator of $G$. + +Fact: $\mathbb{Z}^*_p$ is cyclic + +$|\mathbb{Z}^*_p|=p-1$, so $\exists$ generator $g$, and $\mathbb{Z}$, $\phi(\mathbb{Z}_{13}^*)=12$ + +For example, $2$ is a generator for $\mathbb{Z}_{13}^*$ with $2,4,8,3,6,12,11,9,5,10,7,1$. + +If $g$ is a generator, $f:\mathbb{Z}_p^*\to \mathbb{Z}_p^*$, $f(x)=g^x \mod p$ is onto. + +What type of prime $p$? + +- Large prime. +- If $p-1$ is very factorable, that is very bad. + - Pohlig-Hellman algorithm + - $p=2^n+1$ only need polynomial time to invert +- We want $p=2q+1$, where $q$ is prime. (Sophie Germain primes, or safe primes) + +There are _probably_ infinitely many safe prime and efficient to sample as well. + +If $p$ is safe, $g$ generator. + +$$ +\mathbb{Z}_p^*=\{g,g^2,..,e\} +$$ + +Then $\{g^2,...g^{2q}\}S_{g,p}\subseteq \mathbb{Z}_p^*$ is a subgroup; $g^{2k}\cdot g^{2l}=g^{2(k+l)}\in S_{g,p}$ + +It is cyclic with generator $g^2$. + +It is easy to find a generator. + +- Pick $a\in \mathbb{Z}_p^*$ +- Let $x=a^2$. If $x\neq 1$, it is a generator of subgroup $S_p$ + - $S_p=\{x,x^2,...,x^q\}\mod p$ + +Example: $p=2\cdot 11+1=23$ + +we have a subgroup with generator $4$ and $S_4=\{4,16,18,3,12,2,8,9,13,6,1\}$ + +```python +def get_generator(p): + """ + p should be a prime, or you need to do factorization + """ + g=[] + for i in range(2,p-1): + k=i + sg=[] + step=p + while k!=1 and step>0: + if k==0: + raise ValueError(f"Damn, {i} generates 0 for group {p}") + sg.append(k) + k=(k*i)%p + step-=1 + sg.append(1) + # if len(sg)!=(p-1): continue + g.append((i,[j for j in sg])) + return g +``` + +### (Computational) Diffie-Hellman assumption + +If $p$ is a randomly sampled safe prime. + +Denote safe prime as $\tilde{\Pi}_n=\{p\in \Pi_n:q=\frac{p-1}{2}\in \Pi_{n-1}\}$ + +Then + +$$ +P\left[p\gets \tilde{\Pi_n};a\gets\mathbb{Z}_p^*;g=a^2\neq 1;x\gets \mathbb{Z}_q;y=g^x\mod p:\mathcal{A}(y)=x\right]\leq \epsilon(n) +$$ + +$p\gets \tilde{\Pi_n};a\gets\mathbb{Z}_p^*;g=a^2\neq 1$ is the function condition when we do the encryption on cyclic groups. + +Notes: $f:\Z_q\to \mathbb{Z}_p^*$ is one-to-one, so $f(\mathcal{A}(y))\iff \mathcal{A}(y)=x$ diff --git a/pages/CSE442T/Exam_reviews/CSE442T_E1.md b/content/CSE442T/Exam_reviews/CSE442T_E1.md similarity index 96% rename from pages/CSE442T/Exam_reviews/CSE442T_E1.md rename to content/CSE442T/Exam_reviews/CSE442T_E1.md index 53ff284..acf6dd3 100644 --- a/pages/CSE442T/Exam_reviews/CSE442T_E1.md +++ b/content/CSE442T/Exam_reviews/CSE442T_E1.md @@ -1,215 +1,215 @@ -# System check for exam list - -**The exam will take place in class on Monday, October 21.** - -The topics will cover Chapters 1 and 2, as well as the related probability discussions we've had (caveats below).  Assignments 1 through 3 span this material. - -## Specifics on material: - -NOT "match-making game" in 1.2 (seems fun though) - -NOT the proof of Theorem 31.3 (but definitely the result!) - -NOT 2.4.3 (again, definitely want to know this result, and we have discussed the idea behind it) - -NOT 2.6.5, 2.6.6 - -NOT 2.12, 2.13 - -The probability knowledge/techniques I've expanded on include conditional probability, independence, law of total probability, Bayes' Theorem, union bound, 1-p bound (or "useful bound"), collision - -I expect you to demonstrate understanding of the key definitions, theorems, and proof techniques.  The assignments are designed to reinforce all of these.  However, exam questions will be written with the understanding of the time limitations. - -The exam is "closed-book," with no notes of any kind allowed.  The advantage of this is that some questions might be very basic.  However, I will expect that you will have not just memorized definitions and theorems, but you can also explain their meaning and apply them. - -## Chapter 1 - -### Prove security - -#### Definition 11.1 Shannon secrecy - -$(\mathcal{M},\mathcal{K}, Gen, Enc, Dec)$ (A crypto-system) is said to be private-key encryption scheme that is *Shannon-secrete with respect to distribution $D$ over the message space $\mathcal{M}$* if for all $m'\in \mathcal{M}$ and for all $c$, - -$$ -P[k\gets Gen;m\gets D:m=m'|Enc_k(m)=c]=P[m\gets D:m=m'] -$$ - -(The adversary cannot learn all, part of, any letter of, any function off, or any partial information about the plaintext) - -#### Definition 11.2 Perfect Secrecy - -$(\mathcal{M},\mathcal{K}, Gen, ENc, Dec)$ (A crypto-system) is said to be private-key encryption scheme that is *perfectly secret* if forall $m_1,m_2\in \mathcal{M},\forall c$: - -$$ -P[k\gets Gen:Enc_k(m_1)=c]=P[k\gets Gen:Enc_k(m_2)=c] -$$ - -(For all coding scheme in the crypto system, for any two different message, they are equally likely to be mapped to $c$) - -#### Definition 12.3 - -A private-key encryption scheme is perfectly secret if and only if it is Shannon secret. - -## Chapter 2 - -### Efficient Private-key Encryption - -#### Definition 24.7 - -A triplet of algorithms $(Gen,Enc,Dec)$ is called an efficient private-key encryption scheme if the following holds. - -1. $k\gets Gen(1^n)$ is a p.p.t. such that for every $n\in \mathbb{N}$, it samples a key $k$. -2. $c\gets Enc_k(m)$ is a p.p.t. that given $k$ and $m\in \{0,1\}^n$ produces a ciphertext $c$. -3. $m\gets Dec_c(c)$ is a p.p.t. that given a ciphertext $c$ and key $k$ produces a message $m\in \{0,1\}^n\cup \perp$. -4. For all $n\in \mathbb{N},m\in \{0,1\}^n$ - -$$ -Pr[k\gets Gen(1^n);Dec_k(Enc_k(m))=m]=1 -$$ - -### One-Way functions - -#### Definition 26.1 - -A function $f:\{0,1\}^*\to\{0,1\}^*$ is worst-case one-way if the function is: - -1. Easy to compute. There is a p.p.t $C$ that computes $f(x)$ on all inputs $x\in \{0,1\}^*$, and -2. Hard to invert. There is no adversary $\mathcal{A}$ such that - -$$ -\forall x,P[\mathcal{A}(f(x))\in f^{-1}(f(x))]=1 -$$ - -#### Definition 27.2 Negligible function - -A function $\epsilon(n)$ is negligible if for every $c$. there exists some $n_0$ such that for all $n>n_0$, $\epsilon (n)\leq \frac{1}{n^c}$. - -#### Definition 27.3 Strong One-Way Function - -A function mapping strings to strings $f:\{0,1\}^*\to \{0,1\}^*$ is a strong one-way function if it satisfies the following two conditions: - -1. Easy to compute. There is a p.p.t $C$ that computes $f(x)$ on all inputs $x\in \{0,1\}^*$, and -2. Hard to invert. There is no adversary $\mathcal{A}$ such that - -$$ -P[x\gets\{0,1\}^n;y\gets f(x):f(\mathcal{A}(1^n,y))=y]\leq \epsilon(n) -$$ - -#### Definition 28.4 (Weak One-Way Function) - -A function mapping strings to strings $f:\{0,1\}^*\to \{0,1\}^*$ is a strong one-way function if it satisfies the following two conditions: - -1. Easy to compute. There is a p.p.t $C$ that computes $f(x)$ on all inputs $x\in \{0,1\}^*$, and -2. Hard to invert. There is no adversary $\mathcal{A}$ such that - -$$ -P[x\gets\{0,1\}^n;y\gets f(x):f(\mathcal{A}(1^n,y))=y]\leq 1-\frac{1}{q(n)} -$$ - -#### Notation for prime numbers - -Denote the (finite) set of primes that are smaller than $2^n$ as - -$$ -\Pi_n=\{q|q<2^n\textup{ and } q \textup{ is prime}\} -$$ - -#### Assumption 30.1 (Factoring) - -For every adversary $\mathcal{A}$, there exists a negligible function $\epsilon$ such that - -$$ -P[p\gets \Pi_n;q\gets \Pi_n;N\gets pq:\mathcal{A}(N)\in \{p,q\}]<\epsilon(n) -$$ - -(For every product of random 2 primes, the probability for any adversary to find the prime factors is negligible.) - -(There is no polynomial function that can decompose the product of two $n$ bit prime, the best function is $2^{O(n^{\frac{1}{3}}\log^{\frac{2}{3}}n)}$) - -#### Theorem 35.1 - -For any weak one-way function $f:\{0,1\}^n\to \{0,1\}^*$, there exists a polynomial $m(\cdot)$ such that function - -$$ -f'(x_1,x_2,\dots, x_{m(n)})=(f(x_1),f(x_2),\dots, f(x_{m(n)})). -$$ - -from $f'=(\{0,1\}^n)^{m(n)}\to(\{0,1\}^*)^{m(n)}$ is strong one-way. - -### RSA - -#### Definition 46.7 - -A group $G$ is a set of elements with a binary operator $\oplus:G\times G\to G$ that satisfies the following properties - -1. Closure: $\forall a,b\in G, a\oplus b\in G$ -2. Identity: $\exists i\in G$ such that $\forall a\in G, i\oplus a=a\oplus i=a$ -3. Associativity: $\forall a,b,c\in G,(a\oplus b)\oplus c=a\oplus(b\oplus c)$. -4. Inverse: $\forall a\in G$, there is an element $b\in G$ such that $a\oplus b=b\oplus a=i$ - -#### Definition Euler totient function $\Phi(N)$. - -$$ -\Phi(p)=p-1 -$$ - -if $p$ is prime - -$$ -\Phi(N)=(p-1)(q-1) -$$ - -if $N=pq$ and $p,q$ are primes - -#### Theorem 47.10 - -$\forall a\in \mathbb{Z}_N^*,a^{\Phi(N)}=1\mod N$ - -#### Corollary 48.11 - -$\forall a\in \mathbb{Z}_p^*,a^{p-1}\equiv 1\mod p$. - -#### Corollary 48.12 - -$a^x\mod N=a^{x\mod \Phi(N)}\mod N$ - -## Some other important results - -### Exponent - -$$ -(1-\frac{1}{n})^n\approx e -$$ -when $n$ is large. - -### Primes - -Let $\pi(x)$ be the lower-bounds for prime less than or equal to $x$. - -#### Theorem 31.3 Chebyshev - -For $x>1$,$\pi(x)>\frac{x}{2\log x}$ - -#### Corollary 31.3 - -For $2^n>1$, $p(n)>\frac{1}{n}$ - -(The probability that a uniformly sampled n-bit integer is prime is greater than $\frac{1}{n}$) - -### Modular Arithmetic - -#### Extended Euclid Algorithm - -```python -def eea(a,b)->tuple(int): - # assume a>b - # return x,y such that ax+by=gcd(a,b)=d. - # so y is the modular inverse of b mod a - # so x is the modular inverse of a mod b - # so gcd(a,b)=ax+by - if a%b==0: - return (0,1) - x,y=eea(b,a%b) - return (y,x-y(a//b)) -``` - +# System check for exam list + +**The exam will take place in class on Monday, October 21.** + +The topics will cover Chapters 1 and 2, as well as the related probability discussions we've had (caveats below).  Assignments 1 through 3 span this material. + +## Specifics on material: + +NOT "match-making game" in 1.2 (seems fun though) + +NOT the proof of Theorem 31.3 (but definitely the result!) + +NOT 2.4.3 (again, definitely want to know this result, and we have discussed the idea behind it) + +NOT 2.6.5, 2.6.6 + +NOT 2.12, 2.13 + +The probability knowledge/techniques I've expanded on include conditional probability, independence, law of total probability, Bayes' Theorem, union bound, 1-p bound (or "useful bound"), collision + +I expect you to demonstrate understanding of the key definitions, theorems, and proof techniques.  The assignments are designed to reinforce all of these.  However, exam questions will be written with the understanding of the time limitations. + +The exam is "closed-book," with no notes of any kind allowed.  The advantage of this is that some questions might be very basic.  However, I will expect that you will have not just memorized definitions and theorems, but you can also explain their meaning and apply them. + +## Chapter 1 + +### Prove security + +#### Definition 11.1 Shannon secrecy + +$(\mathcal{M},\mathcal{K}, Gen, Enc, Dec)$ (A crypto-system) is said to be private-key encryption scheme that is *Shannon-secrete with respect to distribution $D$ over the message space $\mathcal{M}$* if for all $m'\in \mathcal{M}$ and for all $c$, + +$$ +P[k\gets Gen;m\gets D:m=m'|Enc_k(m)=c]=P[m\gets D:m=m'] +$$ + +(The adversary cannot learn all, part of, any letter of, any function off, or any partial information about the plaintext) + +#### Definition 11.2 Perfect Secrecy + +$(\mathcal{M},\mathcal{K}, Gen, ENc, Dec)$ (A crypto-system) is said to be private-key encryption scheme that is *perfectly secret* if forall $m_1,m_2\in \mathcal{M},\forall c$: + +$$ +P[k\gets Gen:Enc_k(m_1)=c]=P[k\gets Gen:Enc_k(m_2)=c] +$$ + +(For all coding scheme in the crypto system, for any two different message, they are equally likely to be mapped to $c$) + +#### Definition 12.3 + +A private-key encryption scheme is perfectly secret if and only if it is Shannon secret. + +## Chapter 2 + +### Efficient Private-key Encryption + +#### Definition 24.7 + +A triplet of algorithms $(Gen,Enc,Dec)$ is called an efficient private-key encryption scheme if the following holds. + +1. $k\gets Gen(1^n)$ is a p.p.t. such that for every $n\in \mathbb{N}$, it samples a key $k$. +2. $c\gets Enc_k(m)$ is a p.p.t. that given $k$ and $m\in \{0,1\}^n$ produces a ciphertext $c$. +3. $m\gets Dec_c(c)$ is a p.p.t. that given a ciphertext $c$ and key $k$ produces a message $m\in \{0,1\}^n\cup \perp$. +4. For all $n\in \mathbb{N},m\in \{0,1\}^n$ + +$$ +Pr[k\gets Gen(1^n);Dec_k(Enc_k(m))=m]=1 +$$ + +### One-Way functions + +#### Definition 26.1 + +A function $f:\{0,1\}^*\to\{0,1\}^*$ is worst-case one-way if the function is: + +1. Easy to compute. There is a p.p.t $C$ that computes $f(x)$ on all inputs $x\in \{0,1\}^*$, and +2. Hard to invert. There is no adversary $\mathcal{A}$ such that + +$$ +\forall x,P[\mathcal{A}(f(x))\in f^{-1}(f(x))]=1 +$$ + +#### Definition 27.2 Negligible function + +A function $\epsilon(n)$ is negligible if for every $c$. there exists some $n_0$ such that for all $n>n_0$, $\epsilon (n)\leq \frac{1}{n^c}$. + +#### Definition 27.3 Strong One-Way Function + +A function mapping strings to strings $f:\{0,1\}^*\to \{0,1\}^*$ is a strong one-way function if it satisfies the following two conditions: + +1. Easy to compute. There is a p.p.t $C$ that computes $f(x)$ on all inputs $x\in \{0,1\}^*$, and +2. Hard to invert. There is no adversary $\mathcal{A}$ such that + +$$ +P[x\gets\{0,1\}^n;y\gets f(x):f(\mathcal{A}(1^n,y))=y]\leq \epsilon(n) +$$ + +#### Definition 28.4 (Weak One-Way Function) + +A function mapping strings to strings $f:\{0,1\}^*\to \{0,1\}^*$ is a strong one-way function if it satisfies the following two conditions: + +1. Easy to compute. There is a p.p.t $C$ that computes $f(x)$ on all inputs $x\in \{0,1\}^*$, and +2. Hard to invert. There is no adversary $\mathcal{A}$ such that + +$$ +P[x\gets\{0,1\}^n;y\gets f(x):f(\mathcal{A}(1^n,y))=y]\leq 1-\frac{1}{q(n)} +$$ + +#### Notation for prime numbers + +Denote the (finite) set of primes that are smaller than $2^n$ as + +$$ +\Pi_n=\{q|q<2^n\textup{ and } q \textup{ is prime}\} +$$ + +#### Assumption 30.1 (Factoring) + +For every adversary $\mathcal{A}$, there exists a negligible function $\epsilon$ such that + +$$ +P[p\gets \Pi_n;q\gets \Pi_n;N\gets pq:\mathcal{A}(N)\in \{p,q\}]<\epsilon(n) +$$ + +(For every product of random 2 primes, the probability for any adversary to find the prime factors is negligible.) + +(There is no polynomial function that can decompose the product of two $n$ bit prime, the best function is $2^{O(n^{\frac{1}{3}}\log^{\frac{2}{3}}n)}$) + +#### Theorem 35.1 + +For any weak one-way function $f:\{0,1\}^n\to \{0,1\}^*$, there exists a polynomial $m(\cdot)$ such that function + +$$ +f'(x_1,x_2,\dots, x_{m(n)})=(f(x_1),f(x_2),\dots, f(x_{m(n)})). +$$ + +from $f'=(\{0,1\}^n)^{m(n)}\to(\{0,1\}^*)^{m(n)}$ is strong one-way. + +### RSA + +#### Definition 46.7 + +A group $G$ is a set of elements with a binary operator $\oplus:G\times G\to G$ that satisfies the following properties + +1. Closure: $\forall a,b\in G, a\oplus b\in G$ +2. Identity: $\exists i\in G$ such that $\forall a\in G, i\oplus a=a\oplus i=a$ +3. Associativity: $\forall a,b,c\in G,(a\oplus b)\oplus c=a\oplus(b\oplus c)$. +4. Inverse: $\forall a\in G$, there is an element $b\in G$ such that $a\oplus b=b\oplus a=i$ + +#### Definition Euler totient function $\Phi(N)$. + +$$ +\Phi(p)=p-1 +$$ + +if $p$ is prime + +$$ +\Phi(N)=(p-1)(q-1) +$$ + +if $N=pq$ and $p,q$ are primes + +#### Theorem 47.10 + +$\forall a\in \mathbb{Z}_N^*,a^{\Phi(N)}=1\mod N$ + +#### Corollary 48.11 + +$\forall a\in \mathbb{Z}_p^*,a^{p-1}\equiv 1\mod p$. + +#### Corollary 48.12 + +$a^x\mod N=a^{x\mod \Phi(N)}\mod N$ + +## Some other important results + +### Exponent + +$$ +(1-\frac{1}{n})^n\approx e +$$ +when $n$ is large. + +### Primes + +Let $\pi(x)$ be the lower-bounds for prime less than or equal to $x$. + +#### Theorem 31.3 Chebyshev + +For $x>1$,$\pi(x)>\frac{x}{2\log x}$ + +#### Corollary 31.3 + +For $2^n>1$, $p(n)>\frac{1}{n}$ + +(The probability that a uniformly sampled n-bit integer is prime is greater than $\frac{1}{n}$) + +### Modular Arithmetic + +#### Extended Euclid Algorithm + +```python +def eea(a,b)->tuple(int): + # assume a>b + # return x,y such that ax+by=gcd(a,b)=d. + # so y is the modular inverse of b mod a + # so x is the modular inverse of a mod b + # so gcd(a,b)=ax+by + if a%b==0: + return (0,1) + x,y=eea(b,a%b) + return (y,x-y(a//b)) +``` + diff --git a/pages/CSE442T/Exam_reviews/CSE442T_E2.md b/content/CSE442T/Exam_reviews/CSE442T_E2.md similarity index 100% rename from pages/CSE442T/Exam_reviews/CSE442T_E2.md rename to content/CSE442T/Exam_reviews/CSE442T_E2.md diff --git a/pages/CSE442T/Exam_reviews/_meta.js b/content/CSE442T/Exam_reviews/_meta.js similarity index 100% rename from pages/CSE442T/Exam_reviews/_meta.js rename to content/CSE442T/Exam_reviews/_meta.js diff --git a/pages/CSE442T/_meta.js b/content/CSE442T/_meta.js similarity index 97% rename from pages/CSE442T/_meta.js rename to content/CSE442T/_meta.js index 172950f..1511a17 100644 --- a/pages/CSE442T/_meta.js +++ b/content/CSE442T/_meta.js @@ -1,5 +1,5 @@ export default { - index: "Course Description", + //index: "Course 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index: "Course Description", + //index: "Course Description", "---": { type: 'separator' }, diff --git a/pages/Math3200/index.md b/content/Math3200/index.md similarity index 100% rename from pages/Math3200/index.md rename to content/Math3200/index.md diff --git a/pages/Math401/Math401_N1.md b/content/Math401/Math401_N1.md similarity index 100% rename from pages/Math401/Math401_N1.md rename to content/Math401/Math401_N1.md diff --git a/pages/Math401/Math401_N2.md b/content/Math401/Math401_N2.md similarity index 100% rename from pages/Math401/Math401_N2.md rename to content/Math401/Math401_N2.md diff --git a/pages/Math401/Math401_N3.md b/content/Math401/Math401_N3.md similarity index 100% rename from pages/Math401/Math401_N3.md rename to content/Math401/Math401_N3.md diff --git a/pages/Math401/Math401_P1.md b/content/Math401/Math401_P1.md similarity index 100% rename from pages/Math401/Math401_P1.md rename to content/Math401/Math401_P1.md diff --git a/pages/Math401/Math401_P1_1.md 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b/content/Math4111/Exam_reviews/Math4111_E2.md similarity index 100% rename from pages/Math4111/Exam_reviews/Math4111_E2.md rename to content/Math4111/Exam_reviews/Math4111_E2.md diff --git a/pages/Math4111/Exam_reviews/Math4111_E3.md b/content/Math4111/Exam_reviews/Math4111_E3.md similarity index 100% rename from pages/Math4111/Exam_reviews/Math4111_E3.md rename to content/Math4111/Exam_reviews/Math4111_E3.md diff --git a/pages/Math4111/Exam_reviews/Math4111_Final.md b/content/Math4111/Exam_reviews/Math4111_Final.md similarity index 100% rename from pages/Math4111/Exam_reviews/Math4111_Final.md rename to content/Math4111/Exam_reviews/Math4111_Final.md diff --git a/pages/Math4111/Math4111_L1.md b/content/Math4111/Math4111_L1.md similarity index 96% rename from pages/Math4111/Math4111_L1.md rename to content/Math4111/Math4111_L1.md index 01215b2..fea3b34 100644 --- a/pages/Math4111/Math4111_L1.md +++ b/content/Math4111/Math4111_L1.md @@ -1,97 +1,97 @@ -# Lecture 1 - -## Introduction - -Reading is not recommended before class, it's hard. - -## Chapter 1: The real number and complex number systems - -* Natural numbers: $\mathbb{N}=\{1,2,3,4....\}$ note by some conventions, $0$ is also a natural number - -* IntegersL $\mathbb{Z}=\{...,-2,-1,0,1,2,...\}$ - -* Rational numbers: $\mathbb{Q}=\{\frac{m}{n}:m,n\in\mathbb{Z}\ and\ n\neq 0\}$ - -* Real numbers: $\mathbb{R}$ the topic of chapter - -* Complex numbers: $\mathbb{C}=\{a+bi:a,b\in \mathbb{R}\}$ - -### Theorem ($\sqrt{2}$ is irrational) - -$\exist p\in \mathbb{Q},p^2=2$ is false. - -$\equiv\cancel{\exist} p\in \mathbb{Q}, p^2=2$ - -$\equiv p\in \mathbb{Q},p^2\neq 2$ - -#### Proof - -Suppose for contradiction, $\exist p\in \mathbb{Q}$ such that $p^2=\mathbb{Q}$. - -Let $p=\frac{m}{n}$, where $m,n \in \mathbb{Z}$ are not both even. (reduced form) - -$p^2=2$ and $p=\frac{m}{n}$, so $m^2=2n^2$, so $m^2$ is even, $m$ is even. - -So $m^2$ is divisible by 4, $2n^2$ is divisible by 4. - -So $n^2$ is even. but they are not both even. - -QED - -### Theorem (No closest rational for a irrational number) - -Let $A=\{p\in \mathbb{q}, p>0\ and\ p^2\leq 2\}$, Then $A$ does not have a largest element. - -i.e. $\exist p\in A$ such that $\forall q\in A, q\leq p$ is false. - -> Remark: The book give a very slick proof trying to lean from these kinds of proofs takes some effort. (It is perfectly fine to write that solution this way...) - -#### Thought process - -Let $p\in A,p\in \mathbb{Q}$, $p>0, p^2<2$. - -We want a $\delta\in\mathbb{Q}$ such that $\delta>0$ and $(p+\delta)^2<2$. - -$$ -\begin{aligned} - -(p+\delta)^2&<2\\ -p^2+2p\delta+\delta^2&<2\\ -\delta(2p+\delta)&< 2-p^2\\ -\delta&<\frac{2-p^2}{2p-\delta} -\end{aligned} -$$ - -From $(p+\delta)^2<2$, we know $\delta<2$ (this is a crude bound, $\delta<\sqrt{2}$). - -So one choice can be $\delta=\frac{2-p^2}{2p+2}$ - -#### Proof - -$\forall p\in A$, we can find a $\delta=\frac{2-p^2}{2p+2}$ which is greater than zero ($p^2<2,2-p^2>0,2p+2>0,\delta>0$) and construct a new number $(p+\delta)^2$ such that $p^2<(p+\delta)^2<2$. - -_Here we construct a formula for approximate $\sqrt{2}=\lim_{i\to \infty}p_0=1,p_{i+1}=p_i+\frac{2-p_i^2}{2p_i+2}$_ - -Interesting... - -We can also further optimize the formula by changing the bound of $\delta$ to $\delta< 2-p$, since $(p+\delta)^2<2,p+\delta<2$ - -```python -def sqrt_2(acc): - if acc==0: return 1 - c=sqrt_2(n-1) - return c+((2-c**2)/(2*c+2)) -``` - -### Definition and notations for sets - -Some set notation - -$\Pi\in \mathbb{R}$ - -use $\subset,\subsetneq$ in this class. - -* $A\subset B$, $\forall x\in A, x\in B$ -* $A=B$, $A\subset B$ and $B\subset A$ -* $A\subsetneq$ means $A\subset B$ and $A\neq B$ - +# Lecture 1 + +## Introduction + +Reading is not recommended before class, it's hard. + +## Chapter 1: The real number and complex number systems + +* Natural numbers: $\mathbb{N}=\{1,2,3,4....\}$ note by some conventions, $0$ is also a natural number + +* IntegersL $\mathbb{Z}=\{...,-2,-1,0,1,2,...\}$ + +* Rational numbers: $\mathbb{Q}=\{\frac{m}{n}:m,n\in\mathbb{Z}\ and\ n\neq 0\}$ + +* Real numbers: $\mathbb{R}$ the topic of chapter + +* Complex numbers: $\mathbb{C}=\{a+bi:a,b\in \mathbb{R}\}$ + +### Theorem ($\sqrt{2}$ is irrational) + +$\exist p\in \mathbb{Q},p^2=2$ is false. + +$\equiv\cancel{\exist} p\in \mathbb{Q}, p^2=2$ + +$\equiv p\in \mathbb{Q},p^2\neq 2$ + +#### Proof + +Suppose for contradiction, $\exist p\in \mathbb{Q}$ such that $p^2=\mathbb{Q}$. + +Let $p=\frac{m}{n}$, where $m,n \in \mathbb{Z}$ are not both even. (reduced form) + +$p^2=2$ and $p=\frac{m}{n}$, so $m^2=2n^2$, so $m^2$ is even, $m$ is even. + +So $m^2$ is divisible by 4, $2n^2$ is divisible by 4. + +So $n^2$ is even. but they are not both even. + +QED + +### Theorem (No closest rational for a irrational number) + +Let $A=\{p\in \mathbb{q}, p>0\ and\ p^2\leq 2\}$, Then $A$ does not have a largest element. + +i.e. $\exist p\in A$ such that $\forall q\in A, q\leq p$ is false. + +> Remark: The book give a very slick proof trying to lean from these kinds of proofs takes some effort. (It is perfectly fine to write that solution this way...) + +#### Thought process + +Let $p\in A,p\in \mathbb{Q}$, $p>0, p^2<2$. + +We want a $\delta\in\mathbb{Q}$ such that $\delta>0$ and $(p+\delta)^2<2$. + +$$ +\begin{aligned} + +(p+\delta)^2&<2\\ +p^2+2p\delta+\delta^2&<2\\ +\delta(2p+\delta)&< 2-p^2\\ +\delta&<\frac{2-p^2}{2p-\delta} +\end{aligned} +$$ + +From $(p+\delta)^2<2$, we know $\delta<2$ (this is a crude bound, $\delta<\sqrt{2}$). + +So one choice can be $\delta=\frac{2-p^2}{2p+2}$ + +#### Proof + +$\forall p\in A$, we can find a $\delta=\frac{2-p^2}{2p+2}$ which is greater than zero ($p^2<2,2-p^2>0,2p+2>0,\delta>0$) and construct a new number $(p+\delta)^2$ such that $p^2<(p+\delta)^2<2$. + +_Here we construct a formula for approximate $\sqrt{2}=\lim_{i\to \infty}p_0=1,p_{i+1}=p_i+\frac{2-p_i^2}{2p_i+2}$_ + +Interesting... + +We can also further optimize the formula by changing the bound of $\delta$ to $\delta< 2-p$, since $(p+\delta)^2<2,p+\delta<2$ + +```python +def sqrt_2(acc): + if acc==0: return 1 + c=sqrt_2(n-1) + return c+((2-c**2)/(2*c+2)) +``` + +### Definition and notations for sets + +Some set notation + +$\Pi\in \mathbb{R}$ + +use $\subset,\subsetneq$ in this class. + +* $A\subset B$, $\forall x\in A, x\in B$ +* $A=B$, $A\subset B$ and $B\subset A$ +* $A\subsetneq$ means $A\subset B$ and $A\neq B$ + diff --git a/pages/Math4111/Math4111_L10.md b/content/Math4111/Math4111_L10.md similarity index 98% rename from pages/Math4111/Math4111_L10.md rename to content/Math4111/Math4111_L10.md index dc82dd3..1debee9 100644 --- a/pages/Math4111/Math4111_L10.md +++ b/content/Math4111/Math4111_L10.md @@ -1,66 +1,66 @@ -# Lecture 10 - -## Review - -Recall: If $K\subset \cup_{\alpha\in A} G_{\alpha}$, then we say $\{G_\alpha\}_{\alpha\in A}$ is a cover of $K$. If, in addition, each set $G_{\alpha}$ is open, then we say $\{G_{\alpha}\}_{\alpha\in A}$ is an open cover of $K$. If $\alpha_1,...,\alpha_n\in A$ are such that $K\subset \bigcup _{i=1}^n G_{\alpha_i}$, then we say $\{G_{\alpha_i}\}_{i=1}^n$ is a finite subcover. - -Let $X=\mathbb{R}$. Come up with some examples of covers of $\mathbb{R}$. Try to find a few satisfying each of the following: - -1. A cover of $\mathbb{R}$ which is not an open cover. - $\{[x,x+1]:x\in \mathbb{Z}\}$ -2. An open cover of $\mathbb{R}$ which does have a finite subcover. - $\{\mathbb{R}\}$ is an open cover with finite subcover, itself $\{\mathbb{R}\}$. AND, $\{\mathbb{Q},\mathbb{R}\backslash\mathbb{Q}\}$ is not a subcover of $\{\mathbb{R}\}$ since we need to select subcover from cover set. And not taking the element of sets in the open cover. -3. An open cover of $\mathbb{R}$ which does not have a finite subcover. - $\{(x,x+2):x\in \mathbb{Z}\}$ No finite subcover, infinitely many sets. - - Proof: we proceed by contradiction, suppose we take $\{(n_i,n_i+2):i=1,...,k\}$. The union does not contain $max\{n_1,...,n_k\}+2$ - - $\{\{x\in\mathbb{R}:x0$ such that $B_r(p)0$ such that $B_r(p)0$ such that $S\subset B_r(p)$. - -Consider the following statement: If a set $S\subset X$ is compact, the its is bounded. - -1. Will the proof of this statement involve an arbitrary open cover (one that you, the prover, do not get to choose) or a specific open cover (one that you can choose)? - We should choose a specific cover so that we can construct cover that have a set that is a superset of $S$. -2. Give a proof of the statement. [Suggestion: If you prefer, you could try proving the contrapositive. Both a direct proof and a proof by contrapositive are roughly of the same difficulty.] - -### Continue on compact sets - -#### Lemma - -If $S\in X$ is compact, then $S$ is bounded. - -Proof: - -Fix $p\in X$, then $\{B_n(p)\}_{n\in \mathbb{N}}$ (specific open cover) is an open cover of $S$ (Since $\bigcup_{n\in \mathbb{N}}=X$). Since $S$ is compact, then $\exists$ a finite subcover ${n\in \mathbb{N}}_{i=1}^k=S$, let $r=max(n_1,...n_k)$, Then $S\subset B_r(p)$ - -QED - -#### Definition k-cell - -A 2-cell is a set of the form $[a_1,b_1]\times[a_2,b_2]$ - -#### Theorem 2.39 (K-dimension of Theorem) - -Theorem 2.38 replace with "closed and bounded intervals" to "k-cells". - -Ideas of Proof: - -Apply the Theorem to each dimension separately. - -#### Theorem 2.40 - -Every k-cell is compact. - -We'll prove the case $k=1$ and $I=[0,1]$ (This is to simplify notation. This same ideas are used in the general case) - -Proof: - -That $[0,1]$ is compact. - -(Key idea, divide and conquer) - -Suppose for contradiction that $\exists$ open cover $\{G_a\}_{\alpha\in A}$ of $[0,1]$ with no finite subcovers of $[0,1]$ - -**Step1.** Divide $[0,1]$ in half. $[0,\frac{1}{2}]$ and $[\frac{1}{2},1]$ and at least one of the subintervals cannot be covered by a finite subcollection of $\{G_\alpha\}_{\alpha\in A}$ - -(If both of them could be, combine the two finite subcollections to get a finite subcover of $[0,1]$) - -Let $I_1$ be a subinterval without a finite subcover. - -**Step2.** Divide $I_1$ in half. Let $I_2$ be one of these two subintervals of $I_1$ without a finite subcover. - -**Step3.** etc. - -We obtain a seg of intervals $I_1\subset I_2\subset \dots$ such that - -(a) $[0,1]\supset I_1\supset I_2\supset \dots$ -(b) $\forall n\in \mathbb{N}$, $I_n$ is not covered by a finite subcollection of $\{G_\alpha\}_{\alpha\in A}$ -(c) The length of $I_n$ is $\frac{1}{2^n}$ - -By (a) and **Theorem 2.38**, $\exists x^*\in \bigcap^{\infty}_{n=1} I_n$. - -Since $x^*\in [0,1]$, $\exists \alpha_0$ such that $x^*\in G_{\alpha_0}$ - -Since $G_{\alpha_0}$ is open, $\exist r>0$ such that $B_r(x^*)\subset G_{\alpha_0}$ - -Let $n\in \mathbb{N}$ be such that $\frac{1}{2^n}d(y,z)-r\\ - &=d(y,z)-\frac{1}{2}d(y,z)\\ - &=\frac{1}{2}d(y,z) -\end{aligned} -$$ - -So $B_r(y)\cap S$ is finite. By **Theorem 2.20**, $y\notin S$, this proves the claim so $S'\cap E=\phi$ - -QED +# Lecture 12 + +## Review Questions + +For a metric space $(X,d)$, we say a subset $S\subset X$ si bounded if there exists $p\in X$ and $r>0$ such that $S\subset B_r(p)$. + +Consider the following statement: If a set $S\subset X$ is compact, the its is bounded. + +1. Will the proof of this statement involve an arbitrary open cover (one that you, the prover, do not get to choose) or a specific open cover (one that you can choose)? + We should choose a specific cover so that we can construct cover that have a set that is a superset of $S$. +2. Give a proof of the statement. [Suggestion: If you prefer, you could try proving the contrapositive. Both a direct proof and a proof by contrapositive are roughly of the same difficulty.] + +### Continue on compact sets + +#### Lemma + +If $S\in X$ is compact, then $S$ is bounded. + +Proof: + +Fix $p\in X$, then $\{B_n(p)\}_{n\in \mathbb{N}}$ (specific open cover) is an open cover of $S$ (Since $\bigcup_{n\in \mathbb{N}}=X$). Since $S$ is compact, then $\exists$ a finite subcover ${n\in \mathbb{N}}_{i=1}^k=S$, let $r=max(n_1,...n_k)$, Then $S\subset B_r(p)$ + +QED + +#### Definition k-cell + +A 2-cell is a set of the form $[a_1,b_1]\times[a_2,b_2]$ + +#### Theorem 2.39 (K-dimension of Theorem) + +Theorem 2.38 replace with "closed and bounded intervals" to "k-cells". + +Ideas of Proof: + +Apply the Theorem to each dimension separately. + +#### Theorem 2.40 + +Every k-cell is compact. + +We'll prove the case $k=1$ and $I=[0,1]$ (This is to simplify notation. This same ideas are used in the general case) + +Proof: + +That $[0,1]$ is compact. + +(Key idea, divide and conquer) + +Suppose for contradiction that $\exists$ open cover $\{G_a\}_{\alpha\in A}$ of $[0,1]$ with no finite subcovers of $[0,1]$ + +**Step1.** Divide $[0,1]$ in half. $[0,\frac{1}{2}]$ and $[\frac{1}{2},1]$ and at least one of the subintervals cannot be covered by a finite subcollection of $\{G_\alpha\}_{\alpha\in A}$ + +(If both of them could be, combine the two finite subcollections to get a finite subcover of $[0,1]$) + +Let $I_1$ be a subinterval without a finite subcover. + +**Step2.** Divide $I_1$ in half. Let $I_2$ be one of these two subintervals of $I_1$ without a finite subcover. + +**Step3.** etc. + +We obtain a seg of intervals $I_1\subset I_2\subset \dots$ such that + +(a) $[0,1]\supset I_1\supset I_2\supset \dots$ +(b) $\forall n\in \mathbb{N}$, $I_n$ is not covered by a finite subcollection of $\{G_\alpha\}_{\alpha\in A}$ +(c) The length of $I_n$ is $\frac{1}{2^n}$ + +By (a) and **Theorem 2.38**, $\exists x^*\in \bigcap^{\infty}_{n=1} I_n$. + +Since $x^*\in [0,1]$, $\exists \alpha_0$ such that $x^*\in G_{\alpha_0}$ + +Since $G_{\alpha_0}$ is open, $\exist r>0$ such that $B_r(x^*)\subset G_{\alpha_0}$ + +Let $n\in \mathbb{N}$ be such that $\frac{1}{2^n}d(y,z)-r\\ + &=d(y,z)-\frac{1}{2}d(y,z)\\ + &=\frac{1}{2}d(y,z) +\end{aligned} +$$ + +So $B_r(y)\cap S$ is finite. By **Theorem 2.20**, $y\notin S$, this proves the claim so $S'\cap E=\phi$ + +QED diff --git a/pages/Math4111/Math4111_L13.md b/content/Math4111/Math4111_L13.md similarity index 97% rename from pages/Math4111/Math4111_L13.md rename to content/Math4111/Math4111_L13.md index 49592a7..e1ac53f 100644 --- a/pages/Math4111/Math4111_L13.md +++ b/content/Math4111/Math4111_L13.md @@ -1,143 +1,143 @@ -# Lecture 13 - -## Review - -Consider the metric space $X=\mathbb{R}$ (with the usual metric $d(x,y)=|x-y|$). Let $E=(0,1)$. - -1. Find several examples of sets $Y\subset \mathbb{R}$ such that $E\subset Y$ and $E$ is closed in $Y$. - Example: - 1. $Y=E$, $E$ is closed in $Y$. - We can prove this using normal ways, or - **Theorem 2.23** $E$ is closed in $Y\iff E^c$ is open in $Y$ - $Y\iff E^c=\phi$ and it's open. - 2. $Y=\mathbb{R}\backslash\{0,1\}$ - $Y\backslash E=(-\infty,0)\cup (1,\infty)$ - **Theorem 2.30** $E\subset Y\subset X$, $E$ is open in $Y\iff$ $\exists G$ open in $X$ such that $G\cap Y=E$ - $G\cap Y=Y\backslash E$ - And we know $Y\backslash E$ is open in $Y$. By **Theorem 2.23** $E$ is closed in $Y$. - -2. If $Y$ is as in part 1, we can conclude that $E$ is closed and bounded in $y$. Part of **Theorem 2.41** says: "If a set is closed and bounded, then it is compact." Why doesn't that theorem apply here. - The set is not closed in $\mathbb{R}^k$. - -## New stuffs - -### Connected sets - -#### Definition 2.45 - -$A,B\subset X$, we say $A$ and $B$ are separated in $X$ if $A\cup \overline{B}=\phi$ and $\overline{A}\cup B=\phi$ - -- $E\subset X$ **disconnected** in $X$ if $\exists$ nonempty separated $A,B\subset X$ such that $E=A\cup B$ -- $E\subset X$ is **connected** in $X$ if it is not disconnected. - -Example 2.46 - -$(0,1),(1,2)$ are separated [so $(0,1)\cup (1,2)$ is disconnected] - -$[0,1],(1,2)$ are not separated [so $[0,1]\cup (1,2)=\{1\}$] So this doesn't tell us where $[0,1]\cup (1,2)$ is connected or not. - -#### Theorem 2.47 - -Suppose $E\subset \mathbb{R}$ - -$E$ is connected $\iff \forall (x,y,z)$ with $x,y\in E,x0$ such that $(w-r,w+r)\cap B=\phi$. - -Let $z=w+\frac{r}{2}$, then $x,y,z$ satisfy the desired properties. - -QED - -## Chapter 3: Numerical Sequences and Series - -### Numerical Sequences - -#### Notations - -Rudin use $\{p_n\}$ to denote a sequence $p_1,p_2$. - -To avoid confusion with sets, we use $(p_n)_{n=1}^\infty$ or $(p_n)$ - -#### Definition 3.1 - -Let $(X,d)$ be a metric space. Let $(p_n)$ be a sequence in $X$. - -Let $p\in X$. We say $(p_x)$ **converges** to $p$ if $\forall \epsilon>0,\exists N\in\mathbb{N}$ such that $\forall n\geq N$, $d(p_n,p)<\epsilon$. ($p_n\in B_\epsilon (p)$) - -Notation $\lim_{n\to \infty} p_n=p$, $p_n\to p$ - -We say $(p_n)$ converges if $\exists p\in X$ such that $p_n\to p$. - -i.e. $\exists p\in X$ such that $\forall\epsilon>0,\exists N\in\mathbb{N}$ such that $\forall n\geq N,d(p_n,p)<\epsilon$ - -We say $(p_n)$ **diverges** if $(p_n)$ doesn't converge. - -i.e. $\forall p\in X$, $p_n\cancel{\to} p$ - -i.e. $\forall p\in X$ such that $\exists \epsilon>0,\forall N\in\mathbb{N}$ such that $\exists n\geq N,d(p_n,p)\geq\epsilon$ - -#### Definition 3.2 - -We say a sequence $(p_n)$ is bounded if $\exists x\in X$, $\forall r>0$ such that $\forall n\in \mathbb{N},p_n\in B_r(x)$ - -Example: - -$X=\mathbb{C}$, $s_n=\frac{1}{n}$ - -Then $s_n\to 0$ i.e. $\forall \epsilon>0 \exists N\in \mathbb{N}$ such that $\forall n\geq N$, $|s_n-0|<\epsilon$. - -Proof: - -Let $\epsilon >0$ (arbitrary) - -Let $N\in \mathbb{N}$ be greater than $\frac{1}{\epsilon}$ (by Archimedean property) e.g. $N=\frac{1}{\epsilon}+1$ (we choose $N$) - -Let $n\geq N$ (arbitrary) - -Then $|s_n-q|=\frac{1}{n}\leq \frac{1}{N}\leq \epsilon$ - +# Lecture 13 + +## Review + +Consider the metric space $X=\mathbb{R}$ (with the usual metric $d(x,y)=|x-y|$). Let $E=(0,1)$. + +1. Find several examples of sets $Y\subset \mathbb{R}$ such that $E\subset Y$ and $E$ is closed in $Y$. + Example: + 1. $Y=E$, $E$ is closed in $Y$. + We can prove this using normal ways, or + **Theorem 2.23** $E$ is closed in $Y\iff E^c$ is open in $Y$ + $Y\iff E^c=\phi$ and it's open. + 2. $Y=\mathbb{R}\backslash\{0,1\}$ + $Y\backslash E=(-\infty,0)\cup (1,\infty)$ + **Theorem 2.30** $E\subset Y\subset X$, $E$ is open in $Y\iff$ $\exists G$ open in $X$ such that $G\cap Y=E$ + $G\cap Y=Y\backslash E$ + And we know $Y\backslash E$ is open in $Y$. By **Theorem 2.23** $E$ is closed in $Y$. + +2. If $Y$ is as in part 1, we can conclude that $E$ is closed and bounded in $y$. Part of **Theorem 2.41** says: "If a set is closed and bounded, then it is compact." Why doesn't that theorem apply here. + The set is not closed in $\mathbb{R}^k$. + +## New stuffs + +### Connected sets + +#### Definition 2.45 + +$A,B\subset X$, we say $A$ and $B$ are separated in $X$ if $A\cup \overline{B}=\phi$ and $\overline{A}\cup B=\phi$ + +- $E\subset X$ **disconnected** in $X$ if $\exists$ nonempty separated $A,B\subset X$ such that $E=A\cup B$ +- $E\subset X$ is **connected** in $X$ if it is not disconnected. + +Example 2.46 + +$(0,1),(1,2)$ are separated [so $(0,1)\cup (1,2)$ is disconnected] + +$[0,1],(1,2)$ are not separated [so $[0,1]\cup (1,2)=\{1\}$] So this doesn't tell us where $[0,1]\cup (1,2)$ is connected or not. + +#### Theorem 2.47 + +Suppose $E\subset \mathbb{R}$ + +$E$ is connected $\iff \forall (x,y,z)$ with $x,y\in E,x0$ such that $(w-r,w+r)\cap B=\phi$. + +Let $z=w+\frac{r}{2}$, then $x,y,z$ satisfy the desired properties. + +QED + +## Chapter 3: Numerical Sequences and Series + +### Numerical Sequences + +#### Notations + +Rudin use $\{p_n\}$ to denote a sequence $p_1,p_2$. + +To avoid confusion with sets, we use $(p_n)_{n=1}^\infty$ or $(p_n)$ + +#### Definition 3.1 + +Let $(X,d)$ be a metric space. Let $(p_n)$ be a sequence in $X$. + +Let $p\in X$. We say $(p_x)$ **converges** to $p$ if $\forall \epsilon>0,\exists N\in\mathbb{N}$ such that $\forall n\geq N$, $d(p_n,p)<\epsilon$. ($p_n\in B_\epsilon (p)$) + +Notation $\lim_{n\to \infty} p_n=p$, $p_n\to p$ + +We say $(p_n)$ converges if $\exists p\in X$ such that $p_n\to p$. + +i.e. $\exists p\in X$ such that $\forall\epsilon>0,\exists N\in\mathbb{N}$ such that $\forall n\geq N,d(p_n,p)<\epsilon$ + +We say $(p_n)$ **diverges** if $(p_n)$ doesn't converge. + +i.e. $\forall p\in X$, $p_n\cancel{\to} p$ + +i.e. $\forall p\in X$ such that $\exists \epsilon>0,\forall N\in\mathbb{N}$ such that $\exists n\geq N,d(p_n,p)\geq\epsilon$ + +#### Definition 3.2 + +We say a sequence $(p_n)$ is bounded if $\exists x\in X$, $\forall r>0$ such that $\forall n\in \mathbb{N},p_n\in B_r(x)$ + +Example: + +$X=\mathbb{C}$, $s_n=\frac{1}{n}$ + +Then $s_n\to 0$ i.e. $\forall \epsilon>0 \exists N\in \mathbb{N}$ such that $\forall n\geq N$, $|s_n-0|<\epsilon$. + +Proof: + +Let $\epsilon >0$ (arbitrary) + +Let $N\in \mathbb{N}$ be greater than $\frac{1}{\epsilon}$ (by Archimedean property) e.g. $N=\frac{1}{\epsilon}+1$ (we choose $N$) + +Let $n\geq N$ (arbitrary) + +Then $|s_n-q|=\frac{1}{n}\leq \frac{1}{N}\leq \epsilon$ + QED \ No newline at end of file diff --git a/pages/Math4111/Math4111_L14.md b/content/Math4111/Math4111_L14.md similarity index 96% rename from pages/Math4111/Math4111_L14.md rename to content/Math4111/Math4111_L14.md index 0ab16c7..c827faa 100644 --- a/pages/Math4111/Math4111_L14.md +++ b/content/Math4111/Math4111_L14.md @@ -1,159 +1,159 @@ -# Lecture 14 - -## Review - -Consider the following statement: If sequence $(p_n)$ converges, then its bounded. - -1. Will the proof involve an arbitrary $\epsilon>0$ (one that you, the prover, do nto get to choose) or a specific $\epsilon>0$ (on that you can choose) - We can choose, for example $\epsilon=1$. -2. Give a proof of the statement. - -## Continue on sequence - -### Convergence - -#### Theorem 3.2(c) - -$(p_n)$ converges $\implies(p_n)$ is bounded. - -Proof: - -Suppose $(p_n)$ converges, then $\exists p\in X$ such that $p_n\to p$. Let $\epsilon=1$, then $\exists N\in \mathbb{N}$ such that $\forall n\geq N,d(p_n,p)<1$. Let $r=1+max\{1,d(p_1,p),d(p_2,p),\dots,d(p_{N-1},p)\}$. - -Then $\forall n\in \mathbb{N}, d(p_n,p)\leq r$. - -#### Theorem 3.2 - -Let $(p_n)$ be a sequence in $(X,d)$ - -(a) $p_n\to p\iff \forall r>0,\{n\in \mathbb{N},p_n\notin B_r(p)\}$ is finite -(b) $p_n\to p; p_n\to p'\implies p=p'$ (converging point is unique) -(c) $(p_n)$ converges $\implies(p_n)$ is bounded. -(d) If $E\subset X$ and $p\in \overline{E}$, then $\exist (p_n)\in E$ such that $p_n\to p$. - -Proof: - -(a) We need to show: - -$\forall \epsilon>0 \in N$, $\forall n\geq N,d(p_n,p)<\epsilon$ if and only if $\forall r>0, \{n\in \mathbb{N}:p\notin B_r(p)\}$ is finite. - -$\implies$ - -Suppose $\forall \epsilon>0 \in N$, $\forall n\geq N,d(p_n,p)<\epsilon$. - -**We start with arbitrary $r>0$.** and choose $\epsilon=n$ - -$\exists N$ such that $\forall n\geq \mathbb{N},d(p_n,p)0, \{n\in \mathbb{N}:p\notin B_r(p)\}$ is finite. Choosing $r=\epsilon$. We choose $r=\epsilon$. $\{n\in \mathbb{N}:p\notin B_\epsilon(p)\}<\{1,2,\dots,N-1\}$. - -Let $N=1+max\{n\in \mathbb{N},p_n\notin B_\epsilon(p)\}$ - -Then $\forall n\geq \mathbb{N},p_n\leq B_\epsilon(p)$ - -(b) We'll prove $\forall \epsilon>0,d(p,p')<2\epsilon$ to prove it, let $\epsilon >0$. Then - -$p_n\to p\implies \exists N$ such that $\forall n\geq \mathbb{N},d(p_n,p)<\epsilon$ -$p_n\to p'\implies \exists N'$ such that $\forall n\geq \mathbb{N},d(p_n,p')<\epsilon$ - -Let $n_0=max\{N,N'\}$, then - -$$ -d(p,p')\leq d(p_n,p_{n_0})+d(p_{n_0},p')<2\epsilon -$$ - -And $\forall \epsilon>0,d(p,p')<2\epsilon\implies d(p,p')=0$. So $p=p'$ - -> Remark: We can also prove this with contradiction. Idea $p\neq p'\implies d(p,p')>0$, let $\epsilon=\frac{1}{2}d(p,q')\dots$ - -(d) Suppose $p\in \overline{E}$. Then $\forall n\in \mathbb{N}, B_{\frac{1}{n}}(p)\cap E\neq \phi$. So $\forall n\in \mathbb{N}$, $\exists p_n\in B_{\frac{1}{n}}(p)\cap E$. We'll show $p_n\to p$. - -Let $\epsilon>0$. Choose $N\in \mathbb{N}$ such that $N>\frac{1}{\epsilon}$. Then if $n\geq N$, $d(p_n,p)<\frac{1}{n}\leq \frac{1}{N}\leq \epsilon$ - -QED - -#### Theorem 3.3 - -Let $(s_n), (t_n)$ be sequence in $\mathbb{C}$. Suppose $s_n\to s,t_n\to t$ - -(a) $s_n+t_n\to s+t$ -(b) $cs_n\to cs,c+s_n\to c+s$ -(c) $s_nt_n\to st$ -(d) If $\forall n\in \mathbb{N},s_n\neq 0,s\neq 0$, then $\frac{1}{s_n}\to \frac{1}{s}$ - -Proof: - -(a) We want to prove $\forall \epsilon>0, \exists N$ such that $\forall n\geq N, |(s_n+t_n)-(s+t)|<\epsilon$ - -Let $\epsilon >0$ - -$s_n\to s\implies \exist N_s$ such that $\forall n\geq N_s,|s_n-s|<\frac{\epsilon}{2}$ -$t_n\to t\implies \exist N_t$ such that $\forall n\geq N_t,|t_n-t|<\frac{\epsilon}{2}$ - -Let $N=\max\{N_t,N_s\}$, then if $n\geq N$, - -$$ -\begin{aligned} -|(s_n+t_n)-(s+t)|&=|(s_n+s)-(t_n-t)|\\ -&\leq |s_n-s|+|t_n-t|\\ -&< \frac{\epsilon}{2}+\frac{\epsilon}{2}\\ -&<\epsilon -\end{aligned} -$$ - -(b) exercise - -(c) First we'll prove a special case. - -$$ -s_n\to 0 \textup{ and }t_n\to 0\implies s_nt_n\to 0 -$$ - -Suppose $s_n\to 0$ and $t_n\to 0$. - -Let $\epsilon >0$ - -$s_n\to 0\implies \exist N_s$ such that $\forall n\geq N_s,|s_n-s|<\sqrt{\epsilon}$ -$t_n\to 0\implies \exist N_t$ such that $\forall n\geq N_t,|t_n-t|<\sqrt{\epsilon}$ - -Let $N=\max\{N_t,N_s\}$, then if $n\geq N$, - -$$ -|s_n t_n|< \sqrt{\epsilon}^2=\epsilon -$$ - -Now we prove the general case. - -$$ -s_n\to s \textup{ and }t_n\to t\implies s_nt_n\to st -$$ - -Since - -$$ -s_n t_n=(s_n-s)(t_n-t)+s(t_n-t)+t(s_n-s) -$$ - -So - -$$ -\lim_{n\to \infty}(s_nt_n-st)=\lim_{n\to \infty}(s_n-s)(t_n-t)+\lim_{n\to \infty}s(t_n-t)+\lim_{n\to \infty}t(s_n-s) -$$ - -$\lim_{n\to \infty}(s_n-s)(t_n-t)=0$ by special case - -$\lim_{n\to \infty}s(t_n-t)=0$ by (b) - -$\lim_{n\to \infty}t(s_n-s)=0$ by (b) - -Thought process for (d) - -$$ -\left|\frac{1}{s_n}-\frac{1}{s}\right|=\frac{|s_n-s|}{|s_n||s|}< \epsilon -$$ - +# Lecture 14 + +## Review + +Consider the following statement: If sequence $(p_n)$ converges, then its bounded. + +1. Will the proof involve an arbitrary $\epsilon>0$ (one that you, the prover, do nto get to choose) or a specific $\epsilon>0$ (on that you can choose) + We can choose, for example $\epsilon=1$. +2. Give a proof of the statement. + +## Continue on sequence + +### Convergence + +#### Theorem 3.2(c) + +$(p_n)$ converges $\implies(p_n)$ is bounded. + +Proof: + +Suppose $(p_n)$ converges, then $\exists p\in X$ such that $p_n\to p$. Let $\epsilon=1$, then $\exists N\in \mathbb{N}$ such that $\forall n\geq N,d(p_n,p)<1$. Let $r=1+max\{1,d(p_1,p),d(p_2,p),\dots,d(p_{N-1},p)\}$. + +Then $\forall n\in \mathbb{N}, d(p_n,p)\leq r$. + +#### Theorem 3.2 + +Let $(p_n)$ be a sequence in $(X,d)$ + +(a) $p_n\to p\iff \forall r>0,\{n\in \mathbb{N},p_n\notin B_r(p)\}$ is finite +(b) $p_n\to p; p_n\to p'\implies p=p'$ (converging point is unique) +(c) $(p_n)$ converges $\implies(p_n)$ is bounded. +(d) If $E\subset X$ and $p\in \overline{E}$, then $\exist (p_n)\in E$ such that $p_n\to p$. + +Proof: + +(a) We need to show: + +$\forall \epsilon>0 \in N$, $\forall n\geq N,d(p_n,p)<\epsilon$ if and only if $\forall r>0, \{n\in \mathbb{N}:p\notin B_r(p)\}$ is finite. + +$\implies$ + +Suppose $\forall \epsilon>0 \in N$, $\forall n\geq N,d(p_n,p)<\epsilon$. + +**We start with arbitrary $r>0$.** and choose $\epsilon=n$ + +$\exists N$ such that $\forall n\geq \mathbb{N},d(p_n,p)0, \{n\in \mathbb{N}:p\notin B_r(p)\}$ is finite. Choosing $r=\epsilon$. We choose $r=\epsilon$. $\{n\in \mathbb{N}:p\notin B_\epsilon(p)\}<\{1,2,\dots,N-1\}$. + +Let $N=1+max\{n\in \mathbb{N},p_n\notin B_\epsilon(p)\}$ + +Then $\forall n\geq \mathbb{N},p_n\leq B_\epsilon(p)$ + +(b) We'll prove $\forall \epsilon>0,d(p,p')<2\epsilon$ to prove it, let $\epsilon >0$. Then + +$p_n\to p\implies \exists N$ such that $\forall n\geq \mathbb{N},d(p_n,p)<\epsilon$ +$p_n\to p'\implies \exists N'$ such that $\forall n\geq \mathbb{N},d(p_n,p')<\epsilon$ + +Let $n_0=max\{N,N'\}$, then + +$$ +d(p,p')\leq d(p_n,p_{n_0})+d(p_{n_0},p')<2\epsilon +$$ + +And $\forall \epsilon>0,d(p,p')<2\epsilon\implies d(p,p')=0$. So $p=p'$ + +> Remark: We can also prove this with contradiction. Idea $p\neq p'\implies d(p,p')>0$, let $\epsilon=\frac{1}{2}d(p,q')\dots$ + +(d) Suppose $p\in \overline{E}$. Then $\forall n\in \mathbb{N}, B_{\frac{1}{n}}(p)\cap E\neq \phi$. So $\forall n\in \mathbb{N}$, $\exists p_n\in B_{\frac{1}{n}}(p)\cap E$. We'll show $p_n\to p$. + +Let $\epsilon>0$. Choose $N\in \mathbb{N}$ such that $N>\frac{1}{\epsilon}$. Then if $n\geq N$, $d(p_n,p)<\frac{1}{n}\leq \frac{1}{N}\leq \epsilon$ + +QED + +#### Theorem 3.3 + +Let $(s_n), (t_n)$ be sequence in $\mathbb{C}$. Suppose $s_n\to s,t_n\to t$ + +(a) $s_n+t_n\to s+t$ +(b) $cs_n\to cs,c+s_n\to c+s$ +(c) $s_nt_n\to st$ +(d) If $\forall n\in \mathbb{N},s_n\neq 0,s\neq 0$, then $\frac{1}{s_n}\to \frac{1}{s}$ + +Proof: + +(a) We want to prove $\forall \epsilon>0, \exists N$ such that $\forall n\geq N, |(s_n+t_n)-(s+t)|<\epsilon$ + +Let $\epsilon >0$ + +$s_n\to s\implies \exist N_s$ such that $\forall n\geq N_s,|s_n-s|<\frac{\epsilon}{2}$ +$t_n\to t\implies \exist N_t$ such that $\forall n\geq N_t,|t_n-t|<\frac{\epsilon}{2}$ + +Let $N=\max\{N_t,N_s\}$, then if $n\geq N$, + +$$ +\begin{aligned} +|(s_n+t_n)-(s+t)|&=|(s_n+s)-(t_n-t)|\\ +&\leq |s_n-s|+|t_n-t|\\ +&< \frac{\epsilon}{2}+\frac{\epsilon}{2}\\ +&<\epsilon +\end{aligned} +$$ + +(b) exercise + +(c) First we'll prove a special case. + +$$ +s_n\to 0 \textup{ and }t_n\to 0\implies s_nt_n\to 0 +$$ + +Suppose $s_n\to 0$ and $t_n\to 0$. + +Let $\epsilon >0$ + +$s_n\to 0\implies \exist N_s$ such that $\forall n\geq N_s,|s_n-s|<\sqrt{\epsilon}$ +$t_n\to 0\implies \exist N_t$ such that $\forall n\geq N_t,|t_n-t|<\sqrt{\epsilon}$ + +Let $N=\max\{N_t,N_s\}$, then if $n\geq N$, + +$$ +|s_n t_n|< \sqrt{\epsilon}^2=\epsilon +$$ + +Now we prove the general case. + +$$ +s_n\to s \textup{ and }t_n\to t\implies s_nt_n\to st +$$ + +Since + +$$ +s_n t_n=(s_n-s)(t_n-t)+s(t_n-t)+t(s_n-s) +$$ + +So + +$$ +\lim_{n\to \infty}(s_nt_n-st)=\lim_{n\to \infty}(s_n-s)(t_n-t)+\lim_{n\to \infty}s(t_n-t)+\lim_{n\to \infty}t(s_n-s) +$$ + +$\lim_{n\to \infty}(s_n-s)(t_n-t)=0$ by special case + +$\lim_{n\to \infty}s(t_n-t)=0$ by (b) + +$\lim_{n\to \infty}t(s_n-s)=0$ by (b) + +Thought process for (d) + +$$ +\left|\frac{1}{s_n}-\frac{1}{s}\right|=\frac{|s_n-s|}{|s_n||s|}< \epsilon +$$ + If $n$ is large enough, then... \ No newline at end of file diff --git a/pages/Math4111/Math4111_L15.md b/content/Math4111/Math4111_L15.md similarity index 97% rename from pages/Math4111/Math4111_L15.md rename to content/Math4111/Math4111_L15.md index 5805a53..c31ce41 100644 --- a/pages/Math4111/Math4111_L15.md +++ b/content/Math4111/Math4111_L15.md @@ -1,162 +1,162 @@ -# Lecture 15 - -## Review - -Let $(a_n)_{n=1}^\infty$ and $(b_n)_{n=1}^\infty$ be sequence in $\mathbb{R}$. Let $x_n=(a_n,b_n)\in \mathbb{R}^2$, so $(x_n)_{n=1}^\infty$ be a sequence in $\mathbb{R}^2$. Consider the following statement: - -$$ -a_n\to a\textup{ and }\quad b_n\to b\iff x_n\to (a,b) -$$ - -1.Prove the $\impliedby$ direction. That means you should prove the two things: - (a) If $x_n\to (a,b)$, then $a_n\to a$. (The proof of this begins: Suppose $x_n\to (a,b)$. Let $\epsilon>0$ be arbitrary. Then $\exists N$ such that $\forall n\geq N$) - We begins (with the goal $\forall \epsilon>0,\exists N$ such that $\forall n\geq N,|a_n-a|<\epsilon$). - Proof: - Let $\epsilon>0$ be arbitrary, then $\exists N$ such that $\forall n\geq N,|a_n-a|<\epsilon$. - Then if $n\geq N$, $|a_n-a|\leq \sqrt{|a_n-a|^2}\leq\sqrt{|a_n-a|^2+|b_n-b|^2}=|x_n-(a,b)|<\epsilon$. - QED - (b) If $x_n\to (a,b)$, then $b_n\to b$. - This follows from the same argument from (a) -2. Prove the $\implies$ direction. - Goal: $\forall \epsilon>0,\exists N$ such that $\forall n\geq N,|a_n-a|<\epsilon$. - Proof: - Let $\epsilon>0$ be arbitrary. - Since $a_n\to a$, $\exists N_1$ such that $\forall n\geq N_1,|a_n-a|<\epsilon$. - Since $b_n\to b$, $\exists N_2$ such that $\forall n\geq N_2,|b_n-b|<\epsilon$. - Let $N=\max\{N_1,N_2\}$. Then if $n\geq N$, $|a_n-a|<\epsilon$ and $|b_n-b|<\sqrt{2}\epsilon$. - **Same as last time, we can choose any smaller epsilon.** - Since $a_n\to a$, $\exists N_1$ such that $\forall n\geq N_1,|a_n-a|<\frac{\epsilon}{\sqrt{2}}$. - Since $b_n\to b$, $\exists N_2$ such that $\forall n\geq N_2,|b_n-b|<\frac{\epsilon}{\sqrt{2}}$. - Let $N=\max\{N_1,N_2\}$. Then if $n\geq N$, $|a_n-a|<\epsilon$ and $|b_n-b|<\sqrt{\frac{\epsilon^2}{2}+\frac{\epsilon^2}{2}}=\epsilon$. - QED - -## New Materials - -Continue from **Theorem 3.3** - -Suppose $(s_n),(t_n)$ are sequences in $\mathbb{C}$ and $s_n\to s,t_n\to t$. Then - -(a) $s_n+t_n\to s+t$ -(b) $cs_n\to cs$, $c+s_n\to c+s$ -(c) $s_nt_n\to st$ -(d) If $\forall n\in \mathbb{N},s_n\neq 0, s\neq 0$, then $\frac{1}{s_n}\to \frac{1}{s}$ - -Thought process for (d): - -$$ -\left|\frac{1}{s_n}-\frac{1}{s}\right|=\left|\frac{s-s_n}{s_ns}\right|=\frac{|s-s_n|}{|s||s_n|} -$$ - -We choose large enough $N$ such that $\forall n\geq N,|s_n-s|<\frac{|s|}{2}$. Then by triangle inequality, $|s_n|>\frac{|s|}{2}$. - -$$ -\begin{aligned} -|s|&=|s-s_n+s_n|\\ -|s|&\leq |s-s_n|+|s_n|\\ -|s|&<\frac{|s|}{2}+|s_n|\\ -\frac{|s|}{2}&< |s_n| -\end{aligned} -$$ - -So $\frac{|s_n-s|}{|s||s_n|}<\frac{2|s_n-s|}{|s|^2}$. - -We choose $n$ large enough such that - -$$ -\frac{2|s_n-s|}{|s|^2}<\epsilon -$$ - -Then $|s_n-s|<\frac{\epsilon|s|^2}{2}$. - -Proof: - -Let $\epsilon>0$, since $s_n\to s$ - -$\exists N$ such that $\forall n\geq N,|s_n-s|<\frac{|s|}{2}$. - -$\exists N$ such that $\forall n\geq N,|s_n-s|<\frac{\epsilon|s|^2}{2}$. - -Let $N=\max\{N_1,N_2\}$. Then if $n\geq N$, - -$$ -\left|\frac{1}{s_n}-\frac{1}{s}\right|=\frac{|s-s_n|}{|s||s_n|}<\frac{\frac{\epsilon|s|^2}{2}}{|s|^2}=\epsilon -$$ - -QED - -### Subsequences - -#### Definition 3.5 - -Given a sequence $(p_n)_{n=1}^\infty$, a sequence of $(n_i)_{i=1}^\infty$ is strictly increasing sequence in $\mathbb{N}$. i.e. $n_10, B_r(p)\cap E\backslash \{p\}\neq \phi$ - -- Choose $n_i$ such that $p_{n_i}\in B_i(p)$ -- If $n_1,\dots, n_{i-1}$ have bee chosen, choose $n_i$ such that $n_i>n_{i-1}$ and $p_{n_i}\in B_{\frac{1}{i}}(p)$. Then $p_{n_i}\to p$ - -(b) Since $(p_n)$ is bounded , $\exists M$ such that $\forall n\in N$, $p_n\in \overline{B_M(0)}=\{y\in\mathbb{R}^k:|y|\leq M\}$ - -$\overline{B_M(0)}$ is a closed and bounded set in $\mathbb{R}^k$. - -Then by Theorem 2.41, $\overline{B_M(0)}$ is compact. - -By part (a), $(p_n)$ has a subsequence $(p_{n_i})$ has a subsequence that converges to $B_M(0)$. - -#### Theorem 3.37 - -Let $X$ be a metric space, $(p_n)$ is a sequence in $X$. - -Let $E^*=\{p\in X:\exists\textup{ subsequence }(p_{n_i})\textup{ such that }p_{n_i}\to p\}$. - -Then $E^*$ is closed in $X$. - -Example: - -$X=\mathbb{R}$ - -1. $p_n=\frac{1}{n}$, $E^*=\{0\}$. (Specifically, if $p_n\to p$, then $E^*\to \{p\}$) -2. $p_n=\begin{cases}1,n\textup{ is odd}\\ 0,n\textup{ is even}\end{cases}$, $E^*=\{0,1\}$ -3. $p_n=n$, $E^*=\phi$ -4. $p_n=\sin nx$, $E^*=\{0,1\}$ -5. $p_n=\sin n$, $E^*=[0,1]$ +# Lecture 15 + +## Review + +Let $(a_n)_{n=1}^\infty$ and $(b_n)_{n=1}^\infty$ be sequence in $\mathbb{R}$. Let $x_n=(a_n,b_n)\in \mathbb{R}^2$, so $(x_n)_{n=1}^\infty$ be a sequence in $\mathbb{R}^2$. Consider the following statement: + +$$ +a_n\to a\textup{ and }\quad b_n\to b\iff x_n\to (a,b) +$$ + +1.Prove the $\impliedby$ direction. That means you should prove the two things: + (a) If $x_n\to (a,b)$, then $a_n\to a$. (The proof of this begins: Suppose $x_n\to (a,b)$. Let $\epsilon>0$ be arbitrary. Then $\exists N$ such that $\forall n\geq N$) + We begins (with the goal $\forall \epsilon>0,\exists N$ such that $\forall n\geq N,|a_n-a|<\epsilon$). + Proof: + Let $\epsilon>0$ be arbitrary, then $\exists N$ such that $\forall n\geq N,|a_n-a|<\epsilon$. + Then if $n\geq N$, $|a_n-a|\leq \sqrt{|a_n-a|^2}\leq\sqrt{|a_n-a|^2+|b_n-b|^2}=|x_n-(a,b)|<\epsilon$. + QED + (b) If $x_n\to (a,b)$, then $b_n\to b$. + This follows from the same argument from (a) +2. Prove the $\implies$ direction. + Goal: $\forall \epsilon>0,\exists N$ such that $\forall n\geq N,|a_n-a|<\epsilon$. + Proof: + Let $\epsilon>0$ be arbitrary. + Since $a_n\to a$, $\exists N_1$ such that $\forall n\geq N_1,|a_n-a|<\epsilon$. + Since $b_n\to b$, $\exists N_2$ such that $\forall n\geq N_2,|b_n-b|<\epsilon$. + Let $N=\max\{N_1,N_2\}$. Then if $n\geq N$, $|a_n-a|<\epsilon$ and $|b_n-b|<\sqrt{2}\epsilon$. + **Same as last time, we can choose any smaller epsilon.** + Since $a_n\to a$, $\exists N_1$ such that $\forall n\geq N_1,|a_n-a|<\frac{\epsilon}{\sqrt{2}}$. + Since $b_n\to b$, $\exists N_2$ such that $\forall n\geq N_2,|b_n-b|<\frac{\epsilon}{\sqrt{2}}$. + Let $N=\max\{N_1,N_2\}$. Then if $n\geq N$, $|a_n-a|<\epsilon$ and $|b_n-b|<\sqrt{\frac{\epsilon^2}{2}+\frac{\epsilon^2}{2}}=\epsilon$. + QED + +## New Materials + +Continue from **Theorem 3.3** + +Suppose $(s_n),(t_n)$ are sequences in $\mathbb{C}$ and $s_n\to s,t_n\to t$. Then + +(a) $s_n+t_n\to s+t$ +(b) $cs_n\to cs$, $c+s_n\to c+s$ +(c) $s_nt_n\to st$ +(d) If $\forall n\in \mathbb{N},s_n\neq 0, s\neq 0$, then $\frac{1}{s_n}\to \frac{1}{s}$ + +Thought process for (d): + +$$ +\left|\frac{1}{s_n}-\frac{1}{s}\right|=\left|\frac{s-s_n}{s_ns}\right|=\frac{|s-s_n|}{|s||s_n|} +$$ + +We choose large enough $N$ such that $\forall n\geq N,|s_n-s|<\frac{|s|}{2}$. Then by triangle inequality, $|s_n|>\frac{|s|}{2}$. + +$$ +\begin{aligned} +|s|&=|s-s_n+s_n|\\ +|s|&\leq |s-s_n|+|s_n|\\ +|s|&<\frac{|s|}{2}+|s_n|\\ +\frac{|s|}{2}&< |s_n| +\end{aligned} +$$ + +So $\frac{|s_n-s|}{|s||s_n|}<\frac{2|s_n-s|}{|s|^2}$. + +We choose $n$ large enough such that + +$$ +\frac{2|s_n-s|}{|s|^2}<\epsilon +$$ + +Then $|s_n-s|<\frac{\epsilon|s|^2}{2}$. + +Proof: + +Let $\epsilon>0$, since $s_n\to s$ + +$\exists N$ such that $\forall n\geq N,|s_n-s|<\frac{|s|}{2}$. + +$\exists N$ such that $\forall n\geq N,|s_n-s|<\frac{\epsilon|s|^2}{2}$. + +Let $N=\max\{N_1,N_2\}$. Then if $n\geq N$, + +$$ +\left|\frac{1}{s_n}-\frac{1}{s}\right|=\frac{|s-s_n|}{|s||s_n|}<\frac{\frac{\epsilon|s|^2}{2}}{|s|^2}=\epsilon +$$ + +QED + +### Subsequences + +#### Definition 3.5 + +Given a sequence $(p_n)_{n=1}^\infty$, a sequence of $(n_i)_{i=1}^\infty$ is strictly increasing sequence in $\mathbb{N}$. i.e. $n_10, B_r(p)\cap E\backslash \{p\}\neq \phi$ + +- Choose $n_i$ such that $p_{n_i}\in B_i(p)$ +- If $n_1,\dots, n_{i-1}$ have bee chosen, choose $n_i$ such that $n_i>n_{i-1}$ and $p_{n_i}\in B_{\frac{1}{i}}(p)$. Then $p_{n_i}\to p$ + +(b) Since $(p_n)$ is bounded , $\exists M$ such that $\forall n\in N$, $p_n\in \overline{B_M(0)}=\{y\in\mathbb{R}^k:|y|\leq M\}$ + +$\overline{B_M(0)}$ is a closed and bounded set in $\mathbb{R}^k$. + +Then by Theorem 2.41, $\overline{B_M(0)}$ is compact. + +By part (a), $(p_n)$ has a subsequence $(p_{n_i})$ has a subsequence that converges to $B_M(0)$. + +#### Theorem 3.37 + +Let $X$ be a metric space, $(p_n)$ is a sequence in $X$. + +Let $E^*=\{p\in X:\exists\textup{ subsequence }(p_{n_i})\textup{ such that }p_{n_i}\to p\}$. + +Then $E^*$ is closed in $X$. + +Example: + +$X=\mathbb{R}$ + +1. $p_n=\frac{1}{n}$, $E^*=\{0\}$. (Specifically, if $p_n\to p$, then $E^*\to \{p\}$) +2. $p_n=\begin{cases}1,n\textup{ is odd}\\ 0,n\textup{ is even}\end{cases}$, $E^*=\{0,1\}$ +3. $p_n=n$, $E^*=\phi$ +4. $p_n=\sin nx$, $E^*=\{0,1\}$ +5. $p_n=\sin n$, $E^*=[0,1]$ diff --git a/pages/Math4111/Math4111_L16.md b/content/Math4111/Math4111_L16.md similarity index 97% rename from pages/Math4111/Math4111_L16.md rename to content/Math4111/Math4111_L16.md index 62780b6..1eb2481 100644 --- a/pages/Math4111/Math4111_L16.md +++ b/content/Math4111/Math4111_L16.md @@ -1,152 +1,152 @@ -# Lecture 16 - -## Review - -Let $(s_n)$ be a sequence in $\mathbb{R}$ satisfying the following properties: - -1. It is bounded ($\exists M>0$ such that $\forall n\in \mathbb{N}, |s_n|\leq M$) -2. It is monotonic increasing ($\forall n\in \mathbb{N}, s_n\leq s_{n+1}$) - -Let $E=\{s_n:n\in \mathbb{N}\}$ and $t=sup E$. Prove that $s_n\to t$. [Hint: The proof begins with "Let $\epsilon>0$ be arbitrary." What do we know about $t-\epsilon$?] - -Proof: - -Let $\epsilon>0$ be arbitrary. Then since $t-\epsilon$ is not an upper bound of $E$, $\exists N$ such that $t-\epsilont-\epsilon$. Since $t$ is an upper bound of $E$, $s_nn_1$ (by definition of $E^*$. If $x_2\in E^*$, then there are infinitely many $p\in \mathbb{N}$ such that $p_n\in B_{1/2}(x_2)$). - -Since $x_2\in E^*$, $\exists n_2\in \mathbb{N}$ such that $p_{n_2}\in B_{1/2}(x_2)$. - -By triangle inequality, $d(p_{n_2},q)\leq d(p_{n_2},x_2)+d(x_2,q)<\frac{1}{2}+\frac{1}{2}=1$. - -Step 3: By induction, we can get a sequence $n_1,n_2,\cdots$ such that $\forall i\in \mathbb{N}, d(p_{n_i},q)<\frac{2}{i}$. - -Then $(p_{n_i})$ is a subsequence of $(p_n)$ and $p_{n_i}\to q$. - -QED - -### Cauchy Sequences - -#### Definition 3.8 - -A sequence $(p_n)$ in a metric space $X$ is called a Cauchy sequence if for every $\epsilon>0$, there exists $N\in \mathbb{N}$ such that $\forall m,n\geq N$, $d(p_m,p_n)<\epsilon$. - -*The terms are getting closer to each other.* - -Example: - -$X=\mathbb{Q}$ with the usual metric. Let $(p_n)$ be a sequence - -$$ -3,3.1,3.14,3.141,3.1415,\cdots -$$ - -If $m\leq n$, $|p_m-p_n|<\frac{1}{10^{m}}$. - -Then $(p_n)$ is a Cauchy sequence. Let $\epsilon>0$ be arbitrary. Choose $N$ such that $\frac{1}{10^{N}}>\epsilon$. Then if $m,n\geq N$, then $|p_m-p_n|\leq \frac{1}{10^{m}}<\epsilon$. - -This sequence does not converge in $\mathbb{Q}$. - -$X=\mathbb{R}$ with the usual metric. Let $(p_n)$ be a sequence - -$$ -p_n=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n} -$$ - -This sequence is not bounded above. (by Theorem 3.28), so (as we will prove) it is not a Cauchy sequence. - -The fact that $p_{n+1}-p_n=\frac{1}{n+1}\to 0$ is not relevant to determining whether $(p_n)$ is a Cauchy sequence. - -#### Theorem 3.11 (a) - -$(p_n)$ converges $\implies$ $(p_n)$ is a Cauchy sequence. - -Proof: - -Since $(p_n)$ converges, $\exists p\in X$ such that $p_n\to p$. Let $\epsilon>0$ be arbitrary. Then $\exists N\in \mathbb{N}$ such that $\forall n\geq N$, $d(p_n,p)<\epsilon$. - -If $m,n\geq N$, then $d(p_m,p_n)\leq d(p_m,p)+d(p,p_n)<\epsilon+\epsilon=2\epsilon$. - -*You can also use $\frac{\epsilon}{2}$ instead of $\epsilon$ in the above proof, just for fun.* - -QED - -#### Lemma 3.11 (b) - -If $(p_n)$ is a Cauchy sequence, then $(p_n)$ is bounded above. - -Proof: - -Since $(p_n)$ is a Cauchy sequence, $\exists N\in \mathbb{N}$ such that $\forall m,n\geq N$, $d(p_m,p_n)<1$. - -Let $r=max\{d(p_i,p_j);1\leq i,j\leq N\}+1$. - -Then $\forall n\in \mathbb{N}$, $p_n\in B_r(p_N)$. - -QED - -> Note: This proof is nearly identical to the proof of convergent sequences implies bounded. - -#### Definition 3.9 - -Let $E$ be a nonempty subset of a metric space $X$, and let $S$ be the set of all real numbers of the form $d(p,q)$ for $p,q\in E$. The diameter of $E$, denoted by $diam E$, is defined to be the supremum of $S$. - -Exercise: - -Prove that $(p_n)$ is a Cauchy sequence if and only if $\lim_{N\to \infty}diam\{(p_n):n\geq N\}=0$. - -#### Theorem 3.10 - -(a) $diam E=diam(\overline{E})$ -(b) If $K_n$ is a sequence of nonempty compact sets and $K_1\supset K_2\supset \cdots$, then $\bigcap_{n=1}^{\infty}K_n$ has exactly one point. - -Proof: - -(a) The idea is still, triangle inequality. - -Since $E\subset \overline{E}$, $diam E\leq diam(\overline{E})$. - -Now we want to show that $diam(\overline{E})\leq diam E$. - -Claim: $\forall \epsilon>0$, $2\epsilon+diam E$ is an upper bound of $\{d(p,q):p,q\in \overline{E}\}$. - -Let $p,q\in \overline{E}$. - -Since $p\in \overline{E}$, $\exists p'\in E\cap B_\epsilon(p)$. - -Since $q\in \overline{E}$, $\exists q'\in E\cap B_\epsilon(q)$. - -Then $d(p,q)\leq d(p,p')+d(p',q')+d(q',q)<\epsilon+diam E+\epsilon=diam E+2\epsilon$. - -This proves the claim. - -By definition of supremum, the claim implies that $\forall \epsilon>0$, $diam(\overline{E})\leq 2\epsilon+diam E$. So $diam(\overline{E})\leq diam E$. - -(b) By **Theorem 2.36**, $\bigcap_{n=1}^{\infty}K_n\neq \phi$. Suppose for contradiction that there are at least two distinct points $p,q\in \bigcap_{n=1}^{\infty}K_n$. Then for all $n\in \mathbb{N}$, $x,y\in K_n$ so $diam K_n\geq d(p,q)>0$. Then diameter of $K_n$ does not converge to 0. - -QED - +# Lecture 16 + +## Review + +Let $(s_n)$ be a sequence in $\mathbb{R}$ satisfying the following properties: + +1. It is bounded ($\exists M>0$ such that $\forall n\in \mathbb{N}, |s_n|\leq M$) +2. It is monotonic increasing ($\forall n\in \mathbb{N}, s_n\leq s_{n+1}$) + +Let $E=\{s_n:n\in \mathbb{N}\}$ and $t=sup E$. Prove that $s_n\to t$. [Hint: The proof begins with "Let $\epsilon>0$ be arbitrary." What do we know about $t-\epsilon$?] + +Proof: + +Let $\epsilon>0$ be arbitrary. Then since $t-\epsilon$ is not an upper bound of $E$, $\exists N$ such that $t-\epsilont-\epsilon$. Since $t$ is an upper bound of $E$, $s_nn_1$ (by definition of $E^*$. If $x_2\in E^*$, then there are infinitely many $p\in \mathbb{N}$ such that $p_n\in B_{1/2}(x_2)$). + +Since $x_2\in E^*$, $\exists n_2\in \mathbb{N}$ such that $p_{n_2}\in B_{1/2}(x_2)$. + +By triangle inequality, $d(p_{n_2},q)\leq d(p_{n_2},x_2)+d(x_2,q)<\frac{1}{2}+\frac{1}{2}=1$. + +Step 3: By induction, we can get a sequence $n_1,n_2,\cdots$ such that $\forall i\in \mathbb{N}, d(p_{n_i},q)<\frac{2}{i}$. + +Then $(p_{n_i})$ is a subsequence of $(p_n)$ and $p_{n_i}\to q$. + +QED + +### Cauchy Sequences + +#### Definition 3.8 + +A sequence $(p_n)$ in a metric space $X$ is called a Cauchy sequence if for every $\epsilon>0$, there exists $N\in \mathbb{N}$ such that $\forall m,n\geq N$, $d(p_m,p_n)<\epsilon$. + +*The terms are getting closer to each other.* + +Example: + +$X=\mathbb{Q}$ with the usual metric. Let $(p_n)$ be a sequence + +$$ +3,3.1,3.14,3.141,3.1415,\cdots +$$ + +If $m\leq n$, $|p_m-p_n|<\frac{1}{10^{m}}$. + +Then $(p_n)$ is a Cauchy sequence. Let $\epsilon>0$ be arbitrary. Choose $N$ such that $\frac{1}{10^{N}}>\epsilon$. Then if $m,n\geq N$, then $|p_m-p_n|\leq \frac{1}{10^{m}}<\epsilon$. + +This sequence does not converge in $\mathbb{Q}$. + +$X=\mathbb{R}$ with the usual metric. Let $(p_n)$ be a sequence + +$$ +p_n=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n} +$$ + +This sequence is not bounded above. (by Theorem 3.28), so (as we will prove) it is not a Cauchy sequence. + +The fact that $p_{n+1}-p_n=\frac{1}{n+1}\to 0$ is not relevant to determining whether $(p_n)$ is a Cauchy sequence. + +#### Theorem 3.11 (a) + +$(p_n)$ converges $\implies$ $(p_n)$ is a Cauchy sequence. + +Proof: + +Since $(p_n)$ converges, $\exists p\in X$ such that $p_n\to p$. Let $\epsilon>0$ be arbitrary. Then $\exists N\in \mathbb{N}$ such that $\forall n\geq N$, $d(p_n,p)<\epsilon$. + +If $m,n\geq N$, then $d(p_m,p_n)\leq d(p_m,p)+d(p,p_n)<\epsilon+\epsilon=2\epsilon$. + +*You can also use $\frac{\epsilon}{2}$ instead of $\epsilon$ in the above proof, just for fun.* + +QED + +#### Lemma 3.11 (b) + +If $(p_n)$ is a Cauchy sequence, then $(p_n)$ is bounded above. + +Proof: + +Since $(p_n)$ is a Cauchy sequence, $\exists N\in \mathbb{N}$ such that $\forall m,n\geq N$, $d(p_m,p_n)<1$. + +Let $r=max\{d(p_i,p_j);1\leq i,j\leq N\}+1$. + +Then $\forall n\in \mathbb{N}$, $p_n\in B_r(p_N)$. + +QED + +> Note: This proof is nearly identical to the proof of convergent sequences implies bounded. + +#### Definition 3.9 + +Let $E$ be a nonempty subset of a metric space $X$, and let $S$ be the set of all real numbers of the form $d(p,q)$ for $p,q\in E$. The diameter of $E$, denoted by $diam E$, is defined to be the supremum of $S$. + +Exercise: + +Prove that $(p_n)$ is a Cauchy sequence if and only if $\lim_{N\to \infty}diam\{(p_n):n\geq N\}=0$. + +#### Theorem 3.10 + +(a) $diam E=diam(\overline{E})$ +(b) If $K_n$ is a sequence of nonempty compact sets and $K_1\supset K_2\supset \cdots$, then $\bigcap_{n=1}^{\infty}K_n$ has exactly one point. + +Proof: + +(a) The idea is still, triangle inequality. + +Since $E\subset \overline{E}$, $diam E\leq diam(\overline{E})$. + +Now we want to show that $diam(\overline{E})\leq diam E$. + +Claim: $\forall \epsilon>0$, $2\epsilon+diam E$ is an upper bound of $\{d(p,q):p,q\in \overline{E}\}$. + +Let $p,q\in \overline{E}$. + +Since $p\in \overline{E}$, $\exists p'\in E\cap B_\epsilon(p)$. + +Since $q\in \overline{E}$, $\exists q'\in E\cap B_\epsilon(q)$. + +Then $d(p,q)\leq d(p,p')+d(p',q')+d(q',q)<\epsilon+diam E+\epsilon=diam E+2\epsilon$. + +This proves the claim. + +By definition of supremum, the claim implies that $\forall \epsilon>0$, $diam(\overline{E})\leq 2\epsilon+diam E$. So $diam(\overline{E})\leq diam E$. + +(b) By **Theorem 2.36**, $\bigcap_{n=1}^{\infty}K_n\neq \phi$. Suppose for contradiction that there are at least two distinct points $p,q\in \bigcap_{n=1}^{\infty}K_n$. Then for all $n\in \mathbb{N}$, $x,y\in K_n$ so $diam K_n\geq d(p,q)>0$. Then diameter of $K_n$ does not converge to 0. + +QED + diff --git a/pages/Math4111/Math4111_L17.md b/content/Math4111/Math4111_L17.md similarity index 97% rename from pages/Math4111/Math4111_L17.md rename to content/Math4111/Math4111_L17.md index eaa03e8..18b07e5 100644 --- a/pages/Math4111/Math4111_L17.md +++ b/content/Math4111/Math4111_L17.md @@ -1,123 +1,123 @@ -# Lecture 17 - -## Review - -Given a sequence $(a_n)$ in $\mathbb{R}$, let $E_n=\{a_k:k\geq n\}$. Calculate $diam E_1$, $diam E_2$, $diam E_3$... for the following sequences: - -1. $a_n=0$: $E_n=\{0\}$, $$diam E_1=0, diam E_2=0, diam E_3=0, \ldots$$ -2. $a_n=n$: $E_n=\{n\}$, $$diam E_1=\infty, diam E_2=\infty, diam E_3=\infty, \ldots$$ -3. $a_n=(-1)^n$: $E_n=\{-1,1\}$, $$diam E_1=2, diam E_2=2, diam E_3=2, \ldots$$ -4. $a_n=1/n$: $E_n=\{1/n,1/(n+1),\dots\}$, $$diam E_1=\frac{1}{2}, diam E_2=\frac{1}{3}, diam E_3=\frac{1}{4}, \ldots$$ -5. $a_n=\frac{(-1)^n}{n}$: $E_n=\{-1/n,1/n,\dots\}$, $$diam E_1=\frac{1}{1}+\frac{1}{2}, diam E_2=\frac{1}{2}+\frac{1}{3}, diam E_3=\frac{1}{3}+\frac{1}{4}, \ldots$$ - -## New materials - -### Cauchy sequence - -#### Theorem 3.11 - -(b) If $X$ is a compact metric space, then every Cauchy sequence $(p_n)$ in $X$ converges. - -(c) In $\mathbb{R}^k$, every Cauchy sequence $(p_n)$ converges. - -Proof: - -(b) Let $E_N=\{p_n:n\geq N\}$. Since $(p_n)$ is Cauchy, $\lim_{N\to\infty} diam E_N=0$. By **Theorem 3.10 (a)**, $\lim_{N\to\infty} diam \overline{E_N}=0$. - -Since $X$ is compact, and $\overline{E_N}$ is closed, by **Theorem 2.35**, $\overline{E_N}$ is compact. - -Since $E_1\supset E_2\supset E_3\supset\cdots$, $\overline{E_1}\supset \overline{E_2}\supset \overline{E_3}\supset\cdots$. By **Theorem 3.10(b)**, $\exists p\in X$ such that $p\in\bigcap_{N=1}^{\infty}\overline{E_N}$. - -We claim that $(p_n)$ converges to $p$. Let $\epsilon>0$, there exists $N_0$ such that $\forall N\geq N_0$, $diam \overline{E_N}<\epsilon$. - -For any $n\geq N_0$, $p_n\in \overline{E_{N_0}}$. - -So $d(p_n,p)\leq diam \overline{E_{N_0}}<\epsilon$, by definition of diameter. - -Therefore, $(p_n)$ converges to $p$. - -(c) Let $(p_n)$ be a Cauchy sequence in $\mathbb{R}^k$. - -By **Theorem 3.9**, $(p_n)$ is bounded. So $\exists R>0$ such that $p_n\in B(0,R)$ for all $n$. Moreover $p_n\in \overline{B(0,R)}$. and $\overline{B(0,R)}$ is closed and bounded. Thus by **Theorem 2.41**, $\overline{B(0,R)}$ is compact. - -Note that **Theorem 2.41** only works for $\mathbb{R}^k$. - -So by (b), $(p_n)$ converges to some $p\in \overline{B(0,R)}$. - -QED - -#### Definition 3.12 - -Let $X$ be a metric space. We say $X$ is **complete** if every Cauchy sequence in $X$ converges. - -**Theorem 3.11(b)** can also be rephrased as: - -$X$ is a compact metric space $\implies$ $X$ is complete. - -**Theorem 3.11(c)** can also be rephrased as: - -$\mathbb{R}^k$ is complete. - -> Note: completeness is a property of the "universe" $X$, not a property of any particular sequence in $X$. - -$\mathbb{Q}$ is not complete. $\{3,3.1,3.14,3.141,3.1415,\dots\}$ is a Cauchy sequence in $\mathbb{Q}$ but it does not converge in $\mathbb{Q}$. - -Fact: If $X$ is complete and $E$ is a closed subset of $X$, then $E$ is complete. - -#### Definition 3.13 - -A sequence $(s_n)$ of real numbers is said to be - -- **monotone increasing** if $s_n\leq s_{n+1}$ for all $n$. -- **monotone decreasing** if $s_n\geq s_{n+1}$ for all $n$. -- **strictly monotone increasing** if $s_ns_{n+1}$ for all $n$. -- **monotone** if it is either monotone increasing or monotone decreasing. - -Example: - -1. $s_n=1/n$ is strictly monotone decreasing. -2. $s_n=(-1)^n$ is neither monotone increasing nor monotone decreasing. - -#### Theorem 3.14 - -Suppose $(s_n)$ is monotonic. Then $(s_n)$ converges $\iff$ $(s_n)$ is bounded. - -Proof: - -If $(s_n)$ is monotonic and bounded, then by previous result, $(s_n)$ converges. - -If $(s_n)$ is monotonic and converges, then by **Theorem 3.2(c)**, $(s_n)$ is bounded. - -QED - -### Upper and lower limits - -#### Definition 3.15 (Divergence to $\infty$ or $-\infty$) - -Let $(s_n)$ be a sequence of real numbers with the following properties: - -For every real number $M$ there is an integer $N$ such that $n\geq N$ implies $s_n>M$. We then write $s_n\to\infty$ - -For every real number $M$ there is an integer $N$ such that $n\geq N$ implies $s_n0$, there exists $N_0$ such that $\forall N\geq N_0$, $diam \overline{E_N}<\epsilon$. + +For any $n\geq N_0$, $p_n\in \overline{E_{N_0}}$. + +So $d(p_n,p)\leq diam \overline{E_{N_0}}<\epsilon$, by definition of diameter. + +Therefore, $(p_n)$ converges to $p$. + +(c) Let $(p_n)$ be a Cauchy sequence in $\mathbb{R}^k$. + +By **Theorem 3.9**, $(p_n)$ is bounded. So $\exists R>0$ such that $p_n\in B(0,R)$ for all $n$. Moreover $p_n\in \overline{B(0,R)}$. and $\overline{B(0,R)}$ is closed and bounded. Thus by **Theorem 2.41**, $\overline{B(0,R)}$ is compact. + +Note that **Theorem 2.41** only works for $\mathbb{R}^k$. + +So by (b), $(p_n)$ converges to some $p\in \overline{B(0,R)}$. + +QED + +#### Definition 3.12 + +Let $X$ be a metric space. We say $X$ is **complete** if every Cauchy sequence in $X$ converges. + +**Theorem 3.11(b)** can also be rephrased as: + +$X$ is a compact metric space $\implies$ $X$ is complete. + +**Theorem 3.11(c)** can also be rephrased as: + +$\mathbb{R}^k$ is complete. + +> Note: completeness is a property of the "universe" $X$, not a property of any particular sequence in $X$. + +$\mathbb{Q}$ is not complete. $\{3,3.1,3.14,3.141,3.1415,\dots\}$ is a Cauchy sequence in $\mathbb{Q}$ but it does not converge in $\mathbb{Q}$. + +Fact: If $X$ is complete and $E$ is a closed subset of $X$, then $E$ is complete. + +#### Definition 3.13 + +A sequence $(s_n)$ of real numbers is said to be + +- **monotone increasing** if $s_n\leq s_{n+1}$ for all $n$. +- **monotone decreasing** if $s_n\geq s_{n+1}$ for all $n$. +- **strictly monotone increasing** if $s_ns_{n+1}$ for all $n$. +- **monotone** if it is either monotone increasing or monotone decreasing. + +Example: + +1. $s_n=1/n$ is strictly monotone decreasing. +2. $s_n=(-1)^n$ is neither monotone increasing nor monotone decreasing. + +#### Theorem 3.14 + +Suppose $(s_n)$ is monotonic. Then $(s_n)$ converges $\iff$ $(s_n)$ is bounded. + +Proof: + +If $(s_n)$ is monotonic and bounded, then by previous result, $(s_n)$ converges. + +If $(s_n)$ is monotonic and converges, then by **Theorem 3.2(c)**, $(s_n)$ is bounded. + +QED + +### Upper and lower limits + +#### Definition 3.15 (Divergence to $\infty$ or $-\infty$) + +Let $(s_n)$ be a sequence of real numbers with the following properties: + +For every real number $M$ there is an integer $N$ such that $n\geq N$ implies $s_n>M$. We then write $s_n\to\infty$ + +For every real number $M$ there is an integer $N$ such that $n\geq N$ implies $s_n2\}$ -2. $\{n\in\mathbb{N}:s_n<2\}$ -3. $\{n\in\mathbb{N}:s_n>0\}$ -4. $\{n\in\mathbb{N}:s_n<0\}$ - -For each set, determine if the set $(1)$ must be infinite, or $(2)$ must be finite, or $(3)$ could be either finite or infinite, depending on the sequence $(s_n)$. - -If $\liminf_{n\to\infty} s_n=1$, then $\lim_{n\to\infty} \sup\{s_n,s_{n+1},s_{n+2},\dots\}=1$. - -So 1 must be finite, since if it is infinite, then $\limsup_{n\to\infty} s_n\geq 2$, which contradicts the given $\limsup_{n\to\infty} s_n=1$. - -2 and 3 are infinite. - -since $\liminf_{n\to\infty} s_n=1$, there exists infinitely many $n$ such that $2>s_n>0$. - -4 could be either finite or infinite. - -- $s_n=(-1)^n$ is example for 4 being infinite. -- $s_n=1$ is example for 4 being finite. - -## Continue on Limit Superior and Limit Inferior - -### Limit Superior - -#### Definition 3.16 - -Let $(s_n)$ **be a sequence of real numbers**. - -$S^*$ is the largest possible value that a subsequence of $(s_n)$ can converge to. - -(Normally, we need to be careful about the definition of "largest possible value", but in this case it does exist by **Theorem 3.7**.) - -Abbott's definition: - -$S^*=\limsup_{n\to\infty}\{s_k:k\geq n\}$. - -#### Theorem 3.17 - -Let $(s_n)$ **be a sequence of real numbers**. - -$S^*$ is the unique number satisfying the following: - -1. $\forall xS^*$, $\{n\in\mathbb{N}:s_n\geq x\}$ is finite. (same as saying $\exists N\in\mathbb{N}$ such that $n\geq N\implies s_n Normal squeeze theorem: If $s_n\leq t_n\leq u_n$ for all $n\in\mathbb{N}$, and $\lim_{n\to\infty} s_n=\lim_{n\to\infty} u_n=L$, then $\lim_{n\to\infty} t_n=L$. -> -> Proof: Exercise, hint: $u_n\to L\implies \limsup_{n\to\infty} u_n=\liminf_{n\to\infty} u_n=L$. - -#### Theorem 3.20 - -> Binomial theorem: $(1+x)^n=\sum_{k=0}^n\binom{n}{k}x^k$. - -Special sequences: - -(a) If $p>0$, then $\lim_{n\to\infty}\frac{1}{n^p}=0$. - -We want to find $\frac{1}{n^p}<\epsilon\iff n\geq\frac{1}{\epsilon^{1/p}}$. - -(b) If $p>0$, then $\lim_{n\to\infty}\sqrt[n]{p}=1$. - -We want to find $\sqrt[n]{p}-1<\epsilon\iff p<(1+\epsilon)^n$. - -> Bernoulli's inequality: for $\epsilon>0,n\in\mathbb{N}$, $(1+\epsilon)^n\geq 1+n\epsilon$. - -So it's enough to have $p<1+n\epsilon$ - -So we can choose $N>\frac{p-1}{\epsilon}$. - -Another way of writing this: Let $x_n=\sqrt[n]{p}-1$. - -Then $p=(1+x_n)^n\geq 1+nx_n$. - -So $0\leq x_n\leq\frac{p-1}{n}$. - -By the squeeze theorem, $x_n\to 0$.s - -(c) $\lim_{n\to\infty}\sqrt[n]{n}=1$. - -We want to find $\sqrt[n]{n}-1<\epsilon\iff n<(1+\epsilon)^n$. (this will not work for bernoulli's inequality) - -So it's enough to have $n<\frac{n(n-1)}{2}\epsilon^2\iff n>1+\frac{2}{\epsilon^2}$. So choose $N>1+\frac{2}{\epsilon^2}$. - -(d) If $p>0$ and $\alpha$ is real, then $\lim_{n\to\infty}\frac{n^\alpha}{(1+p)^n}=0$. - -With binomial theorem, $(1+p)^n\geq \binom{n}{k}p^k(k\leq n)$. - -$\binom{n}{k}=\frac{n(n-1)(n-2)\cdots(n-k+1)}{k!}$. - -If $n\geq 2k$, then $n-k+1\geq n-\frac{n}{2}+1\geq\frac{n}{2}$. - -So $\binom{n}{k}\geq\frac{(n/2)^k}{k!}$. - -Continue on next class. +# Lecture 18 + +## Review + +Let $(s_n)$ be a sequence in $\mathbb{R}$, and suppose $\limsup_{n\to\infty} s_n=1$. Consider the following four sets: + +1. $\{n\in\mathbb{N}:s_n>2\}$ +2. $\{n\in\mathbb{N}:s_n<2\}$ +3. $\{n\in\mathbb{N}:s_n>0\}$ +4. $\{n\in\mathbb{N}:s_n<0\}$ + +For each set, determine if the set $(1)$ must be infinite, or $(2)$ must be finite, or $(3)$ could be either finite or infinite, depending on the sequence $(s_n)$. + +If $\liminf_{n\to\infty} s_n=1$, then $\lim_{n\to\infty} \sup\{s_n,s_{n+1},s_{n+2},\dots\}=1$. + +So 1 must be finite, since if it is infinite, then $\limsup_{n\to\infty} s_n\geq 2$, which contradicts the given $\limsup_{n\to\infty} s_n=1$. + +2 and 3 are infinite. + +since $\liminf_{n\to\infty} s_n=1$, there exists infinitely many $n$ such that $2>s_n>0$. + +4 could be either finite or infinite. + +- $s_n=(-1)^n$ is example for 4 being infinite. +- $s_n=1$ is example for 4 being finite. + +## Continue on Limit Superior and Limit Inferior + +### Limit Superior + +#### Definition 3.16 + +Let $(s_n)$ **be a sequence of real numbers**. + +$S^*$ is the largest possible value that a subsequence of $(s_n)$ can converge to. + +(Normally, we need to be careful about the definition of "largest possible value", but in this case it does exist by **Theorem 3.7**.) + +Abbott's definition: + +$S^*=\limsup_{n\to\infty}\{s_k:k\geq n\}$. + +#### Theorem 3.17 + +Let $(s_n)$ **be a sequence of real numbers**. + +$S^*$ is the unique number satisfying the following: + +1. $\forall xS^*$, $\{n\in\mathbb{N}:s_n\geq x\}$ is finite. (same as saying $\exists N\in\mathbb{N}$ such that $n\geq N\implies s_n Normal squeeze theorem: If $s_n\leq t_n\leq u_n$ for all $n\in\mathbb{N}$, and $\lim_{n\to\infty} s_n=\lim_{n\to\infty} u_n=L$, then $\lim_{n\to\infty} t_n=L$. +> +> Proof: Exercise, hint: $u_n\to L\implies \limsup_{n\to\infty} u_n=\liminf_{n\to\infty} u_n=L$. + +#### Theorem 3.20 + +> Binomial theorem: $(1+x)^n=\sum_{k=0}^n\binom{n}{k}x^k$. + +Special sequences: + +(a) If $p>0$, then $\lim_{n\to\infty}\frac{1}{n^p}=0$. + +We want to find $\frac{1}{n^p}<\epsilon\iff n\geq\frac{1}{\epsilon^{1/p}}$. + +(b) If $p>0$, then $\lim_{n\to\infty}\sqrt[n]{p}=1$. + +We want to find $\sqrt[n]{p}-1<\epsilon\iff p<(1+\epsilon)^n$. + +> Bernoulli's inequality: for $\epsilon>0,n\in\mathbb{N}$, $(1+\epsilon)^n\geq 1+n\epsilon$. + +So it's enough to have $p<1+n\epsilon$ + +So we can choose $N>\frac{p-1}{\epsilon}$. + +Another way of writing this: Let $x_n=\sqrt[n]{p}-1$. + +Then $p=(1+x_n)^n\geq 1+nx_n$. + +So $0\leq x_n\leq\frac{p-1}{n}$. + +By the squeeze theorem, $x_n\to 0$.s + +(c) $\lim_{n\to\infty}\sqrt[n]{n}=1$. + +We want to find $\sqrt[n]{n}-1<\epsilon\iff n<(1+\epsilon)^n$. (this will not work for bernoulli's inequality) + +So it's enough to have $n<\frac{n(n-1)}{2}\epsilon^2\iff n>1+\frac{2}{\epsilon^2}$. So choose $N>1+\frac{2}{\epsilon^2}$. + +(d) If $p>0$ and $\alpha$ is real, then $\lim_{n\to\infty}\frac{n^\alpha}{(1+p)^n}=0$. + +With binomial theorem, $(1+p)^n\geq \binom{n}{k}p^k(k\leq n)$. + +$\binom{n}{k}=\frac{n(n-1)(n-2)\cdots(n-k+1)}{k!}$. + +If $n\geq 2k$, then $n-k+1\geq n-\frac{n}{2}+1\geq\frac{n}{2}$. + +So $\binom{n}{k}\geq\frac{(n/2)^k}{k!}$. + +Continue on next class. diff --git a/pages/Math4111/Math4111_L19.md b/content/Math4111/Math4111_L19.md similarity index 96% rename from pages/Math4111/Math4111_L19.md rename to content/Math4111/Math4111_L19.md index 68a8f71..b793a63 100644 --- a/pages/Math4111/Math4111_L19.md +++ b/content/Math4111/Math4111_L19.md @@ -1,194 +1,194 @@ -# Lecture 19 - -## Review - -> Binomial theorem: For $n\in\mathbb{N}$, -> $$(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k$$ - -1. Show that $2^n\geq \binom{n}{4}$ for all $n\geq 4$. (Hint: Expand $(1+1)^n$ using the binomial theorem) - Proof: - $$ - \begin{aligned} - (1+1)^n&=\sum_{k=0}^{n}\binom{n}{k}1^{n-k}1^k\\ - &=\sum_{k=0}^{n}\binom{n}{k}\\ - &=\binom{n}{0}+\binom{n}{1}+\cdots+\binom{n}{n}\\ - &\geq\binom{n}{4} - \end{aligned} - $$ - QED -2. Using part 1, show that $\lim_{n\to\infty}\frac{n^3}{2^n}=0$. - Proof: - $$ - \frac{n^3}{2^n}\leq\frac{n^3}{\binom{n}{4}} - $$ - The value of $\frac{n^3}{\binom{n}{4}}$ is decreasing when $n\geq 4$. - QED - -## New materials - -### Series - -#### Definition 3.21 - -Let $(a_n)_{n=1}^{\infty}$ be a sequence in $\mathbb{C}$. Let $s_n=\sum_{k=1}^{n}a_k$ denotes the sequence of partial sums. - -1. We say the series $\sum_{n=1}^{\infty}a_n$ converges if the sequence of partial sums $(s_n)_{n=1}^{\infty}$ converges. -2. We define the sum of the series $\sum_{n=1}^{\infty}a_n$ to be the limit of the sequence of partial sums, i.e., $$\sum_{n=1}^{\infty}a_n=\lim_{n\to\infty}s_n=\lim_{n\to\infty}\sum_{k=1}^{n}a_k.$$ - -#### Theorem 3.22 (Cauchy criterion for series) - -The series $\sum_{n=1}^{\infty}a_n$ converges if and only if for every $\epsilon>0$, there exists $N\in\mathbb{N}$ such that for all $m,n\in\mathbb{N}$ with $m\geq n\geq N$, -$$ -\left|\sum_{k=n}^{m}a_k\right|<\epsilon. -$$ - -Proof: - -$\sum_{n=1}^{\infty}a_n$ converges if and only if $(s_n)_{n=1}^{\infty}$ converges. - -Since $\mathbb{C}$ is complete, $(s_n)_{n=1}^{\infty}$ converges if and only if $(s_n)_{n=1}^{\infty}$ is Cauchy. - -Since $(s_n)_{n=1}^{\infty}$ is Cauchy, for every $\epsilon>0$, there exists $N\in\mathbb{N}$ such that for all $m,n\in\mathbb{N}$ with $m\geq n\geq N$, -$$ -|s_m-s_n|=\left|\sum_{k=n}^{m}a_k\right|<\epsilon. -$$ - -QED - -Special case of this theorem. - -#### Corollary 3.23 - -If $\sum_{n=1}^{\infty}a_n$ converges, then $\lim_{n\to\infty}a_n=0$. - -Note: the converse is not true. Example: $\sum_{n=1}^{\infty}\frac{1}{n}$ diverges. - -The contrapositive of this corollary is: If $\lim_{n\to\infty}a_n\neq 0$, then $\sum_{n=1}^{\infty}a_n$ diverges. It is useful naming as ``n-th term test for divergence''. - -Observe: - -$\forall n,a_n\geq 0$ - -$(a_n)$ is a non-negative sequence if and only if $(s_n)_{n=1}^{\infty}$ is increasing sequence. - -So if $(a_n)$ is a non-negative sequence, then $\sum_{n=1}^{\infty}a_n$ converges if and only if $(s_n)_{n=1}^{\infty}$ is bounded above. - -#### Theorem 3.25 (Comparison test) - -Let $(a_n)$ be a sequence in $\mathbb{C}$ and $(c_n)$ be a non-negative sequence in $\mathbb{R}$. Suppose $\forall n, |a_n|\leq c_n$. - -(a) If the series $\sum_{n=1}^{\infty}c_n$ converges, then the series $\sum_{n=1}^{\infty}a_n$ converges. -(b) If the series $\sum_{n=1}^{\infty}a_n$ diverges, then the series $\sum_{n=1}^{\infty}c_n$ diverges. - -Proof: - -(a) By **Theorem 3.22**, it's enough to show that for every $\epsilon>0$, there exists $N\in\mathbb{N}$ such that for all $m,n\in\mathbb{N}$ with $m\geq n\geq N$, -$$ -\left|\sum_{k=n}^{m}a_k\right|<\epsilon. -$$ - -Let $\epsilon>0$ be arbitrary. - -Since $\sum_{n=1}^{\infty}c_n$ converges, by **Theorem 3.22**, for the above $\epsilon$, there exists $N\in\mathbb{N}$ such that for all $m,n\in\mathbb{N}$ with $m\geq n\geq N$, -$$ -\left|\sum_{k=n}^{m}c_k\right|\leq \sum_{k=n}^{m}c_k<\epsilon. -$$ - -QED - -#### Theorem 3.26 (Geometric series) - -Let $x\in\mathbb{C}$. - -(a) If $|x|<1$, then the series $\sum_{n=0}^{\infty}x^n$ converges and $\sum_{n=0}^{\infty}x^n=\frac{1}{1-x}$. -(b) If $|x|\geq 1$, then the series $\sum_{n=0}^{\infty}x^n$ diverges. - -Proof: - -(b) If $|x|\geq 1$, then $x^n$ does not converge to 0. So the series $\sum_{n=0}^{\infty}x^n$ diverges. - -(a) Let $s_n=\sum_{k=0}^{n}x^k=1+x+x^2+\cdots+x^n$. - -$xs_n=x+x^2+x^3+\cdots+x^n+x^{n+1}=s_n+x^{n+1}$. - -So $s_n=\frac{1-x^{n+1}}{1-x}$. - -Since $|x|<1$, $x^{n+1}$ converges to 0. So $\lim_{n\to\infty}s_n=\frac{1}{1-x}$. - -QED - -#### Lemma 3.28 - -(a) $\sum_{n=0}^{\infty}\frac{1}{n}$ diverges. -(b) $\sum_{n=0}^{\infty}\frac{1}{n^2}$ converges. - -Proof: - -(a) -$$ -\begin{aligned} -\sum_{n=0}^{\infty}\frac{1}{n}&=\frac{1}{1}+\frac{1}{2}+\left(\frac{1}{3}+\frac{1}{4}\right)+\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)+\cdots\\ -&>\frac{1}{2}+\frac{1}{2}+\left(\frac{1}{4}+\frac{1}{4}\right)+\left(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}\right)+\cdots\\ -&=\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\cdots\\ -&=\infty -\end{aligned} -$$ - -(b) -$$ -\begin{aligned} -\sum_{n=0}^{\infty}\frac{1}{n^2}&=\frac{1}{1}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots\\ -&<\frac{1}{1}+\left(\frac{1}{2^2}+\frac{1}{2^2}\right)+\left(\frac{1}{4^2}+\cdots+\frac{1}{4^2}\right)+\left(\frac{1}{8^2}+\cdots+\frac{1}{8^2}\right)+\cdots\\ -&=\frac{1}{1}+\frac{2}{2^2}+\frac{4}{4^2}+\frac{8}{8^2}+\cdots\\ -&=\frac{1}{1}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+\cdots\\ -&=\frac{1}{1-\frac{1}{2}}\\ -&=\frac{1}{\frac{1}{2}}\\ -&=2 -\end{aligned} -$$ - -> Fun fact: $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$. - -QED - -#### Theorem 3.27 (Cauchy condensation test) - -Suppose $(a_n)$ is a non-negative sequence. The series $\sum_{n=1}^{\infty}a_n$ converges if and only if the series $\sum_{k=0}^{\infty}2^ka_{2^k}$ converges. - -Proof: - -Let $s_n=\sum_{k=1}^{n}a_k$ and $t_k=\sum_{k=0}^{k}2^ka_{2^k}$. - -If $n\leq 2^k$, then - -$$ -\begin{aligned} -s_n&=a_1+a_2+\cdots+a_n\\ -&\leq a_1+(a_2+a_3)+(a_4+a_5+\cdots+a_7)+\cdots+(a_{2^k}+a_{2^k+1}+\cdots+a_{2^{k+1}-1})\\ -&\leq a_1+2a_2+4a_4+\cdots+2^ka_{2^k}\\ -&=t_k. -\end{aligned} -$$ - -If $n\geq 2^{k+1}$, then - -$$ -\begin{aligned} -s_n&=a_1+a_2+\cdots+a_n\\ -&\geq a_1+a_2+(a_3+a_4)+(a_5+a_6+\cdots+a_7)+\cdots+(a_{2^k}+a_{2^k+1}+\cdots+a_{2^{k+1}-1})\\ -&\geq a_1+a_2+2a_4+\cdots+2^{k-1}a_{2^k}\\ -&\geq \frac{1}{2}\left(a_1+2a_2+4a_4+\cdots+2^ka_{2^k}\right)\\ -&=\frac{1}{2}t_k. -\end{aligned} -$$ - -We have shown that - -- If $n\leq 2^k$, then $s_n\leq t_k$. -- If $n\geq 2^{k+1}$, then $s_n\geq \frac{1}{2}t_k$. - -So $(s_n)_{n=1}^{\infty}$ is a bounded above. - -By **Theorem 3.14**, $(s_n)_{n=1}^{\infty}$ converges if and only if $(t_k)_{k=0}^{\infty}$ converges. - -QED +# Lecture 19 + +## Review + +> Binomial theorem: For $n\in\mathbb{N}$, +> $$(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k$$ + +1. Show that $2^n\geq \binom{n}{4}$ for all $n\geq 4$. (Hint: Expand $(1+1)^n$ using the binomial theorem) + Proof: + $$ + \begin{aligned} + (1+1)^n&=\sum_{k=0}^{n}\binom{n}{k}1^{n-k}1^k\\ + &=\sum_{k=0}^{n}\binom{n}{k}\\ + &=\binom{n}{0}+\binom{n}{1}+\cdots+\binom{n}{n}\\ + &\geq\binom{n}{4} + \end{aligned} + $$ + QED +2. Using part 1, show that $\lim_{n\to\infty}\frac{n^3}{2^n}=0$. + Proof: + $$ + \frac{n^3}{2^n}\leq\frac{n^3}{\binom{n}{4}} + $$ + The value of $\frac{n^3}{\binom{n}{4}}$ is decreasing when $n\geq 4$. + QED + +## New materials + +### Series + +#### Definition 3.21 + +Let $(a_n)_{n=1}^{\infty}$ be a sequence in $\mathbb{C}$. Let $s_n=\sum_{k=1}^{n}a_k$ denotes the sequence of partial sums. + +1. We say the series $\sum_{n=1}^{\infty}a_n$ converges if the sequence of partial sums $(s_n)_{n=1}^{\infty}$ converges. +2. We define the sum of the series $\sum_{n=1}^{\infty}a_n$ to be the limit of the sequence of partial sums, i.e., $$\sum_{n=1}^{\infty}a_n=\lim_{n\to\infty}s_n=\lim_{n\to\infty}\sum_{k=1}^{n}a_k.$$ + +#### Theorem 3.22 (Cauchy criterion for series) + +The series $\sum_{n=1}^{\infty}a_n$ converges if and only if for every $\epsilon>0$, there exists $N\in\mathbb{N}$ such that for all $m,n\in\mathbb{N}$ with $m\geq n\geq N$, +$$ +\left|\sum_{k=n}^{m}a_k\right|<\epsilon. +$$ + +Proof: + +$\sum_{n=1}^{\infty}a_n$ converges if and only if $(s_n)_{n=1}^{\infty}$ converges. + +Since $\mathbb{C}$ is complete, $(s_n)_{n=1}^{\infty}$ converges if and only if $(s_n)_{n=1}^{\infty}$ is Cauchy. + +Since $(s_n)_{n=1}^{\infty}$ is Cauchy, for every $\epsilon>0$, there exists $N\in\mathbb{N}$ such that for all $m,n\in\mathbb{N}$ with $m\geq n\geq N$, +$$ +|s_m-s_n|=\left|\sum_{k=n}^{m}a_k\right|<\epsilon. +$$ + +QED + +Special case of this theorem. + +#### Corollary 3.23 + +If $\sum_{n=1}^{\infty}a_n$ converges, then $\lim_{n\to\infty}a_n=0$. + +Note: the converse is not true. Example: $\sum_{n=1}^{\infty}\frac{1}{n}$ diverges. + +The contrapositive of this corollary is: If $\lim_{n\to\infty}a_n\neq 0$, then $\sum_{n=1}^{\infty}a_n$ diverges. It is useful naming as ``n-th term test for divergence''. + +Observe: + +$\forall n,a_n\geq 0$ + +$(a_n)$ is a non-negative sequence if and only if $(s_n)_{n=1}^{\infty}$ is increasing sequence. + +So if $(a_n)$ is a non-negative sequence, then $\sum_{n=1}^{\infty}a_n$ converges if and only if $(s_n)_{n=1}^{\infty}$ is bounded above. + +#### Theorem 3.25 (Comparison test) + +Let $(a_n)$ be a sequence in $\mathbb{C}$ and $(c_n)$ be a non-negative sequence in $\mathbb{R}$. Suppose $\forall n, |a_n|\leq c_n$. + +(a) If the series $\sum_{n=1}^{\infty}c_n$ converges, then the series $\sum_{n=1}^{\infty}a_n$ converges. +(b) If the series $\sum_{n=1}^{\infty}a_n$ diverges, then the series $\sum_{n=1}^{\infty}c_n$ diverges. + +Proof: + +(a) By **Theorem 3.22**, it's enough to show that for every $\epsilon>0$, there exists $N\in\mathbb{N}$ such that for all $m,n\in\mathbb{N}$ with $m\geq n\geq N$, +$$ +\left|\sum_{k=n}^{m}a_k\right|<\epsilon. +$$ + +Let $\epsilon>0$ be arbitrary. + +Since $\sum_{n=1}^{\infty}c_n$ converges, by **Theorem 3.22**, for the above $\epsilon$, there exists $N\in\mathbb{N}$ such that for all $m,n\in\mathbb{N}$ with $m\geq n\geq N$, +$$ +\left|\sum_{k=n}^{m}c_k\right|\leq \sum_{k=n}^{m}c_k<\epsilon. +$$ + +QED + +#### Theorem 3.26 (Geometric series) + +Let $x\in\mathbb{C}$. + +(a) If $|x|<1$, then the series $\sum_{n=0}^{\infty}x^n$ converges and $\sum_{n=0}^{\infty}x^n=\frac{1}{1-x}$. +(b) If $|x|\geq 1$, then the series $\sum_{n=0}^{\infty}x^n$ diverges. + +Proof: + +(b) If $|x|\geq 1$, then $x^n$ does not converge to 0. So the series $\sum_{n=0}^{\infty}x^n$ diverges. + +(a) Let $s_n=\sum_{k=0}^{n}x^k=1+x+x^2+\cdots+x^n$. + +$xs_n=x+x^2+x^3+\cdots+x^n+x^{n+1}=s_n+x^{n+1}$. + +So $s_n=\frac{1-x^{n+1}}{1-x}$. + +Since $|x|<1$, $x^{n+1}$ converges to 0. So $\lim_{n\to\infty}s_n=\frac{1}{1-x}$. + +QED + +#### Lemma 3.28 + +(a) $\sum_{n=0}^{\infty}\frac{1}{n}$ diverges. +(b) $\sum_{n=0}^{\infty}\frac{1}{n^2}$ converges. + +Proof: + +(a) +$$ +\begin{aligned} +\sum_{n=0}^{\infty}\frac{1}{n}&=\frac{1}{1}+\frac{1}{2}+\left(\frac{1}{3}+\frac{1}{4}\right)+\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)+\cdots\\ +&>\frac{1}{2}+\frac{1}{2}+\left(\frac{1}{4}+\frac{1}{4}\right)+\left(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}\right)+\cdots\\ +&=\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\cdots\\ +&=\infty +\end{aligned} +$$ + +(b) +$$ +\begin{aligned} +\sum_{n=0}^{\infty}\frac{1}{n^2}&=\frac{1}{1}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots\\ +&<\frac{1}{1}+\left(\frac{1}{2^2}+\frac{1}{2^2}\right)+\left(\frac{1}{4^2}+\cdots+\frac{1}{4^2}\right)+\left(\frac{1}{8^2}+\cdots+\frac{1}{8^2}\right)+\cdots\\ +&=\frac{1}{1}+\frac{2}{2^2}+\frac{4}{4^2}+\frac{8}{8^2}+\cdots\\ +&=\frac{1}{1}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+\cdots\\ +&=\frac{1}{1-\frac{1}{2}}\\ +&=\frac{1}{\frac{1}{2}}\\ +&=2 +\end{aligned} +$$ + +> Fun fact: $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$. + +QED + +#### Theorem 3.27 (Cauchy condensation test) + +Suppose $(a_n)$ is a non-negative sequence. The series $\sum_{n=1}^{\infty}a_n$ converges if and only if the series $\sum_{k=0}^{\infty}2^ka_{2^k}$ converges. + +Proof: + +Let $s_n=\sum_{k=1}^{n}a_k$ and $t_k=\sum_{k=0}^{k}2^ka_{2^k}$. + +If $n\leq 2^k$, then + +$$ +\begin{aligned} +s_n&=a_1+a_2+\cdots+a_n\\ +&\leq a_1+(a_2+a_3)+(a_4+a_5+\cdots+a_7)+\cdots+(a_{2^k}+a_{2^k+1}+\cdots+a_{2^{k+1}-1})\\ +&\leq a_1+2a_2+4a_4+\cdots+2^ka_{2^k}\\ +&=t_k. +\end{aligned} +$$ + +If $n\geq 2^{k+1}$, then + +$$ +\begin{aligned} +s_n&=a_1+a_2+\cdots+a_n\\ +&\geq a_1+a_2+(a_3+a_4)+(a_5+a_6+\cdots+a_7)+\cdots+(a_{2^k}+a_{2^k+1}+\cdots+a_{2^{k+1}-1})\\ +&\geq a_1+a_2+2a_4+\cdots+2^{k-1}a_{2^k}\\ +&\geq \frac{1}{2}\left(a_1+2a_2+4a_4+\cdots+2^ka_{2^k}\right)\\ +&=\frac{1}{2}t_k. +\end{aligned} +$$ + +We have shown that + +- If $n\leq 2^k$, then $s_n\leq t_k$. +- If $n\geq 2^{k+1}$, then $s_n\geq \frac{1}{2}t_k$. + +So $(s_n)_{n=1}^{\infty}$ is a bounded above. + +By **Theorem 3.14**, $(s_n)_{n=1}^{\infty}$ converges if and only if $(t_k)_{k=0}^{\infty}$ converges. + +QED diff --git a/pages/Math4111/Math4111_L2.md b/content/Math4111/Math4111_L2.md similarity index 97% rename from pages/Math4111/Math4111_L2.md rename to content/Math4111/Math4111_L2.md index 0dc1416..68127d7 100644 --- a/pages/Math4111/Math4111_L2.md +++ b/content/Math4111/Math4111_L2.md @@ -1,106 +1,106 @@ -# Lecture 2 - -Ordered sets, least upper bounds and fields. - -## Warm up - -(a) The statements says: $\forall a\in A, \exists s\in a$ such that $s\geq 7$. - -The negation is $\exist a\in A,\forall s\in a$, such that $s<7$. - -## Ordered sets - -Let $S$ be a set. An order on $S$ is a relation satisfying: - -1. "trichotomy". If $x,y\in S$, then exactly on eof the these statements are hold: $xy$. -2. "transitivity". If $x,y,z\in S$, then $x \beta$ - * So this statement is true for any rational numbers since $\cancel{\exist} a\in E$ such that $x>\beta$. - -### Definition 1.8 - -Least upper bound, LUB, supremum, SUP - -Let $S$ be an ordered set and $E\subset S$. We say $\alpha\in S$ is the LUB of $E$ if - -1. $\alpha$ is the UB of $E$. ($\forall x\in E,x\leq \alpha$) -2. if $\gamma<\alpha$, then $\gamma$ is not UB of $E$. ($\forall \gamma <\alpha, \exist x\in E$ such that $x>\gamma$ ) - -#### Lemma - -Uniqueness of upper bounds. - -If $\alpha$ and $\beta$ are LUBs of $E$, then $\alpha=\beta$. - -Proof: - -Suppose for contradiction $\alpha$ and $\beta$ are both LUB of $E$, then $\alpha\neq\beta$ - -WLOG $\alpha>\beta$ and $\beta>\alpha$. - -QED - -We write $\sup E$ to denote the LUB of $E$. - -This also applies to $GLB$ (greatest lower bound) and infinum of $E$ - -Example: - -1. $S=\mathbb{Q}, E=\{1,2,3\}$ ($E$ is bounded above) - * $\sup E=3$, $\inf E=1$ -2. $S=\mathbb{Q}, E=\{x\in \mathbb{Q}:00, p^2<2\}$. - * $A$ is not empty and bounded above. However, $\sup A$ des not exists. - -If $S=\mathbb{R}, A=\{p\leq \mathbb{Q}:p>0, p^2<2\}$. - * $A$ is not empty and bounded above. However, $\sup A=\sqrt{2}$. - -#### Least upper bound property (LUBP) - -if $\forall E\subset S$ that tis non-empty and bounded above, $\exist Sup E\in S$. - -#### Greatest upper bound property (GLBP) - -S has greatest lower bound property (GLBP) if $\exist E\subset S$ that is non-empty and bounded below, $\exists \inf E\in S$ - -$\mathbb{Q}$ does not have LUBP and GLBP. - -#### Theorem 1.11 - -Let $S$ be an ordered set. Then $S$ has the LUBP $\iff$ $S$ has the GLBP - -Proof: - +# Lecture 2 + +Ordered sets, least upper bounds and fields. + +## Warm up + +(a) The statements says: $\forall a\in A, \exists s\in a$ such that $s\geq 7$. + +The negation is $\exist a\in A,\forall s\in a$, such that $s<7$. + +## Ordered sets + +Let $S$ be a set. An order on $S$ is a relation satisfying: + +1. "trichotomy". If $x,y\in S$, then exactly on eof the these statements are hold: $xy$. +2. "transitivity". If $x,y,z\in S$, then $x \beta$ + * So this statement is true for any rational numbers since $\cancel{\exist} a\in E$ such that $x>\beta$. + +### Definition 1.8 + +Least upper bound, LUB, supremum, SUP + +Let $S$ be an ordered set and $E\subset S$. We say $\alpha\in S$ is the LUB of $E$ if + +1. $\alpha$ is the UB of $E$. ($\forall x\in E,x\leq \alpha$) +2. if $\gamma<\alpha$, then $\gamma$ is not UB of $E$. ($\forall \gamma <\alpha, \exist x\in E$ such that $x>\gamma$ ) + +#### Lemma + +Uniqueness of upper bounds. + +If $\alpha$ and $\beta$ are LUBs of $E$, then $\alpha=\beta$. + +Proof: + +Suppose for contradiction $\alpha$ and $\beta$ are both LUB of $E$, then $\alpha\neq\beta$ + +WLOG $\alpha>\beta$ and $\beta>\alpha$. + +QED + +We write $\sup E$ to denote the LUB of $E$. + +This also applies to $GLB$ (greatest lower bound) and infinum of $E$ + +Example: + +1. $S=\mathbb{Q}, E=\{1,2,3\}$ ($E$ is bounded above) + * $\sup E=3$, $\inf E=1$ +2. $S=\mathbb{Q}, E=\{x\in \mathbb{Q}:00, p^2<2\}$. + * $A$ is not empty and bounded above. However, $\sup A$ des not exists. + +If $S=\mathbb{R}, A=\{p\leq \mathbb{Q}:p>0, p^2<2\}$. + * $A$ is not empty and bounded above. However, $\sup A=\sqrt{2}$. + +#### Least upper bound property (LUBP) + +if $\forall E\subset S$ that tis non-empty and bounded above, $\exist Sup E\in S$. + +#### Greatest upper bound property (GLBP) + +S has greatest lower bound property (GLBP) if $\exist E\subset S$ that is non-empty and bounded below, $\exists \inf E\in S$ + +$\mathbb{Q}$ does not have LUBP and GLBP. + +#### Theorem 1.11 + +Let $S$ be an ordered set. Then $S$ has the LUBP $\iff$ $S$ has the GLBP + +Proof: + Let $S$ be a set with LUBP. (we want to show $S$ has GLBP) \ No newline at end of file diff --git a/pages/Math4111/Math4111_L20.md b/content/Math4111/Math4111_L20.md similarity index 96% rename from pages/Math4111/Math4111_L20.md rename to content/Math4111/Math4111_L20.md index 34dc934..c7dc595 100644 --- a/pages/Math4111/Math4111_L20.md +++ b/content/Math4111/Math4111_L20.md @@ -1,270 +1,270 @@ -# Lecture 20 - -## Review - -Using the binomial theorem, prove that - -$$ -\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\cdots +\frac{1}{n!}\geq \left(1+\frac{1}{n}\right)^n -$$ - -> Binomial theorem: $$(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$$ $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$ - -Proof: -$$ -\begin{aligned} -\left(1+\frac{1}{n}\right)^n &= \sum_{k=0}^{n} \binom{n}{k} \left(1\right)^{n-k} \left(\frac{1}{n}\right)^k \\ -&= \sum_{k=0}^{n} \binom{n}{k} \frac{1}{n^k} \\ -&= \sum_{k=0}^{n} \frac{1}{k!} \prod_{j=1}^{k} \frac{n-j+1}{n} \\ -\end{aligned} -$$ - -Since $j\geq 1$, $\frac{n-j+1}{n} \leq1$. - -$$ -\begin{aligned} -&= \sum_{k=0}^{n} \frac{1}{k!} \prod_{j=1}^{k} \frac{n-j+1}{n} \\ -&\geq \sum_{k=0}^{n} \frac{1}{k!} \\ -\end{aligned} -$$ - -## New material - -### Series - -#### Definition 3.30 - -$$ -e=\sum_{n=0}^{\infty} \frac{1}{n!} -$$ - -#### Lemma 3.30 - -$\sum_{n=0}^{\infty} \frac{1}{n!}$ converges. - -Proof: - -If $n\geq 2$, - -$$ -\begin{aligned} -\frac{1}{n!} &= \frac{1}{n} \cdot \frac{1}{(n-1)!} \\ -&\leq \frac{1}{2} \cdot \frac{1}{2} \cdot \dots \cdot \frac{1}{2} \\ -&= \frac{1}{2^{n-1}} -\end{aligned} -$$ - -$$ -\frac{1}{n!} \leq \frac{1}{2^{n-1}} -$$ - -So $\sum_{n=0}^{\infty} \frac{1}{n!}$ converges. - -#### Theorem 3.31 - -$$ -\lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n = e -$$ - -Proof: - -Let $s_n = \sum_{k=0}^{n} \frac{1}{k!}$, let $t_n = \left(1+\frac{1}{n}\right)^n$. - -Goal: $\lim_{n\to\infty} s_n = \lim_{n\to\infty} t_n$. we already proved $\lim_{n\to\infty} s_n$ exists. But we don't know yet if $\lim_{n\to\infty} t_n$ exists. - -By warmup exercise, $\forall n\geq 0, t_n \leq s_n$. - -So if $\limsup_{n\to\infty} t_n \leq \limsup_{n\to\infty} s_n$, then $\lim_{n\to\infty} t_n$ exists and $\lim_{n\to\infty} t_n = \lim_{n\to\infty} s_n$. - -Now we will show $\limsup_{n\to\infty} t_n \geq e$. - -Ideas: (special case of the argument) - -If $n\geq 2$, then - -$$ -\begin{aligned} -t_n &= \sum_{k=0}^{n} \binom{n}{k} \left(\frac{1}{n}\right)^k \\ -&\geq \binom{n}{0} + \binom{n}{1}\frac{1}{n} + \binom{n}{2}\left(\frac{1}{n}\right)^2 + \cdots + \binom{n}{n}\left(\frac{1}{n}\right)^n \\ -&= 1 + \frac{n}{n} + \frac{n(n-1)}{2n^2} + \cdots + \frac{1}{n^n} \\ -\end{aligned} -$$ - -Let $n\to\infty$, then - -$$ -\liminf_{n\to\infty} t_n \geq 1 + 1 + \frac{1}{2} + \frac{1}{3} + \cdots -$$ - -Fix $m\geq 2$, for any $n\geq m$, - -$$ -t_n \geq \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!}\frac{n}{n}\frac{n-1}{n}\cdots+\frac{1}{m!}\frac{n}{n}\frac{n-1}{n}\cdots\frac{n-m+1}{n} -$$ - -Let $n\to\infty$, then - -$$ -\liminf_{n\to\infty} t_n \geq \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \cdots + \frac{1}{m!}=s_m -$$ - -So $\liminf_{n\to\infty} t_n \geq \lim_{n\to\infty} s_n = e$. - -Therefore, $e\leq \liminf_{n\to\infty} t_n\leq \limsup_{n\to\infty} t_n\leq e$. - -So $\lim_{n\to\infty} t_n$ exists and $\lim_{n\to\infty} t_n = e$. - -QED - -#### Theorem 3.32 - -$e$ is irrational. - -Q: How good is the approximation is $s_n$ to $e$? - -A: Very good actually. -$$ -\begin{aligned} -e-s_n &= \sum_{k=n+1}^{\infty} \frac{1}{k!} \\ -&<\frac{1}{(n+1)!}\left(1+\frac{1}{n+1}+\frac{1}{(n+1)^2}+\cdots\right) \\ -&=\frac{1}{(n+1)!}\sum_{k=0}^{\infty}\left(\frac{1}{n+1}\right)^k \\ -&=\frac{1}{(n+1)!}\frac{1}{1-\frac{1}{n+1}} \\ -&=\frac{1}{n!}\cdot\frac{1}{n} \\ -&<\frac{1}{n!n} -\end{aligned} -$$ - -Proof: - -Suppose $e=\frac{p}{q}$ for some $p,q\in\mathbb{N}$. - -Observe that: - -$$ -s_q=1+1+\frac{1}{2}+\cdots+\frac{1}{q!} -$$ - -So $q! s_q$ is an integer. - -Since $e=\frac{p}{q}$, $q!e$ is an integer, $q!(e-s_q)$ is an integer. - -However, - -$$ -0 $$ \sqrt[n]{|a_n|} \leq \alpha \implies |a_n|\leq \alpha^n$$ - -Given a series $\sum_{n=0}^{\infty} a_n$, put $\alpha = \limsup_{n\to\infty} \sqrt[n]{|a_n|}$. - -Then - -(a) If $\alpha < 1$, then $\sum_{n=0}^{\infty} a_n$ converges. -(b) If $\alpha > 1$, then $\sum_{n=0}^{\infty} a_n$ diverges. -(c) If $\alpha = 1$, the test gives no information - -Proof: - -(a) Suppose $\alpha < 1$. Then $\exists \beta$ such that $\alpha < \beta < 1$. - -By **Theorem 3.17(b)**, $\forall n\geq N, \sqrt[n]{|a_n|} < \beta$. - -So $\forall n\geq N, |a_n| < \beta^n$. - -By comparison test, $\sum_{n=0}^{\infty} a_n$ converges. - -(b) Suppose $\alpha > 1$. By **Theorem 3.17(a)**, $\{n\in \mathbb{N}: \sqrt[n]{|a_n|} > 1\}$ is infinite. - -Thus $a_n\not\to 0$, $\sum_{n=0}^{\infty} a_n$ diverges. - -(c) $\sum_{n=0}^{\infty} \frac{1}{n}$ and $\sum_{n=0}^{\infty} \frac{1}{n^2}$ both have $\alpha = 1$. but the first diverges and the second converges. - -QED - -#### Theorem 3.34 (Ratio test) - -> $$ \left|\frac{a_{n+1}}{a_n}\right| \leq \alpha \implies |a_n|\leq \alpha^n$$ - -Given a series $\sum_{n=0}^{\infty} a_n$, $a_n\in\mathbb{C}\backslash\{0\}$. - -Then - -(a) If $\limsup_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| < 1$, then $\sum_{n=0}^{\infty} a_n$ converges. -(b) If $\left|\frac{a_{n+1}}{a_n}\right| \geq 1$ for all $n\geq n_0$ for some $n_0\in\mathbb{N}$, then $\sum_{n=0}^{\infty} a_n$ diverges. - -Remark: - -1. If $\limsup_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| = 1$, the test gives no information. -2. If $\limsup_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| > 1$, the test gives no information. - -Proof: - -(b) $\forall n\geq n_0, \left|\frac{a_{n+1}}{a_n}\right| \geq 1$. - -So $a_{n_0}\not\to 0$, $\sum_{n=0}^{\infty} a_n$ diverges. - -(a) $\beta \in(\limsup_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|, 1)$. - -By **Theorem 3.17(b)**, $\exists N$ such that $\forall n\geq N, \left|\frac{a_{n+1}}{a_n}\right| < \beta < 1$. - -So, - -$$ -\begin{aligned} -|a_N| &< \beta|a_N|\\ -|a_{N+1}| &< \beta|a_{N+1}|\\ -|a_{N+2}| &< \beta|a_{N+2}|\\ -\end{aligned} -$$ - -i.e. $\forall n\geq N, |a_n| < \beta^{n-N}|a_N|=\beta^n(\beta^{-N}|a_N|)$. - -Since $\sum_{n=N}^{\infty} \beta^n$ converges, by comparison test, $\sum_{n=0}^{\infty} a_n$ converges. - -QED - -We will skip **Theorem 3.37**. One implication is that if ratio test can be applied, then root test can be applied. - -### Power series - -#### Definition 3.38 - -Let $(c_n)$ be a sequence of complex numbers. A power series is a series of the form - -$$ -\sum_{n=0}^{\infty} c_n z^n -$$ - -#### Theorem 3.39 - -Given a power series $\sum_{n=0}^{\infty} c_n z^n$, let $R=\frac{1}{\limsup_{n\to\infty} \sqrt[n]{|c_n|}}$. - -Then - -(a) The series converges absolutely for all $z\in\mathbb{C}$ with $|z| < R$. -(b) The series diverges for all $z\in\mathbb{C}$ with $|z| > R$. -(c) If $0\leq r < R$, then the series converges uniformly on the closed disk $\{z\in\mathbb{C}: |z|\leq r\}$. - -Proof: - -$$ -\begin{aligned} -\limsup_{n\to\infty} \sqrt[n]{|c_n z^n|} &= \limsup_{n\to\infty} \sqrt[n]{|c_n|} \cdot |z| \\ -&= \frac{|z|}{R} -\end{aligned} -$$ - -By root test, the series converges absolutely for all $z\in\mathbb{C}$ with $|z| < R$. - -QED +# Lecture 20 + +## Review + +Using the binomial theorem, prove that + +$$ +\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\cdots +\frac{1}{n!}\geq \left(1+\frac{1}{n}\right)^n +$$ + +> Binomial theorem: $$(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$$ $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$ + +Proof: +$$ +\begin{aligned} +\left(1+\frac{1}{n}\right)^n &= \sum_{k=0}^{n} \binom{n}{k} \left(1\right)^{n-k} \left(\frac{1}{n}\right)^k \\ +&= \sum_{k=0}^{n} \binom{n}{k} \frac{1}{n^k} \\ +&= \sum_{k=0}^{n} \frac{1}{k!} \prod_{j=1}^{k} \frac{n-j+1}{n} \\ +\end{aligned} +$$ + +Since $j\geq 1$, $\frac{n-j+1}{n} \leq1$. + +$$ +\begin{aligned} +&= \sum_{k=0}^{n} \frac{1}{k!} \prod_{j=1}^{k} \frac{n-j+1}{n} \\ +&\geq \sum_{k=0}^{n} \frac{1}{k!} \\ +\end{aligned} +$$ + +## New material + +### Series + +#### Definition 3.30 + +$$ +e=\sum_{n=0}^{\infty} \frac{1}{n!} +$$ + +#### Lemma 3.30 + +$\sum_{n=0}^{\infty} \frac{1}{n!}$ converges. + +Proof: + +If $n\geq 2$, + +$$ +\begin{aligned} +\frac{1}{n!} &= \frac{1}{n} \cdot \frac{1}{(n-1)!} \\ +&\leq \frac{1}{2} \cdot \frac{1}{2} \cdot \dots \cdot \frac{1}{2} \\ +&= \frac{1}{2^{n-1}} +\end{aligned} +$$ + +$$ +\frac{1}{n!} \leq \frac{1}{2^{n-1}} +$$ + +So $\sum_{n=0}^{\infty} \frac{1}{n!}$ converges. + +#### Theorem 3.31 + +$$ +\lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n = e +$$ + +Proof: + +Let $s_n = \sum_{k=0}^{n} \frac{1}{k!}$, let $t_n = \left(1+\frac{1}{n}\right)^n$. + +Goal: $\lim_{n\to\infty} s_n = \lim_{n\to\infty} t_n$. we already proved $\lim_{n\to\infty} s_n$ exists. But we don't know yet if $\lim_{n\to\infty} t_n$ exists. + +By warmup exercise, $\forall n\geq 0, t_n \leq s_n$. + +So if $\limsup_{n\to\infty} t_n \leq \limsup_{n\to\infty} s_n$, then $\lim_{n\to\infty} t_n$ exists and $\lim_{n\to\infty} t_n = \lim_{n\to\infty} s_n$. + +Now we will show $\limsup_{n\to\infty} t_n \geq e$. + +Ideas: (special case of the argument) + +If $n\geq 2$, then + +$$ +\begin{aligned} +t_n &= \sum_{k=0}^{n} \binom{n}{k} \left(\frac{1}{n}\right)^k \\ +&\geq \binom{n}{0} + \binom{n}{1}\frac{1}{n} + \binom{n}{2}\left(\frac{1}{n}\right)^2 + \cdots + \binom{n}{n}\left(\frac{1}{n}\right)^n \\ +&= 1 + \frac{n}{n} + \frac{n(n-1)}{2n^2} + \cdots + \frac{1}{n^n} \\ +\end{aligned} +$$ + +Let $n\to\infty$, then + +$$ +\liminf_{n\to\infty} t_n \geq 1 + 1 + \frac{1}{2} + \frac{1}{3} + \cdots +$$ + +Fix $m\geq 2$, for any $n\geq m$, + +$$ +t_n \geq \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!}\frac{n}{n}\frac{n-1}{n}\cdots+\frac{1}{m!}\frac{n}{n}\frac{n-1}{n}\cdots\frac{n-m+1}{n} +$$ + +Let $n\to\infty$, then + +$$ +\liminf_{n\to\infty} t_n \geq \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \cdots + \frac{1}{m!}=s_m +$$ + +So $\liminf_{n\to\infty} t_n \geq \lim_{n\to\infty} s_n = e$. + +Therefore, $e\leq \liminf_{n\to\infty} t_n\leq \limsup_{n\to\infty} t_n\leq e$. + +So $\lim_{n\to\infty} t_n$ exists and $\lim_{n\to\infty} t_n = e$. + +QED + +#### Theorem 3.32 + +$e$ is irrational. + +Q: How good is the approximation is $s_n$ to $e$? + +A: Very good actually. +$$ +\begin{aligned} +e-s_n &= \sum_{k=n+1}^{\infty} \frac{1}{k!} \\ +&<\frac{1}{(n+1)!}\left(1+\frac{1}{n+1}+\frac{1}{(n+1)^2}+\cdots\right) \\ +&=\frac{1}{(n+1)!}\sum_{k=0}^{\infty}\left(\frac{1}{n+1}\right)^k \\ +&=\frac{1}{(n+1)!}\frac{1}{1-\frac{1}{n+1}} \\ +&=\frac{1}{n!}\cdot\frac{1}{n} \\ +&<\frac{1}{n!n} +\end{aligned} +$$ + +Proof: + +Suppose $e=\frac{p}{q}$ for some $p,q\in\mathbb{N}$. + +Observe that: + +$$ +s_q=1+1+\frac{1}{2}+\cdots+\frac{1}{q!} +$$ + +So $q! s_q$ is an integer. + +Since $e=\frac{p}{q}$, $q!e$ is an integer, $q!(e-s_q)$ is an integer. + +However, + +$$ +0 $$ \sqrt[n]{|a_n|} \leq \alpha \implies |a_n|\leq \alpha^n$$ + +Given a series $\sum_{n=0}^{\infty} a_n$, put $\alpha = \limsup_{n\to\infty} \sqrt[n]{|a_n|}$. + +Then + +(a) If $\alpha < 1$, then $\sum_{n=0}^{\infty} a_n$ converges. +(b) If $\alpha > 1$, then $\sum_{n=0}^{\infty} a_n$ diverges. +(c) If $\alpha = 1$, the test gives no information + +Proof: + +(a) Suppose $\alpha < 1$. Then $\exists \beta$ such that $\alpha < \beta < 1$. + +By **Theorem 3.17(b)**, $\forall n\geq N, \sqrt[n]{|a_n|} < \beta$. + +So $\forall n\geq N, |a_n| < \beta^n$. + +By comparison test, $\sum_{n=0}^{\infty} a_n$ converges. + +(b) Suppose $\alpha > 1$. By **Theorem 3.17(a)**, $\{n\in \mathbb{N}: \sqrt[n]{|a_n|} > 1\}$ is infinite. + +Thus $a_n\not\to 0$, $\sum_{n=0}^{\infty} a_n$ diverges. + +(c) $\sum_{n=0}^{\infty} \frac{1}{n}$ and $\sum_{n=0}^{\infty} \frac{1}{n^2}$ both have $\alpha = 1$. but the first diverges and the second converges. + +QED + +#### Theorem 3.34 (Ratio test) + +> $$ \left|\frac{a_{n+1}}{a_n}\right| \leq \alpha \implies |a_n|\leq \alpha^n$$ + +Given a series $\sum_{n=0}^{\infty} a_n$, $a_n\in\mathbb{C}\backslash\{0\}$. + +Then + +(a) If $\limsup_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| < 1$, then $\sum_{n=0}^{\infty} a_n$ converges. +(b) If $\left|\frac{a_{n+1}}{a_n}\right| \geq 1$ for all $n\geq n_0$ for some $n_0\in\mathbb{N}$, then $\sum_{n=0}^{\infty} a_n$ diverges. + +Remark: + +1. If $\limsup_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| = 1$, the test gives no information. +2. If $\limsup_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| > 1$, the test gives no information. + +Proof: + +(b) $\forall n\geq n_0, \left|\frac{a_{n+1}}{a_n}\right| \geq 1$. + +So $a_{n_0}\not\to 0$, $\sum_{n=0}^{\infty} a_n$ diverges. + +(a) $\beta \in(\limsup_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|, 1)$. + +By **Theorem 3.17(b)**, $\exists N$ such that $\forall n\geq N, \left|\frac{a_{n+1}}{a_n}\right| < \beta < 1$. + +So, + +$$ +\begin{aligned} +|a_N| &< \beta|a_N|\\ +|a_{N+1}| &< \beta|a_{N+1}|\\ +|a_{N+2}| &< \beta|a_{N+2}|\\ +\end{aligned} +$$ + +i.e. $\forall n\geq N, |a_n| < \beta^{n-N}|a_N|=\beta^n(\beta^{-N}|a_N|)$. + +Since $\sum_{n=N}^{\infty} \beta^n$ converges, by comparison test, $\sum_{n=0}^{\infty} a_n$ converges. + +QED + +We will skip **Theorem 3.37**. One implication is that if ratio test can be applied, then root test can be applied. + +### Power series + +#### Definition 3.38 + +Let $(c_n)$ be a sequence of complex numbers. A power series is a series of the form + +$$ +\sum_{n=0}^{\infty} c_n z^n +$$ + +#### Theorem 3.39 + +Given a power series $\sum_{n=0}^{\infty} c_n z^n$, let $R=\frac{1}{\limsup_{n\to\infty} \sqrt[n]{|c_n|}}$. + +Then + +(a) The series converges absolutely for all $z\in\mathbb{C}$ with $|z| < R$. +(b) The series diverges for all $z\in\mathbb{C}$ with $|z| > R$. +(c) If $0\leq r < R$, then the series converges uniformly on the closed disk $\{z\in\mathbb{C}: |z|\leq r\}$. + +Proof: + +$$ +\begin{aligned} +\limsup_{n\to\infty} \sqrt[n]{|c_n z^n|} &= \limsup_{n\to\infty} \sqrt[n]{|c_n|} \cdot |z| \\ +&= \frac{|z|}{R} +\end{aligned} +$$ + +By root test, the series converges absolutely for all $z\in\mathbb{C}$ with $|z| < R$. + +QED diff --git a/pages/Math4111/Math4111_L21.md b/content/Math4111/Math4111_L21.md similarity index 100% rename from pages/Math4111/Math4111_L21.md rename to content/Math4111/Math4111_L21.md diff --git a/pages/Math4111/Math4111_L22.md b/content/Math4111/Math4111_L22.md similarity index 100% rename from pages/Math4111/Math4111_L22.md rename to content/Math4111/Math4111_L22.md diff --git a/pages/Math4111/Math4111_L23.md b/content/Math4111/Math4111_L23.md similarity index 100% rename from pages/Math4111/Math4111_L23.md rename to content/Math4111/Math4111_L23.md diff --git a/pages/Math4111/Math4111_L24.md b/content/Math4111/Math4111_L24.md similarity index 100% rename from pages/Math4111/Math4111_L24.md rename to content/Math4111/Math4111_L24.md diff --git a/pages/Math4111/Math4111_L25.md b/content/Math4111/Math4111_L25.md similarity index 100% rename from pages/Math4111/Math4111_L25.md rename to content/Math4111/Math4111_L25.md diff --git a/pages/Math4111/Math4111_L3.md b/content/Math4111/Math4111_L3.md similarity index 97% rename from pages/Math4111/Math4111_L3.md rename to content/Math4111/Math4111_L3.md index 9414401..1538986 100644 --- a/pages/Math4111/Math4111_L3.md +++ b/content/Math4111/Math4111_L3.md @@ -1,129 +1,129 @@ -# Lecture 3 - -## Review - -Let $S=\mathbb{Z}$. - -1. Let $E=\{x\in S:x>0,x^2<5\}$. What are $sup\ E$ and $\inf\ E$? - - $sup\ E=2,inf\ E=1$ - -2. Can you find a subset $E\subset S$ which is bounded above but not bounded below? - - $E=\{x\in S:x<0\}$ - -3. Does $S$ have the least upper bound property? - - Yes, $\forall E\subset S$ that tis non-empty and bounded above, $\exist \sup E\in S$. - -4. Does $S$ have the greatest lower bound property? - - Yes, $\forall E\subset S$ that tis non-empty and bounded below, $\exist \inf E\in S$. - -## Continue - -### LUBP (The least upper bound property) - -Proof that $LUBP\implies GLBP$. - -Proof: - -Let $S$ be an ordered set with LUBP. Let $B\alpha,\beta$ is not a lower bound of $B$. - - Let $\beta>\alpha$. Since $\alpha$ is an upper bound of $L$, $\beta\notin L$. - - By definition of $L$, $\beta$ is not a lower bound of $B$. - -Thus $\alpha=inf\ B$ - -QED - -### Field - -| | addition | multiplication | -| -------------- | ----------------------------------------------------------- | -------------------------------------------------------------- | -| closure | $\checkmark$ | $\checkmark$ | -| commutativity | $\checkmark$ | $\checkmark$ | -| associativity | $\checkmark$ | $\checkmark$ | -| identity | $\checkmark$ (denoted $0$) | $\checkmark$ (denoted $1$) | -| inverses | $\checkmark$ (denoted $-x$) | $\checkmark$ (exists when $x\neq 0$ denoted $1/x$ or $x^{-1}$) | -| distributivity | $\checkmark$ (distributive of multiplication over addition) || - -Examples: $\mathbb{Q},\mathbb{R},\mathbb{C}$ - -Non-examples: $\mathbb{N}$ fails A4,A5,M5, $\mathbb{Z}$ fails M5 - -Another example of field: $\mathbb{Z}/5\mathbb{Z}=\{1,2,3,4,5\}$, $\forall a,b\in \mathbb{Z}/5\mathbb{Z}$, $a+b=(a+b)\mod 5$, $a\cdot b=(a\cdot b)\mod 5$ - -Some properties of fields: see Proposition 1.14,1.15,1.16 - -Remark: - -1. It's more helpful if you try to prove these yourselves. The proofs are "straightforward". -2. For this course, it's not important to remember which properties are axioms, etc. - -Example of proof: - -#### 1.14(a) $x+y=x+z\implies y=z$ - -Proof: - -$x+y=x+z$, - -$(-x)+(x+y)=(-x)+(x+z)$, - -by A3, $(-x+x)+(y)=(-x+x)+(z)$, - -$0+y=0+z$, - -$y=z$. - -Chain of equalities. - -#### 1.16(a) $\forall x\in \mathbb{F}, 0x=0$ - -1. A4, where 0 is defined. -2. Since $0$ is defined in the addition, identity. The proposition says something about multiplication by 0. The only proposition that relates the addition and multiplication is Distributive law. - -$0x=(0+0)x=0x+0x$, cancel $0x$ on both side we have $0x=0$. - -### Ordered Field (1.17) - -An _ordered field_ is a _field_ $F$ which is also an _ordered set_, such that - -1. $x+y0$ if $x\in F,y\in F,x>0$ and $y>0$. - -#### Prop 1.18 - -If $x>0$ and $y0\implies xz>xy$ - -We define $\mathbb{R}$ to be the unique ordered field with $LUBP$. (The existence and uniqueness are discussed in the appendix of this chapter). - -#### Theorem 1.20 - -1. (Archimedean property) If $x,y\in \mathbb{R}$ and $x>0$, then $\exists n\in \mathbb{N}$ such that $nx>y$. -2. ($\mathbb{Q}$ is dense in $\mathbb{R}$) If $x,y\in \mathbb{R}$ and $x0,x^2<5\}$. What are $sup\ E$ and $\inf\ E$? + + $sup\ E=2,inf\ E=1$ + +2. Can you find a subset $E\subset S$ which is bounded above but not bounded below? + + $E=\{x\in S:x<0\}$ + +3. Does $S$ have the least upper bound property? + + Yes, $\forall E\subset S$ that tis non-empty and bounded above, $\exist \sup E\in S$. + +4. Does $S$ have the greatest lower bound property? + + Yes, $\forall E\subset S$ that tis non-empty and bounded below, $\exist \inf E\in S$. + +## Continue + +### LUBP (The least upper bound property) + +Proof that $LUBP\implies GLBP$. + +Proof: + +Let $S$ be an ordered set with LUBP. Let $B\alpha,\beta$ is not a lower bound of $B$. + + Let $\beta>\alpha$. Since $\alpha$ is an upper bound of $L$, $\beta\notin L$. + + By definition of $L$, $\beta$ is not a lower bound of $B$. + +Thus $\alpha=inf\ B$ + +QED + +### Field + +| | addition | multiplication | +| -------------- | ----------------------------------------------------------- | -------------------------------------------------------------- | +| closure | $\checkmark$ | $\checkmark$ | +| commutativity | $\checkmark$ | $\checkmark$ | +| associativity | $\checkmark$ | $\checkmark$ | +| identity | $\checkmark$ (denoted $0$) | $\checkmark$ (denoted $1$) | +| inverses | $\checkmark$ (denoted $-x$) | $\checkmark$ (exists when $x\neq 0$ denoted $1/x$ or $x^{-1}$) | +| distributivity | $\checkmark$ (distributive of multiplication over addition) || + +Examples: $\mathbb{Q},\mathbb{R},\mathbb{C}$ + +Non-examples: $\mathbb{N}$ fails A4,A5,M5, $\mathbb{Z}$ fails M5 + +Another example of field: $\mathbb{Z}/5\mathbb{Z}=\{1,2,3,4,5\}$, $\forall a,b\in \mathbb{Z}/5\mathbb{Z}$, $a+b=(a+b)\mod 5$, $a\cdot b=(a\cdot b)\mod 5$ + +Some properties of fields: see Proposition 1.14,1.15,1.16 + +Remark: + +1. It's more helpful if you try to prove these yourselves. The proofs are "straightforward". +2. For this course, it's not important to remember which properties are axioms, etc. + +Example of proof: + +#### 1.14(a) $x+y=x+z\implies y=z$ + +Proof: + +$x+y=x+z$, + +$(-x)+(x+y)=(-x)+(x+z)$, + +by A3, $(-x+x)+(y)=(-x+x)+(z)$, + +$0+y=0+z$, + +$y=z$. + +Chain of equalities. + +#### 1.16(a) $\forall x\in \mathbb{F}, 0x=0$ + +1. A4, where 0 is defined. +2. Since $0$ is defined in the addition, identity. The proposition says something about multiplication by 0. The only proposition that relates the addition and multiplication is Distributive law. + +$0x=(0+0)x=0x+0x$, cancel $0x$ on both side we have $0x=0$. + +### Ordered Field (1.17) + +An _ordered field_ is a _field_ $F$ which is also an _ordered set_, such that + +1. $x+y0$ if $x\in F,y\in F,x>0$ and $y>0$. + +#### Prop 1.18 + +If $x>0$ and $y0\implies xz>xy$ + +We define $\mathbb{R}$ to be the unique ordered field with $LUBP$. (The existence and uniqueness are discussed in the appendix of this chapter). + +#### Theorem 1.20 + +1. (Archimedean property) If $x,y\in \mathbb{R}$ and $x>0$, then $\exists n\in \mathbb{N}$ such that $nx>y$. +2. ($\mathbb{Q}$ is dense in $\mathbb{R}$) If $x,y\in \mathbb{R}$ and $x0$, then $\exists n\in \mathbb{N}$ such that $nx>y$. - -Proof - -Suppose the property is false, then $\exist x,y\in \mathbb{R}$ with $x>0$ such that $\forall v\in \mathbb{N}$, nx\leq y$ - -Let $A=\{nx:n\in\mathbb{N}\}$. Then $A\neq\phi$ (Since $x\in A$) and $A$ is bounded above by $y$. Since $\mathbb{R}$ has LUBP, $sup\ A$ exists. Let $\alpha=\sup A$. - -$x>0\implies \alpha-x<\alpha$, $\alpha-x$ is not an upper bound of $A$. (Since $\alpha$ is the LUB of $A$) $\implies \exist m\in \mathbb{N}$ such that $mx>\alpha-x$ by definition of $A$. - -This implies $(m+1)x>\alpha$ - -Since $(m+1)x\in \alpha$, this contradicts the fact that $\alpha$ is an upper bound of $A$. - -QED - -### $\mathbb{Q}$ is dense in $\mathbb{R}$ - -$\mathbb{Q}$ is dense in $\mathbb{R}$ if $x,y\in \mathbb{R}$ and $x1$, and $\exist m_1\in \mathbb{N}$ such that $m_1\cdot 1>nx$, $\exist m_2\in \mathbb{N}$ such that $m_2\cdot 1>-nx$. - -So $-m_21+nx\geq 1+(m-1)=m$ - -QED - -### $\sqrt{2}\in \mathbb{R}$, $(\sqrt[n]{x}\in\mathbb{R})$ - -Notation $\mathbb{R}_{>0}$= the set of positive numbers. - -#### Theorem 1.21 - -$\forall x\in \mathbb{R}_{>0},\forall n\in \mathbb{N},\exist$ unique $y\in \mathbb{R}_{>0}$ such that $y^n=x$. - -(Because of this Theorem we can define $x^{1/x}=y$ and $\sqrt{x}=y$) - -Proof: - -We cna assume $n\geq 2$ (For $n=1,y=x$) - -Step 1 (uniqueness): If $00}: t^n0$ such that $(y+h)^n0$, then $\exists n\in \mathbb{N}$ such that $nx>y$. + +Proof + +Suppose the property is false, then $\exist x,y\in \mathbb{R}$ with $x>0$ such that $\forall v\in \mathbb{N}$, nx\leq y$ + +Let $A=\{nx:n\in\mathbb{N}\}$. Then $A\neq\phi$ (Since $x\in A$) and $A$ is bounded above by $y$. Since $\mathbb{R}$ has LUBP, $sup\ A$ exists. Let $\alpha=\sup A$. + +$x>0\implies \alpha-x<\alpha$, $\alpha-x$ is not an upper bound of $A$. (Since $\alpha$ is the LUB of $A$) $\implies \exist m\in \mathbb{N}$ such that $mx>\alpha-x$ by definition of $A$. + +This implies $(m+1)x>\alpha$ + +Since $(m+1)x\in \alpha$, this contradicts the fact that $\alpha$ is an upper bound of $A$. + +QED + +### $\mathbb{Q}$ is dense in $\mathbb{R}$ + +$\mathbb{Q}$ is dense in $\mathbb{R}$ if $x,y\in \mathbb{R}$ and $x1$, and $\exist m_1\in \mathbb{N}$ such that $m_1\cdot 1>nx$, $\exist m_2\in \mathbb{N}$ such that $m_2\cdot 1>-nx$. + +So $-m_21+nx\geq 1+(m-1)=m$ + +QED + +### $\sqrt{2}\in \mathbb{R}$, $(\sqrt[n]{x}\in\mathbb{R})$ + +Notation $\mathbb{R}_{>0}$= the set of positive numbers. + +#### Theorem 1.21 + +$\forall x\in \mathbb{R}_{>0},\forall n\in \mathbb{N},\exist$ unique $y\in \mathbb{R}_{>0}$ such that $y^n=x$. + +(Because of this Theorem we can define $x^{1/x}=y$ and $\sqrt{x}=y$) + +Proof: + +We cna assume $n\geq 2$ (For $n=1,y=x$) + +Step 1 (uniqueness): If $00}: t^n0$ such that $(y+h)^n0$ such that $x<1/n$ for all $n\in \mathbb{N}$. - -The statement is ambiguous because we can arrange the statement in two ways. - -$\exists x\in \mathbb{R}_{>0}$ such that $\forall n\in \mathbb{N},x\leq \frac{1}{n}$ - -negation: $\forall x\in \mathbb{R}_{>0}$, $\exists n\in \mathbb{N}$, such that $x\leq \frac{1}{n}$. - -The statement is true, let $x=\frac{1}{n+1}$. - -## New Materials - -### Continue on the theorem - -#### Theorem 1.21 - -$\forall x\in \mathbb{R}_{>0},\forall n\in \mathbb{N},\exist$ unique $y\in \mathbb{R}_{>0}$ such that $y^n=x$. - -(Because of this Theorem we can define $x^{1/x}=y$ and $\sqrt{x}=y$) - -Proof: - -We cna assume $n\geq 2$ (For $n=1,y=x$) - -Step 1 (uniqueness): If $00}: t^n0$ such that $(y+h)^n0$ satisfying $h<1$ and $h<\frac{x-y^h}{n(y+h)^{n-1}}$ - -[For actual proof, see the text.] - -Step 2c showing ($y^n\leq x$) - -Suppose for contradiction $y^n>x$ - -Thoughts: Find $k>0$ such that $(y-k)^n>x$. - -Then $y-k$ is an upper bound for $E$, which contradicts the fact that $y$ is the least upper bound of $E$. - -$y^n-(y-k)^n\leq ny^{n-1}k$. - -We want $y^n-ny^{n-1}k\geq x$. - -So want $k\leq \frac{y^n-x}{ny^{n-1}}$ - -[For actual proof, see the text.] - -QED - -### Complex numbers - -1. $=\{a+bi:a,b\in \mathbb{R}\}$. - -Conjugate: $z=a+bi,\bar{z}=a-bi$. - -#### Theorem 1.31 (see text) - -Pure computational proof: boring... - -$z\bar{z}=a^2-(bi)^2=a^2+b^2$ - -You can also use vector sum for representing operation in complex numbers. - -#### Theorem 1.33 (see text) - -More computation and still, boring... - -some fun theorems: - -- $|Re\ z|\leq |z|$ (equal when no imaginary parts) -- $|z+w|\leq |z|+|w|$ (equal when both $z,w$ have no imaginary parts) (Triangle inequality) - -Proof for $|z+w|\leq |z|+|w|$: - -$$ -|z+w|^2=(z+w)(\overline{z+w})=(z+w)(\bar{z}+\bar{w})=|z|^2+|w|^2+z\bar{w}+\bar{z}w -$$ - -Since - -$$ -z\bar{w}+\bar{z}w\leq 2Re(z\bar{w}) -$$ - -$$ -(z+w)(\bar{z}+\bar{w})=|z|^2+|w|^2+z\bar{w}+\bar{z}w\leq |z|^2+|w|^2+2|z||w|\leq |z|+|w| -$$ - -#### Theorem 1.35 Cauchy-Schwarz inequality - -If $\vec{a},\vec{b}\in \mathbb{C}^n$, then - -$$ -|\vec{a}\vec{b}|^2\leq (\vec{a}\vec{a})(\vec{b}\vec{b}) -$$ - -> Remark: The proof is very tricky. - -To help us motivate the proof in text, let's consider the special case of real numbers. - -$$ -(\sum a_j b_j)^2=(\sum a_j^2)(\sum b_j^2) -$$ - -Proof: - -For real numbers: - -Let $A=\sum a_j^2,B=\sum b_j^2, C=\sum a_j b_j$, want to show $C^2\leq AB$ - -Note: if $B=0$, then $b_1=b_2=...=0$, so $C=0$ and we are done, so we may assume $B\neq 0$ so $B>0$. - -Clever step: For any $t\in \mathbb{R}$, - -$$ -0\leq \sum (a_j-t b_j)^2=\sum (a_j^2-2ta_jb_j+t^2b_i^2)=A-2tC+t^2B -$$ - -let $t=C/B$ to get $0\leq A-2(C/B)C+(C/B)^2B=A-\frac{C^2}{B}$ - -to generalize this to $\mathbb{C}$, $A=\sum |a_j|^2,B=\sum |b_j|^2,C=\sum |a_j \bar{b_j}|$. - -QED - -### Euclidean spaces - -Nothing much to say. lol. - -Normal dot product as inner product. - +# Lecture 5 + +## Review + +In each case, determine (with justification) whether the claim or its negation is the true statement. + +(a) For all real numbers satisfying $a0$ such that $x<1/n$ for all $n\in \mathbb{N}$. + +The statement is ambiguous because we can arrange the statement in two ways. + +$\exists x\in \mathbb{R}_{>0}$ such that $\forall n\in \mathbb{N},x\leq \frac{1}{n}$ + +negation: $\forall x\in \mathbb{R}_{>0}$, $\exists n\in \mathbb{N}$, such that $x\leq \frac{1}{n}$. + +The statement is true, let $x=\frac{1}{n+1}$. + +## New Materials + +### Continue on the theorem + +#### Theorem 1.21 + +$\forall x\in \mathbb{R}_{>0},\forall n\in \mathbb{N},\exist$ unique $y\in \mathbb{R}_{>0}$ such that $y^n=x$. + +(Because of this Theorem we can define $x^{1/x}=y$ and $\sqrt{x}=y$) + +Proof: + +We cna assume $n\geq 2$ (For $n=1,y=x$) + +Step 1 (uniqueness): If $00}: t^n0$ such that $(y+h)^n0$ satisfying $h<1$ and $h<\frac{x-y^h}{n(y+h)^{n-1}}$ + +[For actual proof, see the text.] + +Step 2c showing ($y^n\leq x$) + +Suppose for contradiction $y^n>x$ + +Thoughts: Find $k>0$ such that $(y-k)^n>x$. + +Then $y-k$ is an upper bound for $E$, which contradicts the fact that $y$ is the least upper bound of $E$. + +$y^n-(y-k)^n\leq ny^{n-1}k$. + +We want $y^n-ny^{n-1}k\geq x$. + +So want $k\leq \frac{y^n-x}{ny^{n-1}}$ + +[For actual proof, see the text.] + +QED + +### Complex numbers + +1. $=\{a+bi:a,b\in \mathbb{R}\}$. + +Conjugate: $z=a+bi,\bar{z}=a-bi$. + +#### Theorem 1.31 (see text) + +Pure computational proof: boring... + +$z\bar{z}=a^2-(bi)^2=a^2+b^2$ + +You can also use vector sum for representing operation in complex numbers. + +#### Theorem 1.33 (see text) + +More computation and still, boring... + +some fun theorems: + +- $|Re\ z|\leq |z|$ (equal when no imaginary parts) +- $|z+w|\leq |z|+|w|$ (equal when both $z,w$ have no imaginary parts) (Triangle inequality) + +Proof for $|z+w|\leq |z|+|w|$: + +$$ +|z+w|^2=(z+w)(\overline{z+w})=(z+w)(\bar{z}+\bar{w})=|z|^2+|w|^2+z\bar{w}+\bar{z}w +$$ + +Since + +$$ +z\bar{w}+\bar{z}w\leq 2Re(z\bar{w}) +$$ + +$$ +(z+w)(\bar{z}+\bar{w})=|z|^2+|w|^2+z\bar{w}+\bar{z}w\leq |z|^2+|w|^2+2|z||w|\leq |z|+|w| +$$ + +#### Theorem 1.35 Cauchy-Schwarz inequality + +If $\vec{a},\vec{b}\in \mathbb{C}^n$, then + +$$ +|\vec{a}\vec{b}|^2\leq (\vec{a}\vec{a})(\vec{b}\vec{b}) +$$ + +> Remark: The proof is very tricky. + +To help us motivate the proof in text, let's consider the special case of real numbers. + +$$ +(\sum a_j b_j)^2=(\sum a_j^2)(\sum b_j^2) +$$ + +Proof: + +For real numbers: + +Let $A=\sum a_j^2,B=\sum b_j^2, C=\sum a_j b_j$, want to show $C^2\leq AB$ + +Note: if $B=0$, then $b_1=b_2=...=0$, so $C=0$ and we are done, so we may assume $B\neq 0$ so $B>0$. + +Clever step: For any $t\in \mathbb{R}$, + +$$ +0\leq \sum (a_j-t b_j)^2=\sum (a_j^2-2ta_jb_j+t^2b_i^2)=A-2tC+t^2B +$$ + +let $t=C/B$ to get $0\leq A-2(C/B)C+(C/B)^2B=A-\frac{C^2}{B}$ + +to generalize this to $\mathbb{C}$, $A=\sum |a_j|^2,B=\sum |b_j|^2,C=\sum |a_j \bar{b_j}|$. + +QED + +### Euclidean spaces + +Nothing much to say. lol. + +Normal dot product as inner product. + read text... Theorem 1.37 \ No newline at end of file diff --git a/pages/Math4111/Math4111_L6.md b/content/Math4111/Math4111_L6.md similarity index 96% rename from pages/Math4111/Math4111_L6.md rename to content/Math4111/Math4111_L6.md index 0cec817..ae90125 100644 --- a/pages/Math4111/Math4111_L6.md +++ b/content/Math4111/Math4111_L6.md @@ -1,157 +1,157 @@ -# Lecture 6 - -## Review - -Let $A$ and $B$ be subset of $\mathbb{R}$, and consider the function $f:A \to B$ defined by $f(x)=cos(x)$. Find choices for the domain $A$ and co-domain $B$ which make $f$... - -(a) neither injective nor surjective. - -$A=\mathbb{R},B=\mathbb{R}$ - -(b) surjective but not injective. - -$A=[0,\frac{3\pi}{2}],B=[-1,1]$ - -(c) injective but not surjective. - -$A=[0,\pi],B=[-1,1.1]$ - -(d) bijective. - -$A=(0,\pi),B=(-1,1)$ - -> injective: y don't repeat, surjective: there exists x for each y - -## New Materials (Chapter 2: Basic topology (4171). Finite, countable, and uncountable sets) - -### Functions - -#### Definition 2.1/2.2 - -"$f:A\to B$" means "$f$ is a function from $A$ to $B$" - -$A$:"domain", and $B$: "co-domain". - -If $S\subset A$, the **range** of $S$ under $f$ is $f(S)=\{f(x):s\in S\}$ - -The "**image**" of $f$ is $f(A)$. - -If $T\subset B$, **the inverse image** (pre-image) of $T$ under $f$ is - -$$ -f^{-1}(T)=\{x\in A: f(x)\in T\} -$$ - -* $f$ is injective or one-to-one if $\forall x_1,x_2\in A$ such that $f(x_1)=f(x_2)$, then $x_1=x_2$. ($f(x_1)=f(x_2)\implies x_1=x_2 \equiv x_1\neq x_2\implies f(x_1)\neq f(x_2)$) - -* $f$ is surjective or onto if $\forall y\in B, \exists x\in A$ such that $f(x)=y$. ($f(A)=B$) - -* $f$ is bijective if it's both injective and surjective. - -#### Definition 2.3 - -If $\exists$ bijection $f:A\to B$, we say: - -* $A$ and $B$ can be put into 1-1 correspondence -* $A$ and $B$ b oth have the same cardinality -* $A$ and $B$ are equivalent $A\sim B$ - -### Cardinality - -#### Definition 2.4 - -(a) $A$ is finite if $A\neq \phi$ or $\exists n\in \mathbb{N}$ such that $A\sim \{1,2,...,n\}$ - -(b) $A$ is **infinite** if it's not finite - -(c) $A$ is **countable** if $A\sim \mathbb{N}$ - -(d) $A$ is **uncountable** if $A$ is neither finite nor countable - -(e) $A$ is **at most countable** if it's finite or countable - -> Note in some other books call (c) **countable infinite**, and (e) for **countable** - -#### Definition 2.7 - -A **sequence** in $A$ is a function $f:\mathbb{N}\to\mathbb{A}$ - -Note: By conversion, instead of $f(1),...f(n)$, we usually write $x_1,x_2,...,x_3$ and we say $\{x_n\}_{n=1}^{\infty}$ is a sequence. - -#### Theorem 2.8 - -Every infinite subset of countable set $A$ is countable. - -Ideas of proof: if $A$ is countable, so we can list its element in a sequence. and we iterate $E\subset A$ to create a new order function by deleting element $\notin E$ - -#### Definition 2.9 (arbitrary unions and intersections) - -Let $A$ be a set (called the "index set"). For each $\alpha\in A$, let $E_{\alpha}$ be a set. - -Union: $\bigcup_{\alpha \in A}E_{\alpha}=\{x:\exists \alpha \in A$ such that $x\in E_{\alpha}\}$ - -Intersection: $\bigcap_{\alpha \in A}E_{\alpha}=\{x:\exists \alpha \in A$ such that $x\in E_{\alpha}\}$ - -Special notation for special cases: - -$\bigcup^{n}_{m=1}E_m$ and $E_1\cup E_2\cup ...\cup E_n$ are by definition $\bigcup_{\alpha \in \{1,..,n\}}E_{\alpha}$ - -and $\bigcup^{\infty}_{m=1}E_m$ and $E_1\cup E_2\cup ...\cup E_n$ are by definition $\bigcup_{\alpha \in \mathbb{N}}E_{\alpha}$ - -> Note: Despite the $\infty$ symbol this def makes no references to limits, different from infinite sums. - -### Countability - -#### Theorem 2.12 - -"A countable union of countable sets is countable". - -Let $\{E_n\},n=1,2,3,...$ be a sequence of countable sets, and put - -$$ -S=\bigcup^{\infty}_{n=1} E_n -$$ - -The $S$ is countable. - -Proof by infinite grid, and form a new sequence and remove duplicates. - -#### Corollary - -An at most countable union of at most countable sets is at most countable. - -#### Theorem 2.13 - -$A$ is countable, $n\in \mathbb{N}$, - -$\implies A^n=\{(a_{1},...,a_{n}):a_1\in A, a_n\in A\}$, is countable. - -Proof: - -Induct on $n$, - -Base case $n=1$, - -True by assumptions - -Induction step: suppose $A^{n-1}$ is countable. Note $A^n=\{(b,a):b\in A^{n-1},a\in A\}=\bigcup_{b\in A^{n-1}\{(b,a),a\in A\}}$. - -Since $b$ is fixed, so this is in 1-1 correspondence with $A$, so it's countable by Theorem 2.12. - -QED - -#### Theorem 2.14 - -Let $A$ be the set of all sequences for 0s and 1s. Then $A$ is uncountable. - -Proof: - -Let $E\subset A$ be a countable subset. We'll show $A\backslash E\neq \phi$ (i.e.$\exists t\in A$ such that $t\notin E$) - -$E$ is countable so we can list it's elements $S_1,S_2,S_3,...$. - -Then we define a new sequence $t$ which differs from $S_1$'s first bit and $S_2$'s second bit,... - -This is called Cantor's diagonal argument. - -QED +# Lecture 6 + +## Review + +Let $A$ and $B$ be subset of $\mathbb{R}$, and consider the function $f:A \to B$ defined by $f(x)=cos(x)$. Find choices for the domain $A$ and co-domain $B$ which make $f$... + +(a) neither injective nor surjective. + +$A=\mathbb{R},B=\mathbb{R}$ + +(b) surjective but not injective. + +$A=[0,\frac{3\pi}{2}],B=[-1,1]$ + +(c) injective but not surjective. + +$A=[0,\pi],B=[-1,1.1]$ + +(d) bijective. + +$A=(0,\pi),B=(-1,1)$ + +> injective: y don't repeat, surjective: there exists x for each y + +## New Materials (Chapter 2: Basic topology (4171). Finite, countable, and uncountable sets) + +### Functions + +#### Definition 2.1/2.2 + +"$f:A\to B$" means "$f$ is a function from $A$ to $B$" + +$A$:"domain", and $B$: "co-domain". + +If $S\subset A$, the **range** of $S$ under $f$ is $f(S)=\{f(x):s\in S\}$ + +The "**image**" of $f$ is $f(A)$. + +If $T\subset B$, **the inverse image** (pre-image) of $T$ under $f$ is + +$$ +f^{-1}(T)=\{x\in A: f(x)\in T\} +$$ + +* $f$ is injective or one-to-one if $\forall x_1,x_2\in A$ such that $f(x_1)=f(x_2)$, then $x_1=x_2$. ($f(x_1)=f(x_2)\implies x_1=x_2 \equiv x_1\neq x_2\implies f(x_1)\neq f(x_2)$) + +* $f$ is surjective or onto if $\forall y\in B, \exists x\in A$ such that $f(x)=y$. ($f(A)=B$) + +* $f$ is bijective if it's both injective and surjective. + +#### Definition 2.3 + +If $\exists$ bijection $f:A\to B$, we say: + +* $A$ and $B$ can be put into 1-1 correspondence +* $A$ and $B$ b oth have the same cardinality +* $A$ and $B$ are equivalent $A\sim B$ + +### Cardinality + +#### Definition 2.4 + +(a) $A$ is finite if $A\neq \phi$ or $\exists n\in \mathbb{N}$ such that $A\sim \{1,2,...,n\}$ + +(b) $A$ is **infinite** if it's not finite + +(c) $A$ is **countable** if $A\sim \mathbb{N}$ + +(d) $A$ is **uncountable** if $A$ is neither finite nor countable + +(e) $A$ is **at most countable** if it's finite or countable + +> Note in some other books call (c) **countable infinite**, and (e) for **countable** + +#### Definition 2.7 + +A **sequence** in $A$ is a function $f:\mathbb{N}\to\mathbb{A}$ + +Note: By conversion, instead of $f(1),...f(n)$, we usually write $x_1,x_2,...,x_3$ and we say $\{x_n\}_{n=1}^{\infty}$ is a sequence. + +#### Theorem 2.8 + +Every infinite subset of countable set $A$ is countable. + +Ideas of proof: if $A$ is countable, so we can list its element in a sequence. and we iterate $E\subset A$ to create a new order function by deleting element $\notin E$ + +#### Definition 2.9 (arbitrary unions and intersections) + +Let $A$ be a set (called the "index set"). For each $\alpha\in A$, let $E_{\alpha}$ be a set. + +Union: $\bigcup_{\alpha \in A}E_{\alpha}=\{x:\exists \alpha \in A$ such that $x\in E_{\alpha}\}$ + +Intersection: $\bigcap_{\alpha \in A}E_{\alpha}=\{x:\exists \alpha \in A$ such that $x\in E_{\alpha}\}$ + +Special notation for special cases: + +$\bigcup^{n}_{m=1}E_m$ and $E_1\cup E_2\cup ...\cup E_n$ are by definition $\bigcup_{\alpha \in \{1,..,n\}}E_{\alpha}$ + +and $\bigcup^{\infty}_{m=1}E_m$ and $E_1\cup E_2\cup ...\cup E_n$ are by definition $\bigcup_{\alpha \in \mathbb{N}}E_{\alpha}$ + +> Note: Despite the $\infty$ symbol this def makes no references to limits, different from infinite sums. + +### Countability + +#### Theorem 2.12 + +"A countable union of countable sets is countable". + +Let $\{E_n\},n=1,2,3,...$ be a sequence of countable sets, and put + +$$ +S=\bigcup^{\infty}_{n=1} E_n +$$ + +The $S$ is countable. + +Proof by infinite grid, and form a new sequence and remove duplicates. + +#### Corollary + +An at most countable union of at most countable sets is at most countable. + +#### Theorem 2.13 + +$A$ is countable, $n\in \mathbb{N}$, + +$\implies A^n=\{(a_{1},...,a_{n}):a_1\in A, a_n\in A\}$, is countable. + +Proof: + +Induct on $n$, + +Base case $n=1$, + +True by assumptions + +Induction step: suppose $A^{n-1}$ is countable. Note $A^n=\{(b,a):b\in A^{n-1},a\in A\}=\bigcup_{b\in A^{n-1}\{(b,a),a\in A\}}$. + +Since $b$ is fixed, so this is in 1-1 correspondence with $A$, so it's countable by Theorem 2.12. + +QED + +#### Theorem 2.14 + +Let $A$ be the set of all sequences for 0s and 1s. Then $A$ is uncountable. + +Proof: + +Let $E\subset A$ be a countable subset. We'll show $A\backslash E\neq \phi$ (i.e.$\exists t\in A$ such that $t\notin E$) + +$E$ is countable so we can list it's elements $S_1,S_2,S_3,...$. + +Then we define a new sequence $t$ which differs from $S_1$'s first bit and $S_2$'s second bit,... + +This is called Cantor's diagonal argument. + +QED diff --git a/pages/Math4111/Math4111_L7.md b/content/Math4111/Math4111_L7.md similarity index 96% rename from pages/Math4111/Math4111_L7.md rename to content/Math4111/Math4111_L7.md index b4a7423..a9be63d 100644 --- a/pages/Math4111/Math4111_L7.md +++ b/content/Math4111/Math4111_L7.md @@ -1,109 +1,109 @@ -# Lecture 7 - -## Review - -Let $S=\{(x,y,z)\in \mathbb{R}^3:x=1,y=4\}=\{(1,4,z):z\in\mathbb{R}\}$ - -1. How can we describe the set $S$ geometrically in three-dimensional space? - - Just a line -2. Show that $S$ and $\mathbb{R}$ are in one-to-one correspondence. - - We can find a bijective function $f:S\to \mathbb{R}$ -3. Show that for any $(a,b)\in\mathbb{Z}^2$, the set $\{(a,b,z):z\in\mathbb{Z}\}$ is in one-to-one correspondence with $\mathbb{Z}$ - - Use **Theorem 2.13** $A$ is countable, $n\in \mathbb{N} \implies A^n=\{(a_{1},...,a_{n}):a_1\in A, a_n\in A\}$, is countable. - -## New materials - -### Metric spaces - -#### Definition 2.15 - -Let $X$ be a set. A function $d:X\times X\to \mathbb{R}$ is called a distance function or a metric if it satisfies: - -1. Positivity: $\forall p,q\in X,p\neq q\implies d(p,q)>0$, and $\forall p\in X,d(p,p)=0$. -2. Symmetry: $\forall p,q\in X, d(p,q)=d(q,p)$. -3. Triangle inequality: $\forall p,q,r\in X$, $d(p,q)\leq d(p,r)+d(r,q)$ - -We say **$(X,d)$ is a metric space**. If $d$ is understood, $X$ is a metric space. - -Examples: - -The most important example: - -$X\subset \mathbb{R}^k(k\geq 1)$ - -$d(x,y)=|x-y|$ - -And other examples: function spaces... - -An example from graph theory (not needed for this class): - -$d(p,q)$ can be defined by the shortest path fro $p$ to $q$. - -#### Definition 2.17 - -By the *segment* $(a,b)$ we mean the set of all real numbers $x$ such that $a0$. The r-neighborhood of $p$ is $B_r(p)=N_r(o)=\{q\in X: d(p,q)0$ such that $B_r(p)\subset E$. Notation $E^{\circ}=$set of interior points of $E$ -3. $E\subset X$, we say $E$ is **open** if $E\subset E^{\circ}$, i.e. $\forall p\in E, \exists r>0$ such that $B_r(p)\subset E$. - -*Note: is follows from definitions that $E^{\circ}\subset E$ is always true.* - -Example: - -$X=\mathbb{R}^2$($d$ be the euclidean distance) $E=[0,1)\times [0,1)$. - -$E^{\circ}=(0,1)\times (0,1)$ - -So $E=(0,1)\times (0,1)$ is a open set. - -#### Theorem 2.19 - -Let $(X,d)$ be a metric space, $\forall p\in X,\forall r>0$, $B_r(p)$ is an open set. - -*every ball is an open set* - -Proof: - -Let $q\in B_r(p)$. - -Let $h=r-d(p,q)$. - -Since $q\in B_r(p),h>0$. We claim that $B_h(q)$. Then $d(q,s)0, (B_r(p)\cap E)\backslash {p}\neq \phi$. - - Let $E'$ be the set of limit points of $E$. - -2. $E$ is closed if $E'\subset E$ - -Example: - -$X=\mathbb{R}^2$, $E=[0,1)\times [0,1)$. - -$(1,1)$ is a limit point. - -$X=\mathbb{R},E=\{\frac{1}{n},n\in \mathbb{N}\}$ - -$0$ is the only limit point. $E'=\{0\}$ +# Lecture 7 + +## Review + +Let $S=\{(x,y,z)\in \mathbb{R}^3:x=1,y=4\}=\{(1,4,z):z\in\mathbb{R}\}$ + +1. How can we describe the set $S$ geometrically in three-dimensional space? + + Just a line +2. Show that $S$ and $\mathbb{R}$ are in one-to-one correspondence. + + We can find a bijective function $f:S\to \mathbb{R}$ +3. Show that for any $(a,b)\in\mathbb{Z}^2$, the set $\{(a,b,z):z\in\mathbb{Z}\}$ is in one-to-one correspondence with $\mathbb{Z}$ + + Use **Theorem 2.13** $A$ is countable, $n\in \mathbb{N} \implies A^n=\{(a_{1},...,a_{n}):a_1\in A, a_n\in A\}$, is countable. + +## New materials + +### Metric spaces + +#### Definition 2.15 + +Let $X$ be a set. A function $d:X\times X\to \mathbb{R}$ is called a distance function or a metric if it satisfies: + +1. Positivity: $\forall p,q\in X,p\neq q\implies d(p,q)>0$, and $\forall p\in X,d(p,p)=0$. +2. Symmetry: $\forall p,q\in X, d(p,q)=d(q,p)$. +3. Triangle inequality: $\forall p,q,r\in X$, $d(p,q)\leq d(p,r)+d(r,q)$ + +We say **$(X,d)$ is a metric space**. If $d$ is understood, $X$ is a metric space. + +Examples: + +The most important example: + +$X\subset \mathbb{R}^k(k\geq 1)$ + +$d(x,y)=|x-y|$ + +And other examples: function spaces... + +An example from graph theory (not needed for this class): + +$d(p,q)$ can be defined by the shortest path fro $p$ to $q$. + +#### Definition 2.17 + +By the *segment* $(a,b)$ we mean the set of all real numbers $x$ such that $a0$. The r-neighborhood of $p$ is $B_r(p)=N_r(o)=\{q\in X: d(p,q)0$ such that $B_r(p)\subset E$. Notation $E^{\circ}=$set of interior points of $E$ +3. $E\subset X$, we say $E$ is **open** if $E\subset E^{\circ}$, i.e. $\forall p\in E, \exists r>0$ such that $B_r(p)\subset E$. + +*Note: is follows from definitions that $E^{\circ}\subset E$ is always true.* + +Example: + +$X=\mathbb{R}^2$($d$ be the euclidean distance) $E=[0,1)\times [0,1)$. + +$E^{\circ}=(0,1)\times (0,1)$ + +So $E=(0,1)\times (0,1)$ is a open set. + +#### Theorem 2.19 + +Let $(X,d)$ be a metric space, $\forall p\in X,\forall r>0$, $B_r(p)$ is an open set. + +*every ball is an open set* + +Proof: + +Let $q\in B_r(p)$. + +Let $h=r-d(p,q)$. + +Since $q\in B_r(p),h>0$. We claim that $B_h(q)$. Then $d(q,s)0, (B_r(p)\cap E)\backslash {p}\neq \phi$. + + Let $E'$ be the set of limit points of $E$. + +2. $E$ is closed if $E'\subset E$ + +Example: + +$X=\mathbb{R}^2$, $E=[0,1)\times [0,1)$. + +$(1,1)$ is a limit point. + +$X=\mathbb{R},E=\{\frac{1}{n},n\in \mathbb{N}\}$ + +$0$ is the only limit point. $E'=\{0\}$ diff --git a/pages/Math4111/Math4111_L8.md b/content/Math4111/Math4111_L8.md similarity index 96% rename from pages/Math4111/Math4111_L8.md rename to content/Math4111/Math4111_L8.md index 8a61e9c..3cd76ca 100644 --- a/pages/Math4111/Math4111_L8.md +++ b/content/Math4111/Math4111_L8.md @@ -1,150 +1,150 @@ -# Lecture 8 - -## Review - -Let $(X,d)$ be a metric space. Recall that $B_r(x)=\{z\in X:d(x,z)0$, $B_r(p)=\{q\in X:d(p,q)0$, $(B_s(p)\cap E)\backslash \{p\}\neq \phi$ -3. If $p\in E$ and $p$ is not a limit point of $E$, then $p$ is called an **isolated point** of $E$. -4. $E$ is **closed** if $E'\subset E$ -5. $p$ is a **interior point** of $E(p\in E^{\circ})$ if $\exists r>0$ such that $B_r(p)\subset E$. - -## New materials - -### Metric space - -#### Theorem 2.20 - -$p\in E'\implies \forall r>0,B_r(p)\cap E$ is infinite. - -Proof: - -We will prove the contrapositive. - -want to prove $\exists r>0$ such that $B_r(p)\cap E$ is finite $\implies p\notin E'$ ($\exists s>0$ such that $(B_s(p)\cap E)\backslash \{p\}=\phi$) - -Suppose $\exists r>0$ such that $B_r(p)\cap E$ is finite - -let $B_s(p)\cap E)\backslash \{p\}={q_1,...,q_n}$ - -- If $n=0$, then $B_s(p)\cap E)\backslash \{p\}=\phi$, so $p\in E'$ -- If $n\geq 1$, then let $s=min\{d(p,q_m):1\leq m\leq n\}$ - - Each $d(p,q_m)$ is positive and the set is finite, so $s>0$. - -Then $(B_s(p)\cap E)\backslash \{p\}=\phi$, so $p\notin E$ - -QED - -#### Theorem 2.22 De Morgan's law - -$$ -\left(\bigcup_a E_a\right)^c=\bigcap_a(E^c_a) -$$ - -$E^c=X\backslash E$ - -Proof: - -$x\in \cup_{a\in A} E_x\iff \exists a\in A$ such that $x\in E_a$ - -So $x\in \left(\bigcup_a E_a\right)^c\iff \forall a\in A, x\notin E_a\iff \forall a\in A,x\in E_a^c\iff \bigcap_a(E^c_a)$ - -#### Theorem 2.23 - -$E$ is open $\iff$ $E^c$ is closed. - -> Warning: $E$ is open $\cancel{\iff}$ $E$ is closed. -> $E$ is closed $\cancel{\iff}$ $E$ is open. -> -> Example: ->$\phi$, $\R$ is both open and closed. "clopen set" ->$[0,1)$ is not open and not closed. bad... - -Proof: - -$\impliedby$ Suppose $E^c$ is closed. Let $x\in E$, so $x\notin E^c$ - -$E^c$ is closed and $x\notin E^c\implies x\notin (E^c)'\implies \exists r >0$ such that $(B_r(x)\cap E^c)\backslash \{x\}=\phi$ - -So $\phi=(B_r(x)\cap E^c)\backslash \{x\}=B_r(x)\cap E^c$ - -So $B_r(x)\in E$ - -$\implies$ - -Suppose $E$ is open - -$$ -\begin{aligned} - x\in (E^c)'&\implies \forall r>0, (B_r(x)\cap E^c)\backslash \{x\}\neq \phi\\ - &\implies \forall r>0, (B_r(x)\cap E^c)\neq \phi\\ - &\implies \forall r>0, B-r(x)\notin E\\ - &\implies x\notin E^{\circ}\\ - &\implies x\notin E\\ - &\implies x\in E^c -\end{aligned} -$$ - -So $(E^c)'\subset E^c$ - -QED - -#### Theorem 2.24 - -##### An arbitrary union of open sets is open - -Proof: - -Suppose $\forall \alpha, G_\alpha$ is open. Let $x\in \bigcup _{\alpha} G_\alpha$. Then $\exists \alpha_0$ such that $x\in G_{\alpha_0}$. Since $G_{\alpha_0}$ is open, $\exists r>0$ such that $B_r(x)\subset G_{\alpha_0}$ Then $B_r(x)\subset G_{\alpha_0}\subset \bigcup_{\alpha} G_\alpha$ - -QED - -##### A finite intersection of open set is open - -Proof: - -Suppose $\forall i\in \{1,...,n\}$, $G_i$ is open. - -Let $x\in \bigcap^n_{i=1}G_i$, then $\forall i\in \{1,..,n\}$ and $G_i$ is open, so $\exists r_i>0$, such that $B_{r_i}(x)\subset G_i$ - -Let $r=min\{r_1,...,r_n\}$. Then $\forall i\in \{1,...,n\}$. $B_r(x)\subset B_{r_i}(x)\subset G_i$. So $B_r(x)\subset \bigcup_{i=1}^n G_i$ - -QED - -The other two can be proved by **Theorem 2.22,2.23** - -#### Definition 2.26 - -The closure $\bar{E}=E\cup E'$ - -Remark: Using the definition of $E'$, we have, $\bar{E}=\{p\in X,\forall r>0,B_r(p)\cap E\neq \phi\}$ - -#### Definition 2.27 - -$\bar {E}$ is closed. - -Proof: - -We will show $\bar{E}^c$ is open. - -Suppose $p\in \bar{E}^c$. Then by remark, $\exists r>0$ such that $B_r(p)\cap E=\phi$ (a) - -Furthermore,, we claim $B_r(p)\cap E'=\phi$ (b) - -Suppose for contradiction that $\exists q\in B_r(p)\cap E'$ By **Theorem 2.19**, $\exists s>0$ such that $B_s(q)\subset B_r(p)$ - -Since $q\in E',(B_s(q)\cap E)\backslash \{q\}\neq \phi$. This implies $B_r(p)\cap E=\phi$, which contradicts with (a) - -This proves (b) - -So $\bar{E}^c$ is open - +# Lecture 8 + +## Review + +Let $(X,d)$ be a metric space. Recall that $B_r(x)=\{z\in X:d(x,z)0$, $B_r(p)=\{q\in X:d(p,q)0$, $(B_s(p)\cap E)\backslash \{p\}\neq \phi$ +3. If $p\in E$ and $p$ is not a limit point of $E$, then $p$ is called an **isolated point** of $E$. +4. $E$ is **closed** if $E'\subset E$ +5. $p$ is a **interior point** of $E(p\in E^{\circ})$ if $\exists r>0$ such that $B_r(p)\subset E$. + +## New materials + +### Metric space + +#### Theorem 2.20 + +$p\in E'\implies \forall r>0,B_r(p)\cap E$ is infinite. + +Proof: + +We will prove the contrapositive. + +want to prove $\exists r>0$ such that $B_r(p)\cap E$ is finite $\implies p\notin E'$ ($\exists s>0$ such that $(B_s(p)\cap E)\backslash \{p\}=\phi$) + +Suppose $\exists r>0$ such that $B_r(p)\cap E$ is finite + +let $B_s(p)\cap E)\backslash \{p\}={q_1,...,q_n}$ + +- If $n=0$, then $B_s(p)\cap E)\backslash \{p\}=\phi$, so $p\in E'$ +- If $n\geq 1$, then let $s=min\{d(p,q_m):1\leq m\leq n\}$ + + Each $d(p,q_m)$ is positive and the set is finite, so $s>0$. + +Then $(B_s(p)\cap E)\backslash \{p\}=\phi$, so $p\notin E$ + +QED + +#### Theorem 2.22 De Morgan's law + +$$ +\left(\bigcup_a E_a\right)^c=\bigcap_a(E^c_a) +$$ + +$E^c=X\backslash E$ + +Proof: + +$x\in \cup_{a\in A} E_x\iff \exists a\in A$ such that $x\in E_a$ + +So $x\in \left(\bigcup_a E_a\right)^c\iff \forall a\in A, x\notin E_a\iff \forall a\in A,x\in E_a^c\iff \bigcap_a(E^c_a)$ + +#### Theorem 2.23 + +$E$ is open $\iff$ $E^c$ is closed. + +> Warning: $E$ is open $\cancel{\iff}$ $E$ is closed. +> $E$ is closed $\cancel{\iff}$ $E$ is open. +> +> Example: +>$\phi$, $\R$ is both open and closed. "clopen set" +>$[0,1)$ is not open and not closed. bad... + +Proof: + +$\impliedby$ Suppose $E^c$ is closed. Let $x\in E$, so $x\notin E^c$ + +$E^c$ is closed and $x\notin E^c\implies x\notin (E^c)'\implies \exists r >0$ such that $(B_r(x)\cap E^c)\backslash \{x\}=\phi$ + +So $\phi=(B_r(x)\cap E^c)\backslash \{x\}=B_r(x)\cap E^c$ + +So $B_r(x)\in E$ + +$\implies$ + +Suppose $E$ is open + +$$ +\begin{aligned} + x\in (E^c)'&\implies \forall r>0, (B_r(x)\cap E^c)\backslash \{x\}\neq \phi\\ + &\implies \forall r>0, (B_r(x)\cap E^c)\neq \phi\\ + &\implies \forall r>0, B-r(x)\notin E\\ + &\implies x\notin E^{\circ}\\ + &\implies x\notin E\\ + &\implies x\in E^c +\end{aligned} +$$ + +So $(E^c)'\subset E^c$ + +QED + +#### Theorem 2.24 + +##### An arbitrary union of open sets is open + +Proof: + +Suppose $\forall \alpha, G_\alpha$ is open. Let $x\in \bigcup _{\alpha} G_\alpha$. Then $\exists \alpha_0$ such that $x\in G_{\alpha_0}$. Since $G_{\alpha_0}$ is open, $\exists r>0$ such that $B_r(x)\subset G_{\alpha_0}$ Then $B_r(x)\subset G_{\alpha_0}\subset \bigcup_{\alpha} G_\alpha$ + +QED + +##### A finite intersection of open set is open + +Proof: + +Suppose $\forall i\in \{1,...,n\}$, $G_i$ is open. + +Let $x\in \bigcap^n_{i=1}G_i$, then $\forall i\in \{1,..,n\}$ and $G_i$ is open, so $\exists r_i>0$, such that $B_{r_i}(x)\subset G_i$ + +Let $r=min\{r_1,...,r_n\}$. Then $\forall i\in \{1,...,n\}$. $B_r(x)\subset B_{r_i}(x)\subset G_i$. So $B_r(x)\subset \bigcup_{i=1}^n G_i$ + +QED + +The other two can be proved by **Theorem 2.22,2.23** + +#### Definition 2.26 + +The closure $\bar{E}=E\cup E'$ + +Remark: Using the definition of $E'$, we have, $\bar{E}=\{p\in X,\forall r>0,B_r(p)\cap E\neq \phi\}$ + +#### Definition 2.27 + +$\bar {E}$ is closed. + +Proof: + +We will show $\bar{E}^c$ is open. + +Suppose $p\in \bar{E}^c$. Then by remark, $\exists r>0$ such that $B_r(p)\cap E=\phi$ (a) + +Furthermore,, we claim $B_r(p)\cap E'=\phi$ (b) + +Suppose for contradiction that $\exists q\in B_r(p)\cap E'$ By **Theorem 2.19**, $\exists s>0$ such that $B_s(q)\subset B_r(p)$ + +Since $q\in E',(B_s(q)\cap E)\backslash \{q\}\neq \phi$. This implies $B_r(p)\cap E=\phi$, which contradicts with (a) + +This proves (b) + +So $\bar{E}^c$ is open + QED \ No newline at end of file diff --git a/pages/Math4111/Math4111_L9.md b/content/Math4111/Math4111_L9.md similarity index 97% rename from pages/Math4111/Math4111_L9.md rename to content/Math4111/Math4111_L9.md index 2555979..83fe0e4 100644 --- a/pages/Math4111/Math4111_L9.md +++ b/content/Math4111/Math4111_L9.md @@ -1,104 +1,104 @@ -# Lecture 9 - -## Review - -1. Let $X=\mathbb{R}$ (and as usual, let $d(x,y)=|x-y|$). What is the set $B_1(0)$ - $B_1(0)=(-1,1)$ -2. Let $X=[0,5]$ (and let $d$ be as usual). What is the set $B_1(0)$. - $B_1(0)=[0,1)$ -3. Let $X=\mathbb{R}$ and let $E=[0,2)$. Is $E$ open? - No, $0$ is not a interior point. -4. Let $X=[0,5]$ and let $E=[0,2)$. Is $E$ open? - Yes, $0$ is a interior point, we can set radius to $1$ and all the points of $B_1(0)\subset E$ - -## Continue on new materials - -### Metric space - -#### Theorem 2.27 - -If $X$ is a metric spae and $E\subset X$, then. - -1. $\bar{E}$ is closed. -2. $E=\bar{E}$ if and only if $E$ is closed. - Proof: $E=\bar{E}\iff E=E\cup E'\iff E'\subset E\iff E$ is closed. -3. $\bar{E}\subset F$ for every closed set $F\subset X$ such that $E\subset F$. - $F\subset X$ closed, and $E'\subset F'\subset F$, so $\bar{E}=E\cup E'\subset F$ - -#### Theorem 2.28 - -$E\subset \mathbb{R}$ non-$\phi$ and bounded above, $\sup E\in \bar{E}$ - -Proof: - -Let $y\sup E$, To show $y\in \bar{E}$, we need to show $\forall h>0$, $B_h(y)\cap E\neq \phi$ - -Let $h>0$. Since $y-h$ is not an upper bound of $E$, $\exists x\in E$ such that $x>y-h$. - -Since $y$ is an upper bound of $E$, $x\leq y$. So $x\in B_n(y)\cap E$, so $B_h(y)\cap E\neq \phi$. - -QED - -#### Remark 2.29 - -Let $(X,d)$ be a metric space, $E\subset X$. "$E$ is open" is short for "$E$ is open in $X$"/$E$ is open relative to $X$. - -This means $\forall p\in E$, $\exists r>0\{q\in X:d(p,q)0\{q\in Y:d(p,q)0$, then $\{q\in Y:d(p,q)0$ such that $\{q\in X:d(p,q)0$ such that $\{q\in Y:d(p,q)0$, $B_h(y)\cap E\neq \phi$ + +Let $h>0$. Since $y-h$ is not an upper bound of $E$, $\exists x\in E$ such that $x>y-h$. + +Since $y$ is an upper bound of $E$, $x\leq y$. So $x\in B_n(y)\cap E$, so $B_h(y)\cap E\neq \phi$. + +QED + +#### Remark 2.29 + +Let $(X,d)$ be a metric space, $E\subset X$. "$E$ is open" is short for "$E$ is open in $X$"/$E$ is open relative to $X$. + +This means $\forall p\in E$, $\exists r>0\{q\in X:d(p,q)0\{q\in Y:d(p,q)0$, then $\{q\in Y:d(p,q)0$ such that $\{q\in X:d(p,q)0$ such that $\{q\in Y:d(p,q)dim(W)$, then there are no injective maps from $V$ to $W$. - -#### Theorem 3.24 - -Suppose $V,W$ are finite dimensional with $dim(V)0$ - -### Linear Maps and Linear Systems 3EX-1 - -Suppose we have a homogeneous linear system * with $m$ equation and $n$ variables. - -$$ -A_{11} x_1+ ... + A_{1n} x_n=0\\ -...\\ -A_{m1} x_1+ ... + A_{mn} x_n=0 -$$ - -which is equivalent to - -$$ -A\begin{bmatrix} - x_1\\...\\x_n -\end{bmatrix}=\vec{0} -$$ - -also equivalent to - -$$ -T(v)=0,\textup{ for some }T -$$ - -$$ -T(x_1,...,x_n)=(A_{11} x_1+ ... + A_{1n},...,A_{m1} x_1+ ... + A_{mn} x_n),T\in \mathscr{L}(\mathbb{R}^n,\mathbb{R}^m) -$$ - -Solution to * is $null(T)$. - -#### Proposition 3.26 - -A homogeneous linear system with more variables than equations has non-zero solutions. - -Proof: - -Using $T$ as above, note that since $n>m$, use **Theorem 3.22**, implies that $T$ cannot be injective. So, $null (T)$ contains a non-zero vector. - -#### Proposition 3.28 - -An in-homogenous system with more equations than variables has no solutions for some choices of constants. ($A\vec{x}=\vec{b}$ for some $\vec{b}$ this has no solution) - -### Matrices 3A - -#### Definition 3.29 - -For $m,n>0$ and $m\times n$ matrix $A$ is a rectangular array with elements of the $\mathbb{F}$ given by - -$$ -A=\begin{pmatrix} - A_{1,1}& ...&A_{1,n}\\ - ... & & ...\\ - A_{n,1}&...&A_{m,n}\\ -\end{pmatrix} -$$ - -### Operations on matrices - -Addition: - -$$ -A+B=\begin{pmatrix} - A_{1,1}+B_{1,1}& ...&A_{1,n}+B_{1,n}\\ - ... & & ...\\ - A_{n,1}+A_{n,1}&...&A_{m,n}+B_{m,n}\\ -\end{pmatrix} -$$ - -**for $A+B$, $A,B$ need to be the same size** - -Scalar multiplication: - -$$ -\lambda A=\begin{pmatrix} - \lambda A_{1,1}& ...& \lambda A_{1,n}\\ - ... & & ...\\ - \lambda A_{n,1}&...& \lambda A_{m,n}\\ -\end{pmatrix} -$$ - -#### Definition 3.39 - -$\mathbb{F}^{m,n}$ is the set of $m$ by $n$ matrices. - -#### Theorem 3.40 - -$\mathbb{F}^{m,n}$ is a vector space (over $\mathbb{F}$) with $dim(\mathbb{F}^{m,n})=m\times n$ - -### Matrix multiplication 3EX-2 - -Let $A$ be a $m\times n$ matrix and $B$ be an $n\times s$ matrix - -$$ -(A,B)_{i,j}= \sum^n_{r=1} A_{i,r}\cdot B_{r,j} -$$ - -Claim: - -This formula comes from multiplication of linear maps. - -#### Definition 3.44 - -Linear maps to matrices, let $V$, $W$, $Tv_i$ written in terms of $w_i$. - -$$ -M(T)=\begin{pmatrix} - Tv_1\vert Tv_2\vert ...\vert Tv_n -\end{pmatrix} +# Lecture 10 + +## Chapter III Linear maps + +**Assumption: $U,V,W$ are vector spaces (over $\mathbb{F}$)** + +### Vector Space of Linear Maps 3A + +Review + +#### Theorem 3.21 (The Fundamental Theorem of Linear Maps, Rank-nullity Theorem) + +Suppose $V$ is finite dimensional, and $T\in \mathscr{L}(V,W)$, then $range(T)$ is finite dimensional ($W$ don't need to be finite dimensional). and + +$$ +dim(V)=dim(null (T))+dim(range(T)) +$$ + +Proof: + +Let $u_1,...,u_m$ be a basis for $null(T)$, then we extend to a basis of $V$ given by $u_1,...,u_m,v_1,...,v_m$, we have $dim(V)=m+n$. Claim that $Tv_1,...,Tv_n$ forms a basis for $range (T)$. Need to show + +* Linearly independent. (in Homework 3) +* These span $range(T)$. + + Let $w\in range(T)$ the there exists $v\in V$ such that $Tv=W$, $u_1,...,u_m,v_1,...,v_m$ are basis so $\exists a_1,...,a_m,b_1,...,b_n$ such that $v=a_1u_1+...+a_mu_m+b_1v_1+...+b_n v_n$. $Tv=a_1Tu_1+...+a_mTu_m+b_1Tu_1+...+b_nTv_n$. + + Since $u_k\in null(T)$, So $Tv_1,...,Tv_n$ spans range $T$ and so form a basis. Thus $range(T)$ is finite dimensional and $dim(range(T))=n$. So $dim(V)=dim(null (T))+dim(range(T))$ + +#### Theorem 3.22 + +Suppose $V,W$ are finite dimensional with $dim(V)>dim(W)$, then there are no injective maps from $V$ to $W$. + +#### Theorem 3.24 + +Suppose $V,W$ are finite dimensional with $dim(V)0$ + +### Linear Maps and Linear Systems 3EX-1 + +Suppose we have a homogeneous linear system * with $m$ equation and $n$ variables. + +$$ +A_{11} x_1+ ... + A_{1n} x_n=0\\ +...\\ +A_{m1} x_1+ ... + A_{mn} x_n=0 +$$ + +which is equivalent to + +$$ +A\begin{bmatrix} + x_1\\...\\x_n +\end{bmatrix}=\vec{0} +$$ + +also equivalent to + +$$ +T(v)=0,\textup{ for some }T +$$ + +$$ +T(x_1,...,x_n)=(A_{11} x_1+ ... + A_{1n},...,A_{m1} x_1+ ... + A_{mn} x_n),T\in \mathscr{L}(\mathbb{R}^n,\mathbb{R}^m) +$$ + +Solution to * is $null(T)$. + +#### Proposition 3.26 + +A homogeneous linear system with more variables than equations has non-zero solutions. + +Proof: + +Using $T$ as above, note that since $n>m$, use **Theorem 3.22**, implies that $T$ cannot be injective. So, $null (T)$ contains a non-zero vector. + +#### Proposition 3.28 + +An in-homogenous system with more equations than variables has no solutions for some choices of constants. ($A\vec{x}=\vec{b}$ for some $\vec{b}$ this has no solution) + +### Matrices 3A + +#### Definition 3.29 + +For $m,n>0$ and $m\times n$ matrix $A$ is a rectangular array with elements of the $\mathbb{F}$ given by + +$$ +A=\begin{pmatrix} + A_{1,1}& ...&A_{1,n}\\ + ... & & ...\\ + A_{n,1}&...&A_{m,n}\\ +\end{pmatrix} +$$ + +### Operations on matrices + +Addition: + +$$ +A+B=\begin{pmatrix} + A_{1,1}+B_{1,1}& ...&A_{1,n}+B_{1,n}\\ + ... & & ...\\ + A_{n,1}+A_{n,1}&...&A_{m,n}+B_{m,n}\\ +\end{pmatrix} +$$ + +**for $A+B$, $A,B$ need to be the same size** + +Scalar multiplication: + +$$ +\lambda A=\begin{pmatrix} + \lambda A_{1,1}& ...& \lambda A_{1,n}\\ + ... & & ...\\ + \lambda A_{n,1}&...& \lambda A_{m,n}\\ +\end{pmatrix} +$$ + +#### Definition 3.39 + +$\mathbb{F}^{m,n}$ is the set of $m$ by $n$ matrices. + +#### Theorem 3.40 + +$\mathbb{F}^{m,n}$ is a vector space (over $\mathbb{F}$) with $dim(\mathbb{F}^{m,n})=m\times n$ + +### Matrix multiplication 3EX-2 + +Let $A$ be a $m\times n$ matrix and $B$ be an $n\times s$ matrix + +$$ +(A,B)_{i,j}= \sum^n_{r=1} A_{i,r}\cdot B_{r,j} +$$ + +Claim: + +This formula comes from multiplication of linear maps. + +#### Definition 3.44 + +Linear maps to matrices, let $V$, $W$, $Tv_i$ written in terms of $w_i$. + +$$ +M(T)=\begin{pmatrix} + Tv_1\vert Tv_2\vert ...\vert Tv_n +\end{pmatrix} $$ \ No newline at end of file diff --git a/pages/Math429/Math429_L11.md b/content/Math429/Math429_L11.md similarity index 95% rename from pages/Math429/Math429_L11.md rename to content/Math429/Math429_L11.md index 32546b8..ec62239 100644 --- a/pages/Math429/Math429_L11.md +++ b/content/Math429/Math429_L11.md @@ -1,153 +1,153 @@ -# Lecture 11 - -## Chapter III Linear maps - -**Assumption: $U,V,W$ are vector spaces (over $\mathbb{F}$)** - -### Matrices 3C - -#### Definition 3.31 - -Suppose $T\in \mathscr{L}(V,W)$, $v_1,...,v_n$ a basis for $V$ $w_1,...,w_m$ a basis for $W$. Then $M(T)=M(T,(v_1,...,v_n),(w_1,...,w_m))$ is given by $M(T)=A$n where - -$$ -T_{v_k}=A_{1,k}w_1+...+A_{m,k}w_m -$$ - -$$ -\begin{matrix} - & & v_1& & v_2&&...&v_n& -\end{matrix}\\ -\begin{matrix} -w_1\\w_2\\.\\.\\.\\w_m -\end{matrix} -\begin{pmatrix} -A_{1,1} & A_{1,2} &...& A_{1,n}\\ -A_{2,1} & A_{2,2} &...& A_{2,n}\\ -. & . &...&.\\ -. & . &...&.\\ -. & . &...&.\\ -A_{m,1} & A_{m,2} &...& A_{m,n}\\ -\end{pmatrix} -$$ - -Example: - -* $T:\mathbb{F}^2\to \mathbb{F}^3$ - - $T(x,y)=(x+3y,2x+5y,7x+9y)$ - - $M(T)=\begin{pmatrix} - 1&3\\ - 2&5\\ - 7&9 - \end{pmatrix}$ - -* Let $D:\mathscr{P}_3(\mathbb{F})\to \mathscr{P}_2(\mathbb{F})$ be differentiation - - $M(T)=\begin{pmatrix} - 0&1&0&0\\ - 0&0&2&0\\ - 0&0&0&3\\ - \end{pmatrix}$ - -#### Lemma 3.35 - -$S,T\in \mathscr{L}(V,W)$, $M(S+T)=M(S)+M(T)$ - -#### Lemma 3.38 - -$\forall \lambda\in \mathbb{F},T\in \mathscr{L}(V,W)$, $M(\lambda T)=\lambda M(T)$ - -$M:\mathscr{L}(V,W)\to \mathbb{F}^{n,m}$ is a linear map - -#### Matrix multiplication - -#### Definition 3.41 - -$$ -(AB)_{j,k}=\sum^{n}_{r=1} A_{j,r}B_{r,k} -$$ - -#### Theorem 3.42 - -$T\in \mathscr{L}(U,V), S\in\mathscr{L}(V,W)$ then $M(S,T)=M(S)M(T)$ ($dim (U)=p, dim(V)=n, dim(W)=m$) - -Proof: - -Let $w_1,...,v_n$ be a basis for $V$, $w_1,..,w_m$ be a basis for $W$ $u_1,..,u_p$ be a basis of $U$. - -Let $A=M(S),B=M(T)$ - -Compute $M(ST)$ by **Definition 3.31** - -$$ -\begin{aligned} -(ST)u_k&=S(T(u_k))\\ -&=S(\sum^n_{r=1}B_{r,k}v_r)\\ -&=\sum^n_{r=1} B_{r,k}(S_{v_r})\\ -&=\sum^n_{r=1} B_{r,k}(\sum^j_{j=1}A_{j,r} w_j)\\ -&=\sum^n_{r=1} (\sum^j_{j=1}A_{j,r}B_{r,k})w_j\\ -&=\sum^n_{r=1} (M(ST)_{j,k})w_j\\ -\end{aligned} -$$ - -$$ -\begin{aligned} -(M(ST))_{j,k}&=\sum^n_{r=1}A_{j,r}B_{r,k}\\ -&=(AB)_{j,k} -\end{aligned} -$$ - -$$ -M(ST)=AB=M(S)M(T) -$$ - -#### Notation 3.44 - -Suppose $A$ is an $m\times n$ matrix - -then - -1. $A_{j,\cdot}$ denotes the $1\times n$ matrix at the $j$th column. -2. $A_{\cdot,k}$ denotes the $m\times 1$ matrix at the $k$th column. - -#### Proposition 3.46 - -Suppose $A$ is a $m\times n$ matrix and $B$ is a $n\times p$ matrix, then - -$$ -(AB)_{j,k}=(A_{j,\cdot})\cdot (B_{\cdot,k}) -$$ - -Proof: - -$(AB)_{j,k}=A_{j,1}B_{1,k}+...+A_{j,n}B_{n,k}$ - -$(A_{j,\cdot})\cdot (B_{\cdot,k})=(A_{j,\cdot})_{1,1}(B_{\cdot,k})_{1,1}+...+(A_{j,\cdot})_{1,n}(B_{\cdot,k})_{n,1}=A_{j,1}B_{1,k}+...+A_{j,n}B_{n,k}$ - -#### Proposition 3.48 - -Suppose $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix, then - -$$ -(A,B)_{\cdot,k}=A(B_{\cdot,k}) -$$ - -#### Proposition 3.56 - -Let $A$ is an $m\times n$ $b=\begin{pmatrix} - b_1\\...\\b_n -\end{pmatrix}$ a $x\times 1$ matrix. Then $Ab=b_1A_{\cdot,1}+...+b_nA_{\cdot,n}$ - -i.e. $Ab$ is a linear combination of the columns of $A$ - -#### Proposition 3.51 - -Let $C$ be a $m\times c$ matrix and $R$ be a $c\times n$ matrix, then - -1. column $k$ of $CR$ is a linear combination of the columns of $C$ with coefficients given by $R_{\cdot,k}$ - - *putting the propositions together...* - -2. row $j$ of $CR$ is a linear combination of the rows of $R$ with coefficients given by $C_{j,\cdot}$ +# Lecture 11 + +## Chapter III Linear maps + +**Assumption: $U,V,W$ are vector spaces (over $\mathbb{F}$)** + +### Matrices 3C + +#### Definition 3.31 + +Suppose $T\in \mathscr{L}(V,W)$, $v_1,...,v_n$ a basis for $V$ $w_1,...,w_m$ a basis for $W$. Then $M(T)=M(T,(v_1,...,v_n),(w_1,...,w_m))$ is given by $M(T)=A$n where + +$$ +T_{v_k}=A_{1,k}w_1+...+A_{m,k}w_m +$$ + +$$ +\begin{matrix} + & & v_1& & v_2&&...&v_n& +\end{matrix}\\ +\begin{matrix} +w_1\\w_2\\.\\.\\.\\w_m +\end{matrix} +\begin{pmatrix} +A_{1,1} & A_{1,2} &...& A_{1,n}\\ +A_{2,1} & A_{2,2} &...& A_{2,n}\\ +. & . &...&.\\ +. & . &...&.\\ +. & . &...&.\\ +A_{m,1} & A_{m,2} &...& A_{m,n}\\ +\end{pmatrix} +$$ + +Example: + +* $T:\mathbb{F}^2\to \mathbb{F}^3$ + + $T(x,y)=(x+3y,2x+5y,7x+9y)$ + + $M(T)=\begin{pmatrix} + 1&3\\ + 2&5\\ + 7&9 + \end{pmatrix}$ + +* Let $D:\mathscr{P}_3(\mathbb{F})\to \mathscr{P}_2(\mathbb{F})$ be differentiation + + $M(T)=\begin{pmatrix} + 0&1&0&0\\ + 0&0&2&0\\ + 0&0&0&3\\ + \end{pmatrix}$ + +#### Lemma 3.35 + +$S,T\in \mathscr{L}(V,W)$, $M(S+T)=M(S)+M(T)$ + +#### Lemma 3.38 + +$\forall \lambda\in \mathbb{F},T\in \mathscr{L}(V,W)$, $M(\lambda T)=\lambda M(T)$ + +$M:\mathscr{L}(V,W)\to \mathbb{F}^{n,m}$ is a linear map + +#### Matrix multiplication + +#### Definition 3.41 + +$$ +(AB)_{j,k}=\sum^{n}_{r=1} A_{j,r}B_{r,k} +$$ + +#### Theorem 3.42 + +$T\in \mathscr{L}(U,V), S\in\mathscr{L}(V,W)$ then $M(S,T)=M(S)M(T)$ ($dim (U)=p, dim(V)=n, dim(W)=m$) + +Proof: + +Let $w_1,...,v_n$ be a basis for $V$, $w_1,..,w_m$ be a basis for $W$ $u_1,..,u_p$ be a basis of $U$. + +Let $A=M(S),B=M(T)$ + +Compute $M(ST)$ by **Definition 3.31** + +$$ +\begin{aligned} +(ST)u_k&=S(T(u_k))\\ +&=S(\sum^n_{r=1}B_{r,k}v_r)\\ +&=\sum^n_{r=1} B_{r,k}(S_{v_r})\\ +&=\sum^n_{r=1} B_{r,k}(\sum^j_{j=1}A_{j,r} w_j)\\ +&=\sum^n_{r=1} (\sum^j_{j=1}A_{j,r}B_{r,k})w_j\\ +&=\sum^n_{r=1} (M(ST)_{j,k})w_j\\ +\end{aligned} +$$ + +$$ +\begin{aligned} +(M(ST))_{j,k}&=\sum^n_{r=1}A_{j,r}B_{r,k}\\ +&=(AB)_{j,k} +\end{aligned} +$$ + +$$ +M(ST)=AB=M(S)M(T) +$$ + +#### Notation 3.44 + +Suppose $A$ is an $m\times n$ matrix + +then + +1. $A_{j,\cdot}$ denotes the $1\times n$ matrix at the $j$th column. +2. $A_{\cdot,k}$ denotes the $m\times 1$ matrix at the $k$th column. + +#### Proposition 3.46 + +Suppose $A$ is a $m\times n$ matrix and $B$ is a $n\times p$ matrix, then + +$$ +(AB)_{j,k}=(A_{j,\cdot})\cdot (B_{\cdot,k}) +$$ + +Proof: + +$(AB)_{j,k}=A_{j,1}B_{1,k}+...+A_{j,n}B_{n,k}$ + +$(A_{j,\cdot})\cdot (B_{\cdot,k})=(A_{j,\cdot})_{1,1}(B_{\cdot,k})_{1,1}+...+(A_{j,\cdot})_{1,n}(B_{\cdot,k})_{n,1}=A_{j,1}B_{1,k}+...+A_{j,n}B_{n,k}$ + +#### Proposition 3.48 + +Suppose $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix, then + +$$ +(A,B)_{\cdot,k}=A(B_{\cdot,k}) +$$ + +#### Proposition 3.56 + +Let $A$ is an $m\times n$ $b=\begin{pmatrix} + b_1\\...\\b_n +\end{pmatrix}$ a $x\times 1$ matrix. Then $Ab=b_1A_{\cdot,1}+...+b_nA_{\cdot,n}$ + +i.e. $Ab$ is a linear combination of the columns of $A$ + +#### Proposition 3.51 + +Let $C$ be a $m\times c$ matrix and $R$ be a $c\times n$ matrix, then + +1. column $k$ of $CR$ is a linear combination of the columns of $C$ with coefficients given by $R_{\cdot,k}$ + + *putting the propositions together...* + +2. row $j$ of $CR$ is a linear combination of the rows of $R$ with coefficients given by $C_{j,\cdot}$ diff --git a/pages/Math429/Math429_L12.md b/content/Math429/Math429_L12.md similarity index 96% rename from pages/Math429/Math429_L12.md rename to content/Math429/Math429_L12.md index 8ff5799..d35b456 100644 --- a/pages/Math429/Math429_L12.md +++ b/content/Math429/Math429_L12.md @@ -1,106 +1,106 @@ -# Lecture 12 - -## Chapter III Linear maps - -**Assumption: $U,V,W$ are vector spaces (over $\mathbb{F}$)** - -### Matrices 3C - -#### Proposition 3.51 - -Let $C$ be an $m\times c$ matrix and $R$ be a $c\times n$ matrix, then - -1. column $k$ of $CR$ is a linear combination of the columns of $C$ with coefficients given by $R_{\cdot,k}$ - - *putting the propositions together...* - -2. row $j$ of $CR$ is a linear combination of the rows of $R$ with coefficients given by $C_{j,\cdot}$ - -#### Column-Row Factorization and Rank - -#### Definition 3.52 - -Let $A$ be an $m \times n$ matrix, then - -* The column rank of $A$ is the dimension of the span of the columns in $\mathbb{F}^{m,1}$. -* The row range of $A$ is the dimension of the span of the row in $\mathbb{F}^{1,n}$. - -> Transpose: $A^t=A^T$ refers to swapping rows and columns - -#### Theorem 3.56 (Column-Row Factorization) - -Let $A$ be an $m\times j$ matrix with column rank $c$. Then there exists an $m\times c$ matrix $C$ and $c\times x$ matrix $R$ such that $A=CR$ - -Proof: - -Let $V=Span\{A_{\cdot,1},...,A_{\cdot,n}\}$, let $C_{\cdot, 1},...,C_{\cdot, c}$ be a basis of $V$. Since these forms a basis, there exists $R_{j,k}$ such that $A_{i,j}=\sum_{j=1}^c C_{i,j}R_{j,k}$, so $A_{\cdot,j}=\sum_{j=1}^c C_{\cdot,j}R_{j,k}$. This implies that $A=CR$ by construction $C$ is $m\times c$, $R$ is $c\times n$. - -Example: - -$$ -A=\begin{pmatrix} - 1&4&2\\ - 2&5&8\\ - 3&6&4 -\end{pmatrix}=\begin{pmatrix} - 1&4\\ - 2&5\\ - 3&6\\ -\end{pmatrix}\begin{pmatrix} - 1&0&-1\\ - 0&1&2\\ -\end{pmatrix},rank\ A=4 -$$ - -#### Definition 3.58 Rank - -The **rank of a matrix** $A$ is the column rank of $A$ denoted $rank\ A$. - -#### Theorem 3.57 - -Given a matrix $A$ the column rank equals the row rank. - -Proof: - -Note that by **Theorem 3.56**, if $A$ is $m\times n$ and has column rank $c$. $A=CR$ for some $C$ is a $m\times c$ matrix, $R$ is a $c\times n$ matrices, ut the rows of $CR$ are a linear combination of the rows of $R$, and row rank of $R\leq C$. So row rank $A\leq$ column rank of $A$. - -Taking a transpose of matrix, then row rank of $A^T$ (column rank of $A$) $\leq$ column rank of $A^T$ (row rank $A$). - -So column rank is equal to row rank. - -### Invertibility and Isomorphisms 3D - -Invertible Linear Maps - -#### Definition 3.59 - -A linear map $T\in\mathscr{L}(V,W)$ is **invertible** if there exists $S\in \mathscr{L}(W,V)$ such that $ST=I_V$ and $TS=I_W$. Such a $S$ is called an **inverse** of $T$. - -Note: $ST=I_V$ and $TS=I_W$ must **both be true** for inverse map. - -#### Lemma 3.60 - -Every linear map has an unique inverse. - -Proof: Exercise and answer in the book. - -Notation: $T^{-1}$ is the inverse of $T$ - -#### Theorem 3.63 - -A linear map $T:V\to W$ invertible if and only if its injective and surjective. - -Proof: - -$\Rightarrow$ - -$null(T)=\{0\}$ since $T(v)=0\implies (T^{-1}))(T(v))=0\implies range (T)=W$ let $w\in W$ then $T(T^{-1}(w))=w,w\in range (T)$ - -$\Leftarrow$ - -Find $S:W\to V$ a function such that $T(S(v))=v$ by letting $S(v)$ be the unique vector in $v$ such that $T(S(v))=v$. Goal: Show $S:W\to V$ is linear - -$$ -ST(S(w_1)+S(w_2))=S(w_1)+S(w_2)\\ -S(T(S(w_1)))+T(S(w_2))=S(w_1+w_2) -$$ +# Lecture 12 + +## Chapter III Linear maps + +**Assumption: $U,V,W$ are vector spaces (over $\mathbb{F}$)** + +### Matrices 3C + +#### Proposition 3.51 + +Let $C$ be an $m\times c$ matrix and $R$ be a $c\times n$ matrix, then + +1. column $k$ of $CR$ is a linear combination of the columns of $C$ with coefficients given by $R_{\cdot,k}$ + + *putting the propositions together...* + +2. row $j$ of $CR$ is a linear combination of the rows of $R$ with coefficients given by $C_{j,\cdot}$ + +#### Column-Row Factorization and Rank + +#### Definition 3.52 + +Let $A$ be an $m \times n$ matrix, then + +* The column rank of $A$ is the dimension of the span of the columns in $\mathbb{F}^{m,1}$. +* The row range of $A$ is the dimension of the span of the row in $\mathbb{F}^{1,n}$. + +> Transpose: $A^t=A^T$ refers to swapping rows and columns + +#### Theorem 3.56 (Column-Row Factorization) + +Let $A$ be an $m\times j$ matrix with column rank $c$. Then there exists an $m\times c$ matrix $C$ and $c\times x$ matrix $R$ such that $A=CR$ + +Proof: + +Let $V=Span\{A_{\cdot,1},...,A_{\cdot,n}\}$, let $C_{\cdot, 1},...,C_{\cdot, c}$ be a basis of $V$. Since these forms a basis, there exists $R_{j,k}$ such that $A_{i,j}=\sum_{j=1}^c C_{i,j}R_{j,k}$, so $A_{\cdot,j}=\sum_{j=1}^c C_{\cdot,j}R_{j,k}$. This implies that $A=CR$ by construction $C$ is $m\times c$, $R$ is $c\times n$. + +Example: + +$$ +A=\begin{pmatrix} + 1&4&2\\ + 2&5&8\\ + 3&6&4 +\end{pmatrix}=\begin{pmatrix} + 1&4\\ + 2&5\\ + 3&6\\ +\end{pmatrix}\begin{pmatrix} + 1&0&-1\\ + 0&1&2\\ +\end{pmatrix},rank\ A=4 +$$ + +#### Definition 3.58 Rank + +The **rank of a matrix** $A$ is the column rank of $A$ denoted $rank\ A$. + +#### Theorem 3.57 + +Given a matrix $A$ the column rank equals the row rank. + +Proof: + +Note that by **Theorem 3.56**, if $A$ is $m\times n$ and has column rank $c$. $A=CR$ for some $C$ is a $m\times c$ matrix, $R$ is a $c\times n$ matrices, ut the rows of $CR$ are a linear combination of the rows of $R$, and row rank of $R\leq C$. So row rank $A\leq$ column rank of $A$. + +Taking a transpose of matrix, then row rank of $A^T$ (column rank of $A$) $\leq$ column rank of $A^T$ (row rank $A$). + +So column rank is equal to row rank. + +### Invertibility and Isomorphisms 3D + +Invertible Linear Maps + +#### Definition 3.59 + +A linear map $T\in\mathscr{L}(V,W)$ is **invertible** if there exists $S\in \mathscr{L}(W,V)$ such that $ST=I_V$ and $TS=I_W$. Such a $S$ is called an **inverse** of $T$. + +Note: $ST=I_V$ and $TS=I_W$ must **both be true** for inverse map. + +#### Lemma 3.60 + +Every linear map has an unique inverse. + +Proof: Exercise and answer in the book. + +Notation: $T^{-1}$ is the inverse of $T$ + +#### Theorem 3.63 + +A linear map $T:V\to W$ invertible if and only if its injective and surjective. + +Proof: + +$\Rightarrow$ + +$null(T)=\{0\}$ since $T(v)=0\implies (T^{-1}))(T(v))=0\implies range (T)=W$ let $w\in W$ then $T(T^{-1}(w))=w,w\in range (T)$ + +$\Leftarrow$ + +Find $S:W\to V$ a function such that $T(S(v))=v$ by letting $S(v)$ be the unique vector in $v$ such that $T(S(v))=v$. Goal: Show $S:W\to V$ is linear + +$$ +ST(S(w_1)+S(w_2))=S(w_1)+S(w_2)\\ +S(T(S(w_1)))+T(S(w_2))=S(w_1+w_2) +$$ diff --git a/pages/Math429/Math429_L13.md b/content/Math429/Math429_L13.md similarity index 97% rename from pages/Math429/Math429_L13.md rename to content/Math429/Math429_L13.md index d2d94ea..a3e659e 100644 --- a/pages/Math429/Math429_L13.md +++ b/content/Math429/Math429_L13.md @@ -1,102 +1,102 @@ -# Lecture 13 - -## Chapter III Linear maps - -**Assumption: $U,V,W$ are vector spaces (over $\mathbb{F}$)** - -### Matrices 3C - -#### Theorem 3.63 - -A linear map is invertible if and only if it is injective and surjective. - -#### Example - -Consider $T:\mathscr{P}(\mathbb{F})\to \mathscr{P}(\mathbb{F})$, $T(f)=xf$ - -$T$ is injective but not surjective. Since you cannot get constant from multiply $x$. So it is not invertible. - -#### Theorem 3.65 - -Let $V$ and $W$ be finite-dimensional with the same dimension, and $T\in\mathscr{L}(V,W)$, then $T$ is invertible, if and only if $T$ is injective if and only if, $T$ is surjective. - -Proof: - -Suppose $T$ is injective, then $null\ T={0}$, i.e $dim(null\ T)=0$, since $dim\ V=dim\ null\ T+dim\ range\ T$, we have $dim\ V=dim\ range\ T$ but $dim\ V\dim\ W$, so $dim\ W=dim\ range\ T$. Thus $W=range\ T$. This shows that $T\ injective \implies T\ surjective$. - -If $T$ is surjective, then $dim\ range\ T=dim\ W$ but then $dim\ V=dim\ null\ T+dim\ W\implies dim\ null\ T=0$, so $T$ is injective, $T\ surjective\implies T\ injective$. - -#### Theorem 3.68 - -Suppose $V,W$ finite dimensional $dim\ V=dim\ W$, then for $T\in\mathscr{L}(V,W)$ and $S\in \mathscr{L}(W,V)$, then $ST=I\implies TS=I$ - -#### Example 3.67 - -Show that for a polynomial $q$ with degree $m$, there exists a unique polynomial $p$ of degree $m$ such that $((x^2+5x+7)p)''=q$ - -Solution: - -Let $T:\mathscr{P}_m(\mathbb{F})\to \mathscr{P}_m(\mathbb{F})$ given by $T(p)=((x^2+5x+7)p)''$ by $T$ is injective since $(x^2+5x+7)$ has degree $\geq 2$ for $p\neq 0$, therefore, $p$ is surjective. (by **Theorem 3.68**) - -#### Isomorphisms - -#### Definition 3.69 - -An **isomorphism** of vector spaces is a invertible linear map. Two vector spaces $V,W$ are isomorphic if there exists an isomorphism between them. - -Notation: $V\cong W$ means $V$ and $W$ are isomorphic. (Don't use very often, no map is included.) - -Example: - -$\mathscr{P}_m(\mathbb{F})$ and $\mathbb{F}^{m+1}$ are isomorphic. $T:\mathbb{F}^{m+1}\to \mathscr{P}_m(\mathbb{F}): T((a_0,...,a_m))=a_0+a_1x+...+a_n x^n$ - -#### Theorem 3.70 - -Two finite dimensional vector spaces $V,W$ are isomorphic if and only if $dim\ V= dim\ W$ - -Ideas of Proof: - -$\Rightarrow$ use fundamental theorems of linear map - -$\Leftarrow$ Let $v_1,...,v_m\in V$ and $w_1,...,w_n\in W$ be bases. Then define $T:V\to W$ by $T(v_k)=w_k$ for $1\leq k\leq n$ - -Show $T$ is invertible by showing $T$ is injective and surjective. - -#### Theorem 3.71 - -Let $V,W$ be finite dimensional, let $v_1,...,v_n\in V$ and $w_1,...,w_m\in W$ be bases. Then the map - -$$ -M(-,(v_1,...,v_n),(w_1,...,w_m)):\mathscr{L}(V,W)\to \mathbb{F}^{m,n} -$$ - -$T\mapsto M(T)$ or $M(-,(v_1,...,v_n),(w_1,...,w_m))$ is an isomorphism ($M:\mathscr{L}(V,W)\to \mathbb{F}^{m,n}$) - -Sketch of Proof: - -Need to show $M$ is surjective and injective. - -* Injective: i.e need to show if $M(T)=0$, then $T=0$. $M(T)=0\implies Tv_k=0, 1\leq k\leq n$ -* Surjective: i.e let $A\in F^{m,n}$ define $T:V\to W$ given by $Tv_k=\sum_{j=1}^m A_{j,k} w_j$ you cna check that $M(T)=A$ - -#### Corollary 3.72 - -$dim \mathscr{L}(V,W)=(dim\ V)(dim\ W)$ - -#### Definition 3.73 - -$v\in V, v_1,...,v_n$ a basis, then $M(v)=\begin{pmatrix} - b_1\\ - ...\\ - b_n -\end{pmatrix}, v=a_1v_1,...,a_nv_n$ - -#### Proposition 3.75, 3.76 - -$$ -M(T)_{\cdot,k}=M(Tv_k) -$$ - -$$ -M(Tv)=M(T)M(v) +# Lecture 13 + +## Chapter III Linear maps + +**Assumption: $U,V,W$ are vector spaces (over $\mathbb{F}$)** + +### Matrices 3C + +#### Theorem 3.63 + +A linear map is invertible if and only if it is injective and surjective. + +#### Example + +Consider $T:\mathscr{P}(\mathbb{F})\to \mathscr{P}(\mathbb{F})$, $T(f)=xf$ + +$T$ is injective but not surjective. Since you cannot get constant from multiply $x$. So it is not invertible. + +#### Theorem 3.65 + +Let $V$ and $W$ be finite-dimensional with the same dimension, and $T\in\mathscr{L}(V,W)$, then $T$ is invertible, if and only if $T$ is injective if and only if, $T$ is surjective. + +Proof: + +Suppose $T$ is injective, then $null\ T={0}$, i.e $dim(null\ T)=0$, since $dim\ V=dim\ null\ T+dim\ range\ T$, we have $dim\ V=dim\ range\ T$ but $dim\ V\dim\ W$, so $dim\ W=dim\ range\ T$. Thus $W=range\ T$. This shows that $T\ injective \implies T\ surjective$. + +If $T$ is surjective, then $dim\ range\ T=dim\ W$ but then $dim\ V=dim\ null\ T+dim\ W\implies dim\ null\ T=0$, so $T$ is injective, $T\ surjective\implies T\ injective$. + +#### Theorem 3.68 + +Suppose $V,W$ finite dimensional $dim\ V=dim\ W$, then for $T\in\mathscr{L}(V,W)$ and $S\in \mathscr{L}(W,V)$, then $ST=I\implies TS=I$ + +#### Example 3.67 + +Show that for a polynomial $q$ with degree $m$, there exists a unique polynomial $p$ of degree $m$ such that $((x^2+5x+7)p)''=q$ + +Solution: + +Let $T:\mathscr{P}_m(\mathbb{F})\to \mathscr{P}_m(\mathbb{F})$ given by $T(p)=((x^2+5x+7)p)''$ by $T$ is injective since $(x^2+5x+7)$ has degree $\geq 2$ for $p\neq 0$, therefore, $p$ is surjective. (by **Theorem 3.68**) + +#### Isomorphisms + +#### Definition 3.69 + +An **isomorphism** of vector spaces is a invertible linear map. Two vector spaces $V,W$ are isomorphic if there exists an isomorphism between them. + +Notation: $V\cong W$ means $V$ and $W$ are isomorphic. (Don't use very often, no map is included.) + +Example: + +$\mathscr{P}_m(\mathbb{F})$ and $\mathbb{F}^{m+1}$ are isomorphic. $T:\mathbb{F}^{m+1}\to \mathscr{P}_m(\mathbb{F}): T((a_0,...,a_m))=a_0+a_1x+...+a_n x^n$ + +#### Theorem 3.70 + +Two finite dimensional vector spaces $V,W$ are isomorphic if and only if $dim\ V= dim\ W$ + +Ideas of Proof: + +$\Rightarrow$ use fundamental theorems of linear map + +$\Leftarrow$ Let $v_1,...,v_m\in V$ and $w_1,...,w_n\in W$ be bases. Then define $T:V\to W$ by $T(v_k)=w_k$ for $1\leq k\leq n$ + +Show $T$ is invertible by showing $T$ is injective and surjective. + +#### Theorem 3.71 + +Let $V,W$ be finite dimensional, let $v_1,...,v_n\in V$ and $w_1,...,w_m\in W$ be bases. Then the map + +$$ +M(-,(v_1,...,v_n),(w_1,...,w_m)):\mathscr{L}(V,W)\to \mathbb{F}^{m,n} +$$ + +$T\mapsto M(T)$ or $M(-,(v_1,...,v_n),(w_1,...,w_m))$ is an isomorphism ($M:\mathscr{L}(V,W)\to \mathbb{F}^{m,n}$) + +Sketch of Proof: + +Need to show $M$ is surjective and injective. + +* Injective: i.e need to show if $M(T)=0$, then $T=0$. $M(T)=0\implies Tv_k=0, 1\leq k\leq n$ +* Surjective: i.e let $A\in F^{m,n}$ define $T:V\to W$ given by $Tv_k=\sum_{j=1}^m A_{j,k} w_j$ you cna check that $M(T)=A$ + +#### Corollary 3.72 + +$dim \mathscr{L}(V,W)=(dim\ V)(dim\ W)$ + +#### Definition 3.73 + +$v\in V, v_1,...,v_n$ a basis, then $M(v)=\begin{pmatrix} + b_1\\ + ...\\ + b_n +\end{pmatrix}, v=a_1v_1,...,a_nv_n$ + +#### Proposition 3.75, 3.76 + +$$ +M(T)_{\cdot,k}=M(Tv_k) +$$ + +$$ +M(Tv)=M(T)M(v) $$ \ No newline at end of file diff --git a/pages/Math429/Math429_L14.md b/content/Math429/Math429_L14.md similarity index 96% rename from pages/Math429/Math429_L14.md rename to content/Math429/Math429_L14.md index b6008bf..8a93a3b 100644 --- a/pages/Math429/Math429_L14.md +++ b/content/Math429/Math429_L14.md @@ -1,132 +1,132 @@ -# Lecture 14 - -## Chapter III Linear maps - -**Assumption: $U,V,W$ are vector spaces (over $\mathbb{F}$)** - -### Matrices 3C - -Review - -#### Proposition 3.76 - -$$ -M(Tv)=M(T)M(v) -$$ - -#### Theorem 3.78 - -Let $V,W$ be finite dimensional vector space, and $T\in \mathscr{L}(V,W)$ then $dim\ range\ T=column\ rank (M(T))=rank(M(T))$ - -Proof: - -$range=Span\{Tv_1,...,Tv_n\}$ compare to $Span\{M(T)_{\cdot,1},...,M(T)_{\cdot, n}\}=Span\{M(T)M(v_1),...,M(T)M(v_n)\}=Span\{M(Tv_1),...,M(Tv_n)\}$ - -Since $M$ is a isomorphism, then the theorem makes sense. - -#### Change of Basis - -#### Definition 3.79, 3.80 - -The identity matrix - -$$ -I=\begin{pmatrix} - 1.& 0\\ - 0& '1\\ -\end{pmatrix} -$$ - -The inverse matrix of an invertible matrix $A$ denoted $A^{-1}$ is the matrix such that - -$$ -AA^{-1}=I=A^{-1}A -$$ - -Question: Let $u_1,...,u_n$ and $v_1,...,v_n$ be two bases for $V$. What is $M(I,(u_1,...,u_n),(v_1,...,v_n)),I\in \mathscr{L}(V)$ - -#### Proposition 3.82 - -Let $u_1,...,u_n$ and $v_1,...,v_n$ be bases of $V$, then $M(I,(u_1,...,u_n),(v_1,...,v_n)),I\in \mathscr{L}(V)$ and $M(I,(v_1,...,v_n),(u_1,...,u_n)),I\in \mathscr{L}(V)$ are inverse to each other. - -Proof: - -$$ -M(I,(u_1,...,u_n),(v-1,...,v_n)),I\in \mathscr{L}(V) M(I,(v-1,...,v_n),(u_1,...,u_n))=M(I,(u_1,...,u_n),(u_1,...,u_n)) -$$ - -#### Theorem 3.84 Change of Basis - -Let $u_1,...,u_n$ and $v_1,...,v_n$ be two bases for $V$ and $T\in \mathscr{L}(v), A=M(T,(u_1,...,u_n)), B=M(T,(v_1,...,v_n)), C=M(I,(u_1,...,u_n),(v_1,...,v_n))$, then $A=C^{-1}BC$ - -#### Theorem 3.86 - -Let $T\in \mathscr{L}(v)$ be an invertible linear map, then $M(T^{-1})=M(T)^{-1}$ - -### Products and Quotients of Vector Spaces 3E - -Goals: To construct vectors spaces from other vector spaces. - -#### Definition 3.87 - -Suppose $V_1,...,V_m$ vectors spaces over some field $\mathbb{F}$, then the product is given by - -$$ -V_1\times ...\times V_n=\{(v_1,v_2,...,v_n)\vert v_1\in V_1, v_2\in V_2,...,v_n\in V_n\} -$$ - -with addition given by - -$$ -(v_1,...,v_n)+(u_1,...,u_n)=(v_1+u_1,...,v_n+u_n) -$$ - -and scalar multiplication - -$$ -\lambda (v_1,...,v_n)=(\lambda v_1,...,\lambda v_n),\lambda \in \mathbb{F} -$$ - -#### Theorem 3.89 - -If $v_1,...,v_n$ are vectors paces over $\mathbb{F}$ then $V_1\times ...\times V_n$ is a vector space over $\mathbb{F}$ - -Example: - -$V=\mathscr{P}_2(\mathbb{R})\times \mathbb{R}^2=\{(p,v)\vert p\in \mathscr{P}_2(\mathbb{R}), v\in \mathbb{R}^2\}=\{(a_0+a_1x+a_2x,(b,c))\vert a_0,a_1,a_2,b,c\in \mathbb{R}\}$ - -A basis for $V$ would be $(1,(0,0)),(x,(0,0)),(x^2,(0,0)),(0,(1,0)),(0,(0,1))$ - -#### Theorem 3.92 - -$$ -dim(V_1\times ...\times V_n)=dim(V_1)+...+dim(V_n) -$$ - -Sketch of proof: - -take a basis for each $V_k$, make them vectors in the product then combine the entire list of vector to be basis. - -Example: - -$\mathbb{R}^2\times \mathbb{R}^3=\{((a,b),(c,d,e))\vert a,b,c,d,e\in \R\}$ - -$\mathbb{R}^2\times \mathbb{R}^3\cong \mathbb{R}^5,((a,b),(c,d,e))\mapsto(a,b,c,d,e)$ - -#### Theorem 3.93 - -Let $V_1,...,V_m\subseteq V$, define $\Gamma: V_1\times...\times V_m\to V_1+...+V_m$. $\Gamma(v_1,...,v_n)=v_1+...+v_n$ then $\Gamma$ is always surjective. And it is injective if and only if $V_1+...+V_m$ is a direct sum. - -Sketch of the proof: - -injective $\iff null\ T\{ (0,...,0) \} \iff$ the only way to write $0=v_1,...,v_m$ is $v_1=...=v_n=0 \iff$ then $V_1+...+V_m$ is a direct sum - -#### Theorem 3.94 - -$V_1+...+V_m$ is a direct sum if and only if $dim(V_1+...+V_m)=dim(V_1)+...+dim(V_m)$ - -Proof: - -Use $\Gamma$ above is an isomorphism $\iff$ $V_1+...+V_m$ is a direct sum - -Use $\Gamma$ above is an isomorphism $\implies dim(V_1+...+V_m)=dim(V_1)+...+dim(V_m)$ +# Lecture 14 + +## Chapter III Linear maps + +**Assumption: $U,V,W$ are vector spaces (over $\mathbb{F}$)** + +### Matrices 3C + +Review + +#### Proposition 3.76 + +$$ +M(Tv)=M(T)M(v) +$$ + +#### Theorem 3.78 + +Let $V,W$ be finite dimensional vector space, and $T\in \mathscr{L}(V,W)$ then $dim\ range\ T=column\ rank (M(T))=rank(M(T))$ + +Proof: + +$range=Span\{Tv_1,...,Tv_n\}$ compare to $Span\{M(T)_{\cdot,1},...,M(T)_{\cdot, n}\}=Span\{M(T)M(v_1),...,M(T)M(v_n)\}=Span\{M(Tv_1),...,M(Tv_n)\}$ + +Since $M$ is a isomorphism, then the theorem makes sense. + +#### Change of Basis + +#### Definition 3.79, 3.80 + +The identity matrix + +$$ +I=\begin{pmatrix} + 1.& 0\\ + 0& '1\\ +\end{pmatrix} +$$ + +The inverse matrix of an invertible matrix $A$ denoted $A^{-1}$ is the matrix such that + +$$ +AA^{-1}=I=A^{-1}A +$$ + +Question: Let $u_1,...,u_n$ and $v_1,...,v_n$ be two bases for $V$. What is $M(I,(u_1,...,u_n),(v_1,...,v_n)),I\in \mathscr{L}(V)$ + +#### Proposition 3.82 + +Let $u_1,...,u_n$ and $v_1,...,v_n$ be bases of $V$, then $M(I,(u_1,...,u_n),(v_1,...,v_n)),I\in \mathscr{L}(V)$ and $M(I,(v_1,...,v_n),(u_1,...,u_n)),I\in \mathscr{L}(V)$ are inverse to each other. + +Proof: + +$$ +M(I,(u_1,...,u_n),(v-1,...,v_n)),I\in \mathscr{L}(V) M(I,(v-1,...,v_n),(u_1,...,u_n))=M(I,(u_1,...,u_n),(u_1,...,u_n)) +$$ + +#### Theorem 3.84 Change of Basis + +Let $u_1,...,u_n$ and $v_1,...,v_n$ be two bases for $V$ and $T\in \mathscr{L}(v), A=M(T,(u_1,...,u_n)), B=M(T,(v_1,...,v_n)), C=M(I,(u_1,...,u_n),(v_1,...,v_n))$, then $A=C^{-1}BC$ + +#### Theorem 3.86 + +Let $T\in \mathscr{L}(v)$ be an invertible linear map, then $M(T^{-1})=M(T)^{-1}$ + +### Products and Quotients of Vector Spaces 3E + +Goals: To construct vectors spaces from other vector spaces. + +#### Definition 3.87 + +Suppose $V_1,...,V_m$ vectors spaces over some field $\mathbb{F}$, then the product is given by + +$$ +V_1\times ...\times V_n=\{(v_1,v_2,...,v_n)\vert v_1\in V_1, v_2\in V_2,...,v_n\in V_n\} +$$ + +with addition given by + +$$ +(v_1,...,v_n)+(u_1,...,u_n)=(v_1+u_1,...,v_n+u_n) +$$ + +and scalar multiplication + +$$ +\lambda (v_1,...,v_n)=(\lambda v_1,...,\lambda v_n),\lambda \in \mathbb{F} +$$ + +#### Theorem 3.89 + +If $v_1,...,v_n$ are vectors paces over $\mathbb{F}$ then $V_1\times ...\times V_n$ is a vector space over $\mathbb{F}$ + +Example: + +$V=\mathscr{P}_2(\mathbb{R})\times \mathbb{R}^2=\{(p,v)\vert p\in \mathscr{P}_2(\mathbb{R}), v\in \mathbb{R}^2\}=\{(a_0+a_1x+a_2x,(b,c))\vert a_0,a_1,a_2,b,c\in \mathbb{R}\}$ + +A basis for $V$ would be $(1,(0,0)),(x,(0,0)),(x^2,(0,0)),(0,(1,0)),(0,(0,1))$ + +#### Theorem 3.92 + +$$ +dim(V_1\times ...\times V_n)=dim(V_1)+...+dim(V_n) +$$ + +Sketch of proof: + +take a basis for each $V_k$, make them vectors in the product then combine the entire list of vector to be basis. + +Example: + +$\mathbb{R}^2\times \mathbb{R}^3=\{((a,b),(c,d,e))\vert a,b,c,d,e\in \R\}$ + +$\mathbb{R}^2\times \mathbb{R}^3\cong \mathbb{R}^5,((a,b),(c,d,e))\mapsto(a,b,c,d,e)$ + +#### Theorem 3.93 + +Let $V_1,...,V_m\subseteq V$, define $\Gamma: V_1\times...\times V_m\to V_1+...+V_m$. $\Gamma(v_1,...,v_n)=v_1+...+v_n$ then $\Gamma$ is always surjective. And it is injective if and only if $V_1+...+V_m$ is a direct sum. + +Sketch of the proof: + +injective $\iff null\ T\{ (0,...,0) \} \iff$ the only way to write $0=v_1,...,v_m$ is $v_1=...=v_n=0 \iff$ then $V_1+...+V_m$ is a direct sum + +#### Theorem 3.94 + +$V_1+...+V_m$ is a direct sum if and only if $dim(V_1+...+V_m)=dim(V_1)+...+dim(V_m)$ + +Proof: + +Use $\Gamma$ above is an isomorphism $\iff$ $V_1+...+V_m$ is a direct sum + +Use $\Gamma$ above is an isomorphism $\implies dim(V_1+...+V_m)=dim(V_1)+...+dim(V_m)$ diff --git a/pages/Math429/Math429_L15.md b/content/Math429/Math429_L15.md similarity index 95% rename from pages/Math429/Math429_L15.md rename to content/Math429/Math429_L15.md index 40aaab3..5ed38d4 100644 --- a/pages/Math429/Math429_L15.md +++ b/content/Math429/Math429_L15.md @@ -1,136 +1,136 @@ -# Lecture 15 - -## Chapter III Linear maps - -**Assumption: $U,V,W$ are vector spaces (over $\mathbb{F}$)** - -### Products and Quotients of Vector Spaces 3E - -Quotient Space - -Idea: For a vector space $V$ and a subspace $U$. Construct a new vector space $V/U$ which is elements of $V$ up to equivalence by $U$. - -#### Definition 3.97 - -For $v\in V$ and $U$ a subspace of $V$. Then $v+U=\{v+u\vert u\in U\}$ is the translate of $U$ by $v$. (also called a coset of $U$) - -Example - -Let $U\subseteq \mathbb{R}^2$ be $U=\{(x,2x)\vert x\in \mathbb{R}\}$, $v=(5,3)\in\mathbb{R}^2$, $v+U=\{(x+3.5, 2x)\vert x\in \R\}$ - -Describe the solutions to $(p(x))'=x^2$, $p(x)=\frac{1}{3}x^3+c$. Let $u\in \mathscr{P}(\mathbb{R})$ be the constant functions then the set of solutions to $(p(x))'=x^2$ is $\frac{1}{3}x^3+U$ - -#### Definition 3.99 - -Suppose $U$ is a subspace of $V$, then the **quotient space** $V/U$ is given by - -$$ -V/U=\{v+U\vert v\in V\} -$$ - -This is not subset of $V$. - -Example: - -Let $U\subseteq \mathbb{R}^2$ be $U=\{(x,2x)\vert x\in \mathbb{R}\}$, then $\mathbb{R}^2/U$ is the set of all lines of slope $2$ in $\mathbb{R}^2$ - -#### Lemma 3.101 - -Let $U$ be a subspace of $V$ and $v,w\in V$ then the following are equivalent - -a) $v-w\in U$ -b) $v+U=w+U$ -c) $(v+U)\cap(w+U)\neq \phi$ - -Proof: - -* $a\implies b$ - -Suppose $v-w\in U$, we wish to show that $v+U=w+U$. - -Let $u\in U$ then $v+u=w+((v-w)+u)\in w+U$ - -So $v+U\in w+U$ and by symmetry, $w+U\subseteq v+U$ so $v+U=w+U$ - -* $b\implies c$ - -$u\neq \phi \implies v+U=w+U\neq \phi$ - -* $c\implies a$ - -Suppose $(v+U)\cap (w+U)\neq\phi$ So let $u_1,u_2\in U$ be such that $v+u_1=w+u_2$ but then $v-w=u_2-u_1\in U$ - -#### Definition 3.102 - -Let $U\subseteq V$ be a subspace, define the following: - -* $(v+U)+(w+U)=(v+w)+U$ -* $\lambda (v+U)=(\lambda v)+U$ - -#### Theorem 3.103 - -Let $U\in V$ be a subspace, then $V/U$ is a vector space. - -Proof: - -Assume for now that Definition 3.102 is well defined. - -* commutativity: by commutativity on $V$. -* associativity: by associativity on $V$. -* distributive: law by $V$. -* additive identity: $0+U$. -* additive inverse: $-v+U$. -* multiplicative identity: $1(v+U)=v+U$ - -Why is 3.102 well defined. - -Let $v_1,v_2,w_1,w_2\in V$ such that $v_1+U=v_2+U$ and $w_1+U=w_2+U$ - -Note by lemma 3.101 - -$v_1-v_2\in U$ and $w_1-w_2\in U \implies$ - -$(v_1+w_1)-(v_2+w_2)\in U \implies$ - -$(v_1+w_1)+U=(v_2+w_2)+U=(v_1+U)+(w_1+U)=(v_2+U)+(w_2+U)$ - -same idea for scalar multiplication. - -#### Definition 3.104 - -Let $U\subseteq V$. The quotient map is - -$$ -\pi:V\to V/U, \pi (v)=v+U -$$ - -#### Lemma 3.104.1 - -$\pi$ is a linear map - -#### Theorem 3.105 - -Let $V$ be finite dimensional $U\subseteq V$ then $dim(V/U)=dim\ V-dim\ U$ - -Proof: - -Note $null\ pi=U$, since if $\pi(v)=0=0+u\iff v\in U$ - -By the Fundamental Theorem of Linear Maps says - -$$ -dim\ (range\ \pi)+dim\ (null\ T)=dim\ V -$$ - -but $\pi$ is surjective, so we are done. - -#### Theorem 3.106 - -Suppose $T\in \mathscr{L}(V,W)$ then, - -Define $\tilde{T}:V/null\ T\to \tilde{W}$ by $\tilde{T}(v+null\ T)$ Then we have the following. - -1. $\tilde{T}\circ\pi =T$ -2. $\tilde{T}$ is injective -3. $range \tilde{T}=range\ T$ -4. $V/null\ T$ and $range\ T$ are isomorphic +# Lecture 15 + +## Chapter III Linear maps + +**Assumption: $U,V,W$ are vector spaces (over $\mathbb{F}$)** + +### Products and Quotients of Vector Spaces 3E + +Quotient Space + +Idea: For a vector space $V$ and a subspace $U$. Construct a new vector space $V/U$ which is elements of $V$ up to equivalence by $U$. + +#### Definition 3.97 + +For $v\in V$ and $U$ a subspace of $V$. Then $v+U=\{v+u\vert u\in U\}$ is the translate of $U$ by $v$. (also called a coset of $U$) + +Example + +Let $U\subseteq \mathbb{R}^2$ be $U=\{(x,2x)\vert x\in \mathbb{R}\}$, $v=(5,3)\in\mathbb{R}^2$, $v+U=\{(x+3.5, 2x)\vert x\in \R\}$ + +Describe the solutions to $(p(x))'=x^2$, $p(x)=\frac{1}{3}x^3+c$. Let $u\in \mathscr{P}(\mathbb{R})$ be the constant functions then the set of solutions to $(p(x))'=x^2$ is $\frac{1}{3}x^3+U$ + +#### Definition 3.99 + +Suppose $U$ is a subspace of $V$, then the **quotient space** $V/U$ is given by + +$$ +V/U=\{v+U\vert v\in V\} +$$ + +This is not subset of $V$. + +Example: + +Let $U\subseteq \mathbb{R}^2$ be $U=\{(x,2x)\vert x\in \mathbb{R}\}$, then $\mathbb{R}^2/U$ is the set of all lines of slope $2$ in $\mathbb{R}^2$ + +#### Lemma 3.101 + +Let $U$ be a subspace of $V$ and $v,w\in V$ then the following are equivalent + +a) $v-w\in U$ +b) $v+U=w+U$ +c) $(v+U)\cap(w+U)\neq \phi$ + +Proof: + +* $a\implies b$ + +Suppose $v-w\in U$, we wish to show that $v+U=w+U$. + +Let $u\in U$ then $v+u=w+((v-w)+u)\in w+U$ + +So $v+U\in w+U$ and by symmetry, $w+U\subseteq v+U$ so $v+U=w+U$ + +* $b\implies c$ + +$u\neq \phi \implies v+U=w+U\neq \phi$ + +* $c\implies a$ + +Suppose $(v+U)\cap (w+U)\neq\phi$ So let $u_1,u_2\in U$ be such that $v+u_1=w+u_2$ but then $v-w=u_2-u_1\in U$ + +#### Definition 3.102 + +Let $U\subseteq V$ be a subspace, define the following: + +* $(v+U)+(w+U)=(v+w)+U$ +* $\lambda (v+U)=(\lambda v)+U$ + +#### Theorem 3.103 + +Let $U\in V$ be a subspace, then $V/U$ is a vector space. + +Proof: + +Assume for now that Definition 3.102 is well defined. + +* commutativity: by commutativity on $V$. +* associativity: by associativity on $V$. +* distributive: law by $V$. +* additive identity: $0+U$. +* additive inverse: $-v+U$. +* multiplicative identity: $1(v+U)=v+U$ + +Why is 3.102 well defined. + +Let $v_1,v_2,w_1,w_2\in V$ such that $v_1+U=v_2+U$ and $w_1+U=w_2+U$ + +Note by lemma 3.101 + +$v_1-v_2\in U$ and $w_1-w_2\in U \implies$ + +$(v_1+w_1)-(v_2+w_2)\in U \implies$ + +$(v_1+w_1)+U=(v_2+w_2)+U=(v_1+U)+(w_1+U)=(v_2+U)+(w_2+U)$ + +same idea for scalar multiplication. + +#### Definition 3.104 + +Let $U\subseteq V$. The quotient map is + +$$ +\pi:V\to V/U, \pi (v)=v+U +$$ + +#### Lemma 3.104.1 + +$\pi$ is a linear map + +#### Theorem 3.105 + +Let $V$ be finite dimensional $U\subseteq V$ then $dim(V/U)=dim\ V-dim\ U$ + +Proof: + +Note $null\ pi=U$, since if $\pi(v)=0=0+u\iff v\in U$ + +By the Fundamental Theorem of Linear Maps says + +$$ +dim\ (range\ \pi)+dim\ (null\ T)=dim\ V +$$ + +but $\pi$ is surjective, so we are done. + +#### Theorem 3.106 + +Suppose $T\in \mathscr{L}(V,W)$ then, + +Define $\tilde{T}:V/null\ T\to \tilde{W}$ by $\tilde{T}(v+null\ T)$ Then we have the following. + +1. $\tilde{T}\circ\pi =T$ +2. $\tilde{T}$ is injective +3. $range \tilde{T}=range\ T$ +4. $V/null\ T$ and $range\ T$ are isomorphic diff --git a/pages/Math429/Math429_L16.md b/content/Math429/Math429_L16.md similarity index 96% rename from pages/Math429/Math429_L16.md rename to content/Math429/Math429_L16.md index a784111..0e3a6d0 100644 --- a/pages/Math429/Math429_L16.md +++ b/content/Math429/Math429_L16.md @@ -1,125 +1,125 @@ -# Lecture 16 - -## Chapter IV Polynomials - -**$\mathbb{F}$ denotes $\mathbb{R}$ or $\mathbb{C}$** - ---- - -Review - -### Products and Quotients of Vector Spaces 3E - -#### Theorem 3.107 - -Let $T\in \mathscr{L}(V,W)$, then define $\tilde{T}:V/null\ T\to W$, given by $\tilde{T}(v+null\ T)=Tv$ - -a) $\tilde{T}\circ \pi=T$ where $\pi: V/null\ T$ - -b) $\tilde{T}$ is injective - -c) $range\ T=range\ \tilde{T}$ - -d) $V/null\ T$ and $range\ T$ are isomorphic - -Example: - -Consider $D:\mathscr{P}_M(\mathbb{F})\to \mathscr{P}_{m-1}(\mathbb{F})$ be differentiation map - -$D$ is surjective by $D$ is not injective $null\ D=${constant polynomials} - -$\tilde{D}:\mathscr{P}_M(\mathbb{F})/$ constant polynomials $\to \mathscr{P}_{m-1}(\mathbb{F})$ - -This map ($\tilde{D}$) is injective since $range\ \tilde{D}=range\ D=\mathscr{P}_{m-1}(\mathbb{F})$ - -$\tilde{D}^{-1}:\mathscr{P}_{m-1}(\mathbb{F})\to \mathscr{P}_M(\mathbb{F})/$ constant polynomials (anti-derivative) - ---- - -New materials - -### Complex numbers 1A - -#### Definition 1.1 - -Complex numbers - -$z=a+bi$ is a complex number for $a,b\in \mathbb{R}$, ($Re\ z=a,Im\ z=b$) - -$\bar{z}=a-bi$ complex conjugate $|z|=\sqrt{a^2+b^2}$ - -#### Properties 1.n - -1. $z+\bar{z}=2a$ -2. $z-\bar{z}=2b$ -3. $z\bar{z}=|z|^2$ -4. $\overline{z+w}=\bar{z}+\bar{w}$ -5. $\overline{zw}=\bar{z}\bar{w}$ -6. $\bar{\bar{z}}=z$ -7. $|a|\leq |z|$ -8. $|b|\leq |z|$ -9. $|\bar{z}|=|z|$ -10. $|zw|=|z||w|$ -11. $|z+w|\leq |z|+|w|$ - -### Polynomials 4A - -$$ -p(x)=\sum_{i=0}^{n}a_i x^i -$$ - -#### Lemma 4.6 - -If $p$ is a polynomial and $\lambda$ is a zero of $p$, then $p(x)=(x-\lambda)q(x)$ for some polynomial $q(x)$ with $deg\ q=deg\ p -1$ - -#### Lemma 4.8 - -If $m=deg\ p,p\neq 0$ then $p$ has at most $m$ zeros. - -Sketch of Proof: - -Induction using 4.6 - -### Division Algorithm 4B - -#### Theorem 4.9 - -Suppose $p,s\in \mathscr{P}(\mathbb{F}),s\neq 0$. Then there exists a unique $q,r\in \mathscr{P}(\mathbb{F})$ such that $p=sq+r$, and $deg\ r\leq deg\ s$ - -Proof: - -Let $n=deg\ p,m=deg\ s$ if $n< m$, we are done $q=0,r=p$. - -Otherwise ($n\leq m$) consider $1,z,...,z^{m-1},s,zs,...,z^{r-m}s$. is a basis of $\mathscr{P}_n(\mathbb{F})$. - -Then there exists a unique $a_1,...,a_n\in\mathbb{F}$ such that $p(z)=a_0+a_1z+...+a_{m-1}z^{m-1}+a_m s+...+ a_n z^{n-m}s=(a_0+a_1z+...+a_{m-1}z^{m-1})+s(a_m +...+a_n z^{n-m})$ - -let $r=(a_0+a_1z+...+a_{m-1}z^{m-1}), q=(a_m +...+a_n z^{n-m})$ then we are done. - -### Zeros of polynomial over $\mathbb{C}$ 4C - -#### Theorem 4.12 Fundamental Theorem of Algorithm - -Every non-constant polynomial over $\mathbb{C}$ has at least one root. - -#### Theorem 4.13 - -If $p\in \mathscr{P}(\mathbb{C})$ then $p$ has a unique factorization up to order as $p(z)=c(z-\lambda_1)(z-\lambda_m)$ for $c,\lambda_1,...,\lambda_m\in \mathbb{C}$ - -Sketch of Proof: - -(4.12)+(4.6) - -### Zeros of polynomial over $\mathbb{R}$ 4D - -#### Proposition 4.14 - -If $p\in \mathscr{P}(\mathbb{C})$ with real coefficients, then if $p(\lambda )=0$ then $p(\bar{\lambda})=0$ - -#### Theorem 4.16 Fundamental Theorem of Algorithm for real numbers - -If $p$ is a non-constant polynomial over $\mathbb{R}$ the $p$ has a unique factorization - -$p(x)=c(x-\lambda_1)...(x-\lambda_m)(x^2+b_1 x+c_1)...(x^2+b_m x+c_m)$ - -with $b_k^2\leq 4c_k$ +# Lecture 16 + +## Chapter IV Polynomials + +**$\mathbb{F}$ denotes $\mathbb{R}$ or $\mathbb{C}$** + +--- + +Review + +### Products and Quotients of Vector Spaces 3E + +#### Theorem 3.107 + +Let $T\in \mathscr{L}(V,W)$, then define $\tilde{T}:V/null\ T\to W$, given by $\tilde{T}(v+null\ T)=Tv$ + +a) $\tilde{T}\circ \pi=T$ where $\pi: V/null\ T$ + +b) $\tilde{T}$ is injective + +c) $range\ T=range\ \tilde{T}$ + +d) $V/null\ T$ and $range\ T$ are isomorphic + +Example: + +Consider $D:\mathscr{P}_M(\mathbb{F})\to \mathscr{P}_{m-1}(\mathbb{F})$ be differentiation map + +$D$ is surjective by $D$ is not injective $null\ D=${constant polynomials} + +$\tilde{D}:\mathscr{P}_M(\mathbb{F})/$ constant polynomials $\to \mathscr{P}_{m-1}(\mathbb{F})$ + +This map ($\tilde{D}$) is injective since $range\ \tilde{D}=range\ D=\mathscr{P}_{m-1}(\mathbb{F})$ + +$\tilde{D}^{-1}:\mathscr{P}_{m-1}(\mathbb{F})\to \mathscr{P}_M(\mathbb{F})/$ constant polynomials (anti-derivative) + +--- + +New materials + +### Complex numbers 1A + +#### Definition 1.1 + +Complex numbers + +$z=a+bi$ is a complex number for $a,b\in \mathbb{R}$, ($Re\ z=a,Im\ z=b$) + +$\bar{z}=a-bi$ complex conjugate $|z|=\sqrt{a^2+b^2}$ + +#### Properties 1.n + +1. $z+\bar{z}=2a$ +2. $z-\bar{z}=2b$ +3. $z\bar{z}=|z|^2$ +4. $\overline{z+w}=\bar{z}+\bar{w}$ +5. $\overline{zw}=\bar{z}\bar{w}$ +6. $\bar{\bar{z}}=z$ +7. $|a|\leq |z|$ +8. $|b|\leq |z|$ +9. $|\bar{z}|=|z|$ +10. $|zw|=|z||w|$ +11. $|z+w|\leq |z|+|w|$ + +### Polynomials 4A + +$$ +p(x)=\sum_{i=0}^{n}a_i x^i +$$ + +#### Lemma 4.6 + +If $p$ is a polynomial and $\lambda$ is a zero of $p$, then $p(x)=(x-\lambda)q(x)$ for some polynomial $q(x)$ with $deg\ q=deg\ p -1$ + +#### Lemma 4.8 + +If $m=deg\ p,p\neq 0$ then $p$ has at most $m$ zeros. + +Sketch of Proof: + +Induction using 4.6 + +### Division Algorithm 4B + +#### Theorem 4.9 + +Suppose $p,s\in \mathscr{P}(\mathbb{F}),s\neq 0$. Then there exists a unique $q,r\in \mathscr{P}(\mathbb{F})$ such that $p=sq+r$, and $deg\ r\leq deg\ s$ + +Proof: + +Let $n=deg\ p,m=deg\ s$ if $n< m$, we are done $q=0,r=p$. + +Otherwise ($n\leq m$) consider $1,z,...,z^{m-1},s,zs,...,z^{r-m}s$. is a basis of $\mathscr{P}_n(\mathbb{F})$. + +Then there exists a unique $a_1,...,a_n\in\mathbb{F}$ such that $p(z)=a_0+a_1z+...+a_{m-1}z^{m-1}+a_m s+...+ a_n z^{n-m}s=(a_0+a_1z+...+a_{m-1}z^{m-1})+s(a_m +...+a_n z^{n-m})$ + +let $r=(a_0+a_1z+...+a_{m-1}z^{m-1}), q=(a_m +...+a_n z^{n-m})$ then we are done. + +### Zeros of polynomial over $\mathbb{C}$ 4C + +#### Theorem 4.12 Fundamental Theorem of Algorithm + +Every non-constant polynomial over $\mathbb{C}$ has at least one root. + +#### Theorem 4.13 + +If $p\in \mathscr{P}(\mathbb{C})$ then $p$ has a unique factorization up to order as $p(z)=c(z-\lambda_1)(z-\lambda_m)$ for $c,\lambda_1,...,\lambda_m\in \mathbb{C}$ + +Sketch of Proof: + +(4.12)+(4.6) + +### Zeros of polynomial over $\mathbb{R}$ 4D + +#### Proposition 4.14 + +If $p\in \mathscr{P}(\mathbb{C})$ with real coefficients, then if $p(\lambda )=0$ then $p(\bar{\lambda})=0$ + +#### Theorem 4.16 Fundamental Theorem of Algorithm for real numbers + +If $p$ is a non-constant polynomial over $\mathbb{R}$ the $p$ has a unique factorization + +$p(x)=c(x-\lambda_1)...(x-\lambda_m)(x^2+b_1 x+c_1)...(x^2+b_m x+c_m)$ + +with $b_k^2\leq 4c_k$ diff --git a/pages/Math429/Math429_L17.md b/content/Math429/Math429_L17.md similarity index 96% rename from pages/Math429/Math429_L17.md rename to content/Math429/Math429_L17.md index 22683cf..f024163 100644 --- a/pages/Math429/Math429_L17.md +++ b/content/Math429/Math429_L17.md @@ -1,105 +1,105 @@ -# Lecture 17 - -## Chapter III Linear maps - -**Assumption: $U,V,W$ are vector spaces (over $\mathbb{F}$)** - -### Duality 3F - -#### Definition 3.108 - -A **linear functional** on $V$ is a linear map from $V$ to $\mathbb{F}$. - -#### Definition 3.110 - -The **dual space** of V denoted by $V'$ (or in some books $\check{V},V^*$) is given by $V'=\mathscr{L}(V,\mathbb{F})$. - -The elements of $V'$ are also called **linear functional**. - -#### Theorem 3.111 - -The $dim\ V'=dim\ V$. - -Proof: - -$dim\ \mathscr{L}(V,\mathbb{F})=dim\ V\cdot dim\ \mathbb{F}$ - -#### Definition 3.112 - -If $v_1,...,v_n$ is a basis for $V$, then the **dual basis** of $v_1,..,v_n$ is $\psi_1,...,\psi_n\in V'$ where - -$$ -\psi_j(v_k)=\begin{cases} - 1 \textup{ if }k=i\\ - 0 \textup{ if }k\neq i -\end{cases} -$$ - -Example: - -$V=\mathbb{R}^3$ $e_1,e_2,e_3$ the standard basis, the dual basis $\psi_1,\psi_2,\psi_3$ is given by $\psi_1 (x,y,z)=x,\psi_2 (x,y,z)=y,\psi_3 (x,y,z)=z$ - -#### Theorem 3.116 - -When $v_1,...,v_n$ a basis of $V$ the dual basis $\psi_1,...,\psi_n\in V'$ is a basis - -Sketch of Proof: - -$dim\ V'=dim\ V=n$, $\psi_1,...,\psi_n\in V'$ are linearly independent. - -#### Theorem 3.114 - -Given $v_1,...,v_n$ a basis of $V$, and $\psi_1,...,\psi_n\in V'$ be dual basis of $V'$. then for $v\in V$, - -$$ -v=\psi_1(v)v_1+...+\psi_n(v)v_n -$$ - -Proof: - -Let $V=a_1 v_1+...+a_n v_n$, consider $\psi_k(v)$, by definition $\psi_k(v)=\psi_k(a_1 v_1+...+a_n v_n)=a_1\psi_k( v_1)+...+a_n\psi_k( v_n)=a_k$ - -#### Definition 3.118 - -Suppose $T\in \mathscr{L}(V,W)$. The **dual map** $T'\in \mathcal{R}( W', V')$ defined by $T'(\psi)=\psi\circ T$. ($\psi\in W'=\mathcal{R}(W,\mathbb{F}), T'(\psi) \in V'=\mathscr{L}(V,\mathbb{F})$) - -Example: - -$T:\mathscr{P}_2(\mathbb{F})\to \mathscr{P}_3(\mathbb{F}),T(f)=xf$ - -$$ -T'(\mathscr{P}_3(\mathbb{F}))'\to (\mathscr{P}_2(\mathbb{F}))',T'(\psi)(f)=\psi(T(f))=\psi(xf) -$$ - -Suppose $\psi(f)=f'(1)\to T(\psi)(f)=(xf)'(1)=f(1)+(xf')(1)=f(1)+f'(1)$ - -#### Theorem 3.120 - -Suppose $T\in \mathscr{L}(V,W)$ - -a) $(S+T)'=S'+T', \forall S\in \mathscr{L}(V,W)$ -b) $(\lambda T)'=\lambda T', \forall \lambda\in \mathbb{F}$ -c) $(ST)'=T'S', \forall S\in \mathscr{L}(V,W)$ - -Goal: find $range\ T'$ and $null\ T'$ - -#### Definition 3.121 - -Let $U\subseteq V$ be a subspace. The **annihilator** of $U$, denoted by $U^0$ is given by $U^0=\{ \psi\in V'\vert \psi(u)=0\forall u\in U\}$ - -#### Proposition 3.124 - -Given $U\subseteq V$ be a subspace. The **annihilator** of $U$, $U^0\subseteq V'$ is a subspace. - -$$ -dim\ U^0=dim\ V-dim\ U=(dim\ V')-dim\ U -$$ - -Sketch of proof: - -look at $i:U\to V,i(u)=u$, compute $i':V'\to U'$ look at $null\ i'=U^0$ - -#### Theorem 3.128, 3.130 - -a) $null\ T'=(range\ T)^0$, $dim (null\ T')=dim\ null\ T+dim\ W-dim\ V$ -b) $range\ T'=(null\ T)^0$, $dim (range\ T')=dim (range\ T)$ +# Lecture 17 + +## Chapter III Linear maps + +**Assumption: $U,V,W$ are vector spaces (over $\mathbb{F}$)** + +### Duality 3F + +#### Definition 3.108 + +A **linear functional** on $V$ is a linear map from $V$ to $\mathbb{F}$. + +#### Definition 3.110 + +The **dual space** of V denoted by $V'$ (or in some books $\check{V},V^*$) is given by $V'=\mathscr{L}(V,\mathbb{F})$. + +The elements of $V'$ are also called **linear functional**. + +#### Theorem 3.111 + +The $dim\ V'=dim\ V$. + +Proof: + +$dim\ \mathscr{L}(V,\mathbb{F})=dim\ V\cdot dim\ \mathbb{F}$ + +#### Definition 3.112 + +If $v_1,...,v_n$ is a basis for $V$, then the **dual basis** of $v_1,..,v_n$ is $\psi_1,...,\psi_n\in V'$ where + +$$ +\psi_j(v_k)=\begin{cases} + 1 \textup{ if }k=i\\ + 0 \textup{ if }k\neq i +\end{cases} +$$ + +Example: + +$V=\mathbb{R}^3$ $e_1,e_2,e_3$ the standard basis, the dual basis $\psi_1,\psi_2,\psi_3$ is given by $\psi_1 (x,y,z)=x,\psi_2 (x,y,z)=y,\psi_3 (x,y,z)=z$ + +#### Theorem 3.116 + +When $v_1,...,v_n$ a basis of $V$ the dual basis $\psi_1,...,\psi_n\in V'$ is a basis + +Sketch of Proof: + +$dim\ V'=dim\ V=n$, $\psi_1,...,\psi_n\in V'$ are linearly independent. + +#### Theorem 3.114 + +Given $v_1,...,v_n$ a basis of $V$, and $\psi_1,...,\psi_n\in V'$ be dual basis of $V'$. then for $v\in V$, + +$$ +v=\psi_1(v)v_1+...+\psi_n(v)v_n +$$ + +Proof: + +Let $V=a_1 v_1+...+a_n v_n$, consider $\psi_k(v)$, by definition $\psi_k(v)=\psi_k(a_1 v_1+...+a_n v_n)=a_1\psi_k( v_1)+...+a_n\psi_k( v_n)=a_k$ + +#### Definition 3.118 + +Suppose $T\in \mathscr{L}(V,W)$. The **dual map** $T'\in \mathcal{R}( W', V')$ defined by $T'(\psi)=\psi\circ T$. ($\psi\in W'=\mathcal{R}(W,\mathbb{F}), T'(\psi) \in V'=\mathscr{L}(V,\mathbb{F})$) + +Example: + +$T:\mathscr{P}_2(\mathbb{F})\to \mathscr{P}_3(\mathbb{F}),T(f)=xf$ + +$$ +T'(\mathscr{P}_3(\mathbb{F}))'\to (\mathscr{P}_2(\mathbb{F}))',T'(\psi)(f)=\psi(T(f))=\psi(xf) +$$ + +Suppose $\psi(f)=f'(1)\to T(\psi)(f)=(xf)'(1)=f(1)+(xf')(1)=f(1)+f'(1)$ + +#### Theorem 3.120 + +Suppose $T\in \mathscr{L}(V,W)$ + +a) $(S+T)'=S'+T', \forall S\in \mathscr{L}(V,W)$ +b) $(\lambda T)'=\lambda T', \forall \lambda\in \mathbb{F}$ +c) $(ST)'=T'S', \forall S\in \mathscr{L}(V,W)$ + +Goal: find $range\ T'$ and $null\ T'$ + +#### Definition 3.121 + +Let $U\subseteq V$ be a subspace. The **annihilator** of $U$, denoted by $U^0$ is given by $U^0=\{ \psi\in V'\vert \psi(u)=0\forall u\in U\}$ + +#### Proposition 3.124 + +Given $U\subseteq V$ be a subspace. The **annihilator** of $U$, $U^0\subseteq V'$ is a subspace. + +$$ +dim\ U^0=dim\ V-dim\ U=(dim\ V')-dim\ U +$$ + +Sketch of proof: + +look at $i:U\to V,i(u)=u$, compute $i':V'\to U'$ look at $null\ i'=U^0$ + +#### Theorem 3.128, 3.130 + +a) $null\ T'=(range\ T)^0$, $dim (null\ T')=dim\ null\ T+dim\ W-dim\ V$ +b) $range\ T'=(null\ T)^0$, $dim (range\ T')=dim (range\ T)$ diff --git a/pages/Math429/Math429_L18.md b/content/Math429/Math429_L18.md similarity index 96% rename from pages/Math429/Math429_L18.md rename to content/Math429/Math429_L18.md index f6fe39e..a371cd4 100644 --- a/pages/Math429/Math429_L18.md +++ b/content/Math429/Math429_L18.md @@ -1,113 +1,113 @@ -# Lecture 18 - -## Chapter III Linear maps - -**Assumption: $U,V,W$ are vector spaces (over $\mathbb{F}$)** - -### Duality 3F - ---- - -Review - -#### Theorem 3.128, 3.130 - -Let $V,W$ be a finite dimensional vector space, $T\in \mathscr{L}(V,W)$ - -a) $null\ T'=(range\ T)^0$, $dim (null\ T')=dim\ null\ T+dim\ W-dim\ V$ -b) $range\ T'=(null\ T)^0$, $dim (range\ T')=dim (range\ T)$ -c) dim(range\ T')= dim(range\ T) - ---- - -New materials - -#### Theorem 3.129, 3.131 - -Let $V,W$ be a finite dimensional vector space, $T\in \mathscr{L}(V,W)$ - -a) $T$ is injective $\iff T'$ is surjective -b) $T$ is surjective $\iff T'$ is injective - -Proof: - -$T$ is injective $\iff null\ T=\{0\}\iff range\ T'=V'\iff T'$ surjective - -$T$ is surjective $\iff range\ T=W\iff null\ T'=0\iff T'$ injective - -#### Theorem 3.132 - -Let $V,W$ be a finite dimensional vector space, $T\in \mathscr{L}(V,W)$ - -Then $M(T')=(M(T))^T$. Where the basis for $M(T)'$ are the dual basis to the ones for $M(T)$ - -#### Theorem 3.133 - -$col\ rank\ A=row\ rank\ A$ - -Proof: $col\ rank\ A=col\ rank\ (M(T))=dim\ range\ T=dim\ range\ T'=dim\ range\ T'=col\ rank\ (M(T'))=col\ rank\ (M(T)^T)=row\ rank\ (M(T))$ - -## Chapter V Eigenvalue and Eigenvectors - -### Invariant Subspaces 5A - -Goal: Study maps in $\mathscr{L}(V)$ (linear operations) - -Question: Given $T\in \mathscr{L}(V)$ when can I restrict to $U\subseteq V$ such that $T\vert_U\in \mathscr{L}(U)$ - -#### Definition 5.2 - -Suppose $T\in \mathscr{L}(V)$ and $U\subseteq V$ a subspace is said to be invariant under $T$ if $Tu\in U,\forall u\in U$ - -Example: - -For any $T\in \mathscr{L}(V)$, the following are invariance subspaces. - -1. $\{0\}$ -2. $V$ -3. $null\ T$, $v\in null\ T\implies Tv=0\in null\ T$ -4. $range\ T$, $v\in range\ T\subseteq V \implies Tv\in range\ T$ - -#### Definition 5.5 - -Suppose $T\in\mathscr{L}(V)$, then for $\lambda \in \mathbb{F}$ is an **eigenvalue** of $T$ if $\exists v\in V$ such that $v\neq 0$ and $Tv=\lambda v$. - -#### Definition 5.8 - -Suppose $T\in\mathscr{L}(V)$ and $\lambda \in \mathbb{F}$ is an eigenvalue of $T$. The $v\in V$ is an **eigenvector** of $T$ corresponding to $\lambda$ if $v\neq 0$ and $Tv=\lambda v$ - -Note: if $\lambda$ is an eigenvalue of $T$ and $v$ an eigenvector corresponding to $\lambda$, then $U=Span(V)$ is an invariant subspace. and $T\vert_U$ is multiplication by $\lambda$ - -#### Proposition 5.7 - -$V$ is finite dimensional $T\in \mathscr{L},\lambda\in \mathbb{F}$ then the following are equivalent: (TFAE) - -a) $\lambda$ is an eigenvalue -b) $T-\lambda I$ is not injective -c) $T-\lambda I$ is not surjective -d) $T-\lambda I$ is not invertible - -Proof: - -(a)$\iff$ (b) $\lambda$ is an eigenvalue $\iff \exists v\in V$ such that $Tv=\lambda v\iff \exists v\in V, v\neq 0, (T-\lambda I)v=0$ - -Example: - -$T(x,y)=(-y,x)$ what are the eigenvalues of $T$. - -If $\mathbb{F}=\mathbb{R}$ rotation by $90\degree$, so no eigenvalues. - -what if $\mathbb{F}=\mathbb{C}$? we can solve the system $T(x,y)=\lambda (x,y),(-y,x)=\lambda (x,y)$ - -$$ --y=\lambda x \\ -x=\lambda y -$$ - -So - -$$ --1=\lambda ^2,\lambda =\plusmn i -$$ - +# Lecture 18 + +## Chapter III Linear maps + +**Assumption: $U,V,W$ are vector spaces (over $\mathbb{F}$)** + +### Duality 3F + +--- + +Review + +#### Theorem 3.128, 3.130 + +Let $V,W$ be a finite dimensional vector space, $T\in \mathscr{L}(V,W)$ + +a) $null\ T'=(range\ T)^0$, $dim (null\ T')=dim\ null\ T+dim\ W-dim\ V$ +b) $range\ T'=(null\ T)^0$, $dim (range\ T')=dim (range\ T)$ +c) dim(range\ T')= dim(range\ T) + +--- + +New materials + +#### Theorem 3.129, 3.131 + +Let $V,W$ be a finite dimensional vector space, $T\in \mathscr{L}(V,W)$ + +a) $T$ is injective $\iff T'$ is surjective +b) $T$ is surjective $\iff T'$ is injective + +Proof: + +$T$ is injective $\iff null\ T=\{0\}\iff range\ T'=V'\iff T'$ surjective + +$T$ is surjective $\iff range\ T=W\iff null\ T'=0\iff T'$ injective + +#### Theorem 3.132 + +Let $V,W$ be a finite dimensional vector space, $T\in \mathscr{L}(V,W)$ + +Then $M(T')=(M(T))^T$. Where the basis for $M(T)'$ are the dual basis to the ones for $M(T)$ + +#### Theorem 3.133 + +$col\ rank\ A=row\ rank\ A$ + +Proof: $col\ rank\ A=col\ rank\ (M(T))=dim\ range\ T=dim\ range\ T'=dim\ range\ T'=col\ rank\ (M(T'))=col\ rank\ (M(T)^T)=row\ rank\ (M(T))$ + +## Chapter V Eigenvalue and Eigenvectors + +### Invariant Subspaces 5A + +Goal: Study maps in $\mathscr{L}(V)$ (linear operations) + +Question: Given $T\in \mathscr{L}(V)$ when can I restrict to $U\subseteq V$ such that $T\vert_U\in \mathscr{L}(U)$ + +#### Definition 5.2 + +Suppose $T\in \mathscr{L}(V)$ and $U\subseteq V$ a subspace is said to be invariant under $T$ if $Tu\in U,\forall u\in U$ + +Example: + +For any $T\in \mathscr{L}(V)$, the following are invariance subspaces. + +1. $\{0\}$ +2. $V$ +3. $null\ T$, $v\in null\ T\implies Tv=0\in null\ T$ +4. $range\ T$, $v\in range\ T\subseteq V \implies Tv\in range\ T$ + +#### Definition 5.5 + +Suppose $T\in\mathscr{L}(V)$, then for $\lambda \in \mathbb{F}$ is an **eigenvalue** of $T$ if $\exists v\in V$ such that $v\neq 0$ and $Tv=\lambda v$. + +#### Definition 5.8 + +Suppose $T\in\mathscr{L}(V)$ and $\lambda \in \mathbb{F}$ is an eigenvalue of $T$. The $v\in V$ is an **eigenvector** of $T$ corresponding to $\lambda$ if $v\neq 0$ and $Tv=\lambda v$ + +Note: if $\lambda$ is an eigenvalue of $T$ and $v$ an eigenvector corresponding to $\lambda$, then $U=Span(V)$ is an invariant subspace. and $T\vert_U$ is multiplication by $\lambda$ + +#### Proposition 5.7 + +$V$ is finite dimensional $T\in \mathscr{L},\lambda\in \mathbb{F}$ then the following are equivalent: (TFAE) + +a) $\lambda$ is an eigenvalue +b) $T-\lambda I$ is not injective +c) $T-\lambda I$ is not surjective +d) $T-\lambda I$ is not invertible + +Proof: + +(a)$\iff$ (b) $\lambda$ is an eigenvalue $\iff \exists v\in V$ such that $Tv=\lambda v\iff \exists v\in V, v\neq 0, (T-\lambda I)v=0$ + +Example: + +$T(x,y)=(-y,x)$ what are the eigenvalues of $T$. + +If $\mathbb{F}=\mathbb{R}$ rotation by $90\degree$, so no eigenvalues. + +what if $\mathbb{F}=\mathbb{C}$? we can solve the system $T(x,y)=\lambda (x,y),(-y,x)=\lambda (x,y)$ + +$$ +-y=\lambda x \\ +x=\lambda y +$$ + +So + +$$ +-1=\lambda ^2,\lambda =\plusmn i +$$ + when $\lambda =-i$, $v=(1,i)$, $\lambda=i$, $v=(1,-i)$ \ No newline at end of file diff --git a/pages/Math429/Math429_L19.md b/content/Math429/Math429_L19.md similarity index 96% rename from pages/Math429/Math429_L19.md rename to content/Math429/Math429_L19.md index 4b4b56b..87b09f3 100644 --- a/pages/Math429/Math429_L19.md +++ b/content/Math429/Math429_L19.md @@ -1,118 +1,118 @@ -# Lecture 19 - -## Chapter V Eigenvalue and Eigenvectors - -### Invariant Subspaces 5A - -#### Proposition 5.11 - -Suppose $T\in \mathscr{L}(V)$, let $v_1,...,v_n$ be eigenvectors for distinct eigenvalues $\lambda_1,...,\lambda_m$. Then $v_1,...,v_n$ is linearly independent. - -Proof: - -Suppose $v_1,...,v_m$ is linearly dependent, we can assume that $v_1,...,v_{m-1}$ is linearly independent. So let $a_1,...,a_{m}$ not all $=0$. such that $a_1v_1+...+a_nv_m=0$, then we apply $(T-\lambda_m I)$ (map $v_n$ to 0) - -$$ -(T-\lambda_m I)v_k=(\lambda_k-\lambda_m)v_k -$$ - -so - -$$ -(T-\lambda_m I)=a_1(\lambda_1-\lambda_m)v_1+...+a_{m-1}(\lambda_{m-1}-\lambda_{m})v_m -$$ - -but not all of the $a_1,...,a_{m-1}$ are zero and $\lambda_k-\lambda_m\neq 0$ for $1\leq k\leq \lambda$ so they must be linearly independent. - -#### Theorem 5.12 - -Suppose $dim\ V=n$ and $T\in \mathscr{L}(V)$ then $T$ has at most $n$ distinct eigenvalues - -Proof: - -Since $dim\ V=n$ no linearly independent list has length than $n$ so by **Proposition 5.11**, there are at most $n$ distinct eigenvalues. - -#### Polynomials on operators - -$p(z)=z+3z+z^3\in \mathscr{P}(\mathbb{R})$ - -let $T=\begin{pmatrix} - 1&1\\ - 0&1 -\end{pmatrix}\in \mathscr{L}(\mathbb{R}^2)$ - -$P(T)=2I+3T+T^3=2I+3T+\begin{pmatrix} - 1&3\\ - 0&1 -\end{pmatrix}=\begin{pmatrix} - 6&4\\ - 0&6 -\end{pmatrix}$ - -#### Notation - -$T^m=TT...TT$ (m times) $T$ must be an operator within the same space - -$T^0=I$ - -$T^{-m}=(T^{-1})^m$ (where $T$ is invertible) - -if $p\in \mathscr{P}(\mathbb{F})$ with $p(z)=\sum_{i=0}^na_iz^i$ and $T\in \mathscr{L}(V)$ $V$ is a vector space over $\mathbb{F}$ - -$$ -p(T)\sum_{i=0}^na_iT^i -$$ - -#### Lemma 5.17 - -Given $p,q\in \mathscr{P}(\mathbb{F})$, $T\in \mathscr{L}(V)$ - -then - -a) $(pq)T=p(T)q(T)$ -b) $p(T)q(T)=q(T)p(T)$ - -#### Theorem 5.18 - -Suppose $T\in \mathscr{L}(V),p\in \mathscr{P}(\mathbb{F})$, then $null\ (P(T))$ and $range\ (P(T))$ are invariant with respect to $T$. - -### 5B The Minimal Polynomial - -#### Theorem 5.15 - -Every operator on **finite dimensional complex vector space** has at least on eigenvalues. - -Proof: - -Let $dim\ V=n,T\in \mathscr{L}(V), v\in V$ be a nonzero vector. - -Now consider $v,Tv,T^2 v,...,T^n v$. Since this list is of length $n+1$, there is a linear dependence. Let $m$ be the smallest integer such that $v,Tv,..T^m v$ is linearly dependent, then - -$$ -a_0 v+a_1Tv+...+a_m T^m v=0 -$$ - -Let $p(z)=a_0+a_1 z+...+a_m z^m$, then $p(T)(v)=0,p(z)\neq 0$ - -$p(z)$ factors as $(z-\lambda) q(z)$ where $degree\ q< degree\ p$ - -$$ -p(T)(v)=((T-\lambda I)q(T))(v)=0 -$$ - -$$ -(T-\lambda I)(q(T)(v))=0 -$$ - -but $m$ was minimal so that $p(z)=a_0+a_1 z+...+a_m z^m$ were linearly independent, so $q(T)(v)\neq 0$, so $\lambda$ is an eigenvalue with eigenvector $q(T)(v)$ - -#### Definition 5.24 - -Suppose $V$ is finite dimensional $T\in\mathscr{L}(V),p\in \mathscr{P}(\mathbb{F})$, then the **minimal polynomial** is the unique monic (the coefficient of the highest degree is 1) polynomial of minimal degree such that $p(T)=0$ - -#### Theorem 5.27 - -Let $V$ be finite dimensional, and $T\in\mathscr{L}(V)$, $p(z)$ the minimal polynomial. - -1. The roots of $p(z)$ are exactly the eigenvalues of $T$. -2. If $\mathbb{F}=\mathbb{C}$, $p(z)=(z-\lambda_1)...(z-\lambda_m)$ where $\lambda_1,...,\lambda_m$ are all the eigenvalues. +# Lecture 19 + +## Chapter V Eigenvalue and Eigenvectors + +### Invariant Subspaces 5A + +#### Proposition 5.11 + +Suppose $T\in \mathscr{L}(V)$, let $v_1,...,v_n$ be eigenvectors for distinct eigenvalues $\lambda_1,...,\lambda_m$. Then $v_1,...,v_n$ is linearly independent. + +Proof: + +Suppose $v_1,...,v_m$ is linearly dependent, we can assume that $v_1,...,v_{m-1}$ is linearly independent. So let $a_1,...,a_{m}$ not all $=0$. such that $a_1v_1+...+a_nv_m=0$, then we apply $(T-\lambda_m I)$ (map $v_n$ to 0) + +$$ +(T-\lambda_m I)v_k=(\lambda_k-\lambda_m)v_k +$$ + +so + +$$ +(T-\lambda_m I)=a_1(\lambda_1-\lambda_m)v_1+...+a_{m-1}(\lambda_{m-1}-\lambda_{m})v_m +$$ + +but not all of the $a_1,...,a_{m-1}$ are zero and $\lambda_k-\lambda_m\neq 0$ for $1\leq k\leq \lambda$ so they must be linearly independent. + +#### Theorem 5.12 + +Suppose $dim\ V=n$ and $T\in \mathscr{L}(V)$ then $T$ has at most $n$ distinct eigenvalues + +Proof: + +Since $dim\ V=n$ no linearly independent list has length than $n$ so by **Proposition 5.11**, there are at most $n$ distinct eigenvalues. + +#### Polynomials on operators + +$p(z)=z+3z+z^3\in \mathscr{P}(\mathbb{R})$ + +let $T=\begin{pmatrix} + 1&1\\ + 0&1 +\end{pmatrix}\in \mathscr{L}(\mathbb{R}^2)$ + +$P(T)=2I+3T+T^3=2I+3T+\begin{pmatrix} + 1&3\\ + 0&1 +\end{pmatrix}=\begin{pmatrix} + 6&4\\ + 0&6 +\end{pmatrix}$ + +#### Notation + +$T^m=TT...TT$ (m times) $T$ must be an operator within the same space + +$T^0=I$ + +$T^{-m}=(T^{-1})^m$ (where $T$ is invertible) + +if $p\in \mathscr{P}(\mathbb{F})$ with $p(z)=\sum_{i=0}^na_iz^i$ and $T\in \mathscr{L}(V)$ $V$ is a vector space over $\mathbb{F}$ + +$$ +p(T)\sum_{i=0}^na_iT^i +$$ + +#### Lemma 5.17 + +Given $p,q\in \mathscr{P}(\mathbb{F})$, $T\in \mathscr{L}(V)$ + +then + +a) $(pq)T=p(T)q(T)$ +b) $p(T)q(T)=q(T)p(T)$ + +#### Theorem 5.18 + +Suppose $T\in \mathscr{L}(V),p\in \mathscr{P}(\mathbb{F})$, then $null\ (P(T))$ and $range\ (P(T))$ are invariant with respect to $T$. + +### 5B The Minimal Polynomial + +#### Theorem 5.15 + +Every operator on **finite dimensional complex vector space** has at least on eigenvalues. + +Proof: + +Let $dim\ V=n,T\in \mathscr{L}(V), v\in V$ be a nonzero vector. + +Now consider $v,Tv,T^2 v,...,T^n v$. Since this list is of length $n+1$, there is a linear dependence. Let $m$ be the smallest integer such that $v,Tv,..T^m v$ is linearly dependent, then + +$$ +a_0 v+a_1Tv+...+a_m T^m v=0 +$$ + +Let $p(z)=a_0+a_1 z+...+a_m z^m$, then $p(T)(v)=0,p(z)\neq 0$ + +$p(z)$ factors as $(z-\lambda) q(z)$ where $degree\ q< degree\ p$ + +$$ +p(T)(v)=((T-\lambda I)q(T))(v)=0 +$$ + +$$ +(T-\lambda I)(q(T)(v))=0 +$$ + +but $m$ was minimal so that $p(z)=a_0+a_1 z+...+a_m z^m$ were linearly independent, so $q(T)(v)\neq 0$, so $\lambda$ is an eigenvalue with eigenvector $q(T)(v)$ + +#### Definition 5.24 + +Suppose $V$ is finite dimensional $T\in\mathscr{L}(V),p\in \mathscr{P}(\mathbb{F})$, then the **minimal polynomial** is the unique monic (the coefficient of the highest degree is 1) polynomial of minimal degree such that $p(T)=0$ + +#### Theorem 5.27 + +Let $V$ be finite dimensional, and $T\in\mathscr{L}(V)$, $p(z)$ the minimal polynomial. + +1. The roots of $p(z)$ are exactly the eigenvalues of $T$. +2. If $\mathbb{F}=\mathbb{C}$, $p(z)=(z-\lambda_1)...(z-\lambda_m)$ where $\lambda_1,...,\lambda_m$ are all the eigenvalues. diff --git a/pages/Math429/Math429_L2.md b/content/Math429/Math429_L2.md similarity index 100% rename from pages/Math429/Math429_L2.md rename to content/Math429/Math429_L2.md diff --git a/pages/Math429/Math429_L20.md b/content/Math429/Math429_L20.md similarity index 97% rename from pages/Math429/Math429_L20.md rename to content/Math429/Math429_L20.md index 9776182..1591373 100644 --- a/pages/Math429/Math429_L20.md +++ b/content/Math429/Math429_L20.md @@ -1,76 +1,76 @@ -# Lecture 20 - -## Chapter V Eigenvalue and Eigenvectors - -### Minimal polynomial 5B - -#### Definition 5.24 - -Suppose $V$ is finite dimensional, and $T\in \mathscr{L}(V)$ is a linear operator, then the **minimal polynomial** of $T$ is the unique monic polynomial $p$ of smallest degree satisfying the $p(T)=0$. - -#### Theorem 5.22 - -Suppose $V$ is finite dimensional $T\in \mathscr{L}(V)$, then there exists a unique monic polynomial $p\in \mathscr{P}(\mathbb{F})$ of smallest degree such that $p(T)=0$. Furthermore $deg\ p \leq dim\ V$ - -Proof: - -Induct on $dim\ V$ to prove existence. - -* Base case: $dim\ V=0$, i.e $V={0}$. Then any linear operator on $V$ is $0$ including the $I$. So use $p(z)=1$ then $p(T)=I=0$. - -* Inductive step: Suppose the existence holds for all vector spaces with dimension $< dim\ V$. and $dim V\neq 0$, Toke $v\in V,v\neq 0$. Then the list $v,Tv,Tv^2,...,T^n v,n= dim\ V$ is linearly dependent. - - then we take the smallest $m$ such that $v,Tv,...,T^m v$ is linearly dependent, then there exists $c_0,...,c_{n-1}$ such that $c_0 v+c_1T_v+...+c_{m-1} T^{m-1}+T^mv=0$ - - Now we define $p(z)=c_0+c_1z+...+c_{m-1}z^{m-1}+z_m,p(T)v=0$, by ($c_0 v+c_1T_v+...+c_{m-1} T^{m-1}+T^mv=0$) - - Moreover, $p(T)(T^k v)$ let $q(z)=z^k$, then $p(T)(T^k)=p(T)q(T)(v)=0$, so $T^k v\in null(p(T))$, thus since $v,Tv,..,T^{m-1}v$ are linearly independent, thus $dim\ null\ (p(T))\geq m$. - -Note that $dim\ range\ (p(T))\leq dim\ V-m$ is invariant with respect to $T$. - -So consider $T\vert _{range\ (p(T))}$, so by the inductive hypothesis, there exists $S\in \mathscr{P}(\mathbb{F})$ with $deg\ p\leq dim\ range\ (p(T))$ such that $S(T\vert_{range\ (p(T))})$. Now consider $(SP)\in \mathscr{P}(\mathbb{F})$ to see this let $v\in V$. then $(SP)(T)(v)=(S(T)p(T))(v)=S(T)(p(T)v)=S(T)0=0$ - -$deg\ S p=deg\ S+deg\ p\leq dim\ V$ - -uniqueness: Let $p$ be the minimal polynomial, then let $q\in \mathscr{L}(\mathbb{F})$ monic with $q(T)=0$ and $deg\ q=deg\ p$ the $(p-q)(T)=0$ and $deg(p-q)\leq deg\ p$ but then $p-q=0 \implies p=q$ - -### Finding Minimal polynomials - -Idea: Choose $v\in V,v\neq 0$ find $m$ such that $v,Tv,...,T^{dim\ V} v$ - -Find constant (if they exists) such that $v_0v+c_1Tv+...+c_{dim\ V-1} T^{dim\ V-1}+ T^{dim\ V}=0$ - -then if the solution is unique (not always true). then $p(z)=v_0v+c_1Tv+...+c_{dim\ V-1} T^{dim\ V-1}+ T^{dim\ V}$ is the minimal polynomial. - -Example: - -Suppose $T\in \mathscr{L}(\mathbb{R}^5)$ with $M(T)=\begin{pmatrix} - 0&0&0&0&-3\\ - 1&0&0&0&6\\ - 0&1&0&0&0\\ - 0&0&1&0&0\\ - 0&0&0&1&0\\ -\end{pmatrix}$ - -let $v=e_1,Tv=e_2,T^2v=e_3,T^3 v=e_4, T^4v=e_5, T^5v=-3e_1+6e_2$ - -now $T^5v-6Tv+3v=0$ this is unique so $p(z)=z^5-6z+3$ is the minimal polynomial. - -#### Theorem 5.27 - -If $V$ is finite dimensional and $T\in\mathscr{L}(V)$, with minimal polynomial $p$, then the zeros of $p$ are (exactly) their eigenvalues. - -#### Theorem 5.29 - -$T\in \mathscr{L}(V)$, $p$ the minimal polynomial and $q\in\mathscr{P}(\mathbb{F})$, such that $q(T)=0$, the $p$ divides $q$. - -#### Corollary 5.31 - -If $T\in \mathscr{L}(V)$ with minimal polynomial $p$ $U\subseteq V$ (invariant subspace), then $p$ is a multiple of $T\vert_U$ divides $p$. - -#### Theorem 5.32 - -$T$ is not invertible $\iff$ The minimal polynomial has $0$ as a constant term. - - - +# Lecture 20 + +## Chapter V Eigenvalue and Eigenvectors + +### Minimal polynomial 5B + +#### Definition 5.24 + +Suppose $V$ is finite dimensional, and $T\in \mathscr{L}(V)$ is a linear operator, then the **minimal polynomial** of $T$ is the unique monic polynomial $p$ of smallest degree satisfying the $p(T)=0$. + +#### Theorem 5.22 + +Suppose $V$ is finite dimensional $T\in \mathscr{L}(V)$, then there exists a unique monic polynomial $p\in \mathscr{P}(\mathbb{F})$ of smallest degree such that $p(T)=0$. Furthermore $deg\ p \leq dim\ V$ + +Proof: + +Induct on $dim\ V$ to prove existence. + +* Base case: $dim\ V=0$, i.e $V={0}$. Then any linear operator on $V$ is $0$ including the $I$. So use $p(z)=1$ then $p(T)=I=0$. + +* Inductive step: Suppose the existence holds for all vector spaces with dimension $< dim\ V$. and $dim V\neq 0$, Toke $v\in V,v\neq 0$. Then the list $v,Tv,Tv^2,...,T^n v,n= dim\ V$ is linearly dependent. + + then we take the smallest $m$ such that $v,Tv,...,T^m v$ is linearly dependent, then there exists $c_0,...,c_{n-1}$ such that $c_0 v+c_1T_v+...+c_{m-1} T^{m-1}+T^mv=0$ + + Now we define $p(z)=c_0+c_1z+...+c_{m-1}z^{m-1}+z_m,p(T)v=0$, by ($c_0 v+c_1T_v+...+c_{m-1} T^{m-1}+T^mv=0$) + + Moreover, $p(T)(T^k v)$ let $q(z)=z^k$, then $p(T)(T^k)=p(T)q(T)(v)=0$, so $T^k v\in null(p(T))$, thus since $v,Tv,..,T^{m-1}v$ are linearly independent, thus $dim\ null\ (p(T))\geq m$. + +Note that $dim\ range\ (p(T))\leq dim\ V-m$ is invariant with respect to $T$. + +So consider $T\vert _{range\ (p(T))}$, so by the inductive hypothesis, there exists $S\in \mathscr{P}(\mathbb{F})$ with $deg\ p\leq dim\ range\ (p(T))$ such that $S(T\vert_{range\ (p(T))})$. Now consider $(SP)\in \mathscr{P}(\mathbb{F})$ to see this let $v\in V$. then $(SP)(T)(v)=(S(T)p(T))(v)=S(T)(p(T)v)=S(T)0=0$ + +$deg\ S p=deg\ S+deg\ p\leq dim\ V$ + +uniqueness: Let $p$ be the minimal polynomial, then let $q\in \mathscr{L}(\mathbb{F})$ monic with $q(T)=0$ and $deg\ q=deg\ p$ the $(p-q)(T)=0$ and $deg(p-q)\leq deg\ p$ but then $p-q=0 \implies p=q$ + +### Finding Minimal polynomials + +Idea: Choose $v\in V,v\neq 0$ find $m$ such that $v,Tv,...,T^{dim\ V} v$ + +Find constant (if they exists) such that $v_0v+c_1Tv+...+c_{dim\ V-1} T^{dim\ V-1}+ T^{dim\ V}=0$ + +then if the solution is unique (not always true). then $p(z)=v_0v+c_1Tv+...+c_{dim\ V-1} T^{dim\ V-1}+ T^{dim\ V}$ is the minimal polynomial. + +Example: + +Suppose $T\in \mathscr{L}(\mathbb{R}^5)$ with $M(T)=\begin{pmatrix} + 0&0&0&0&-3\\ + 1&0&0&0&6\\ + 0&1&0&0&0\\ + 0&0&1&0&0\\ + 0&0&0&1&0\\ +\end{pmatrix}$ + +let $v=e_1,Tv=e_2,T^2v=e_3,T^3 v=e_4, T^4v=e_5, T^5v=-3e_1+6e_2$ + +now $T^5v-6Tv+3v=0$ this is unique so $p(z)=z^5-6z+3$ is the minimal polynomial. + +#### Theorem 5.27 + +If $V$ is finite dimensional and $T\in\mathscr{L}(V)$, with minimal polynomial $p$, then the zeros of $p$ are (exactly) their eigenvalues. + +#### Theorem 5.29 + +$T\in \mathscr{L}(V)$, $p$ the minimal polynomial and $q\in\mathscr{P}(\mathbb{F})$, such that $q(T)=0$, the $p$ divides $q$. + +#### Corollary 5.31 + +If $T\in \mathscr{L}(V)$ with minimal polynomial $p$ $U\subseteq V$ (invariant subspace), then $p$ is a multiple of $T\vert_U$ divides $p$. + +#### Theorem 5.32 + +$T$ is not invertible $\iff$ The minimal polynomial has $0$ as a constant term. + + + diff --git a/pages/Math429/Math429_L21.md b/content/Math429/Math429_L21.md similarity index 97% rename from pages/Math429/Math429_L21.md rename to content/Math429/Math429_L21.md index 3d71803..0d0a128 100644 --- a/pages/Math429/Math429_L21.md +++ b/content/Math429/Math429_L21.md @@ -1,77 +1,77 @@ -# Lecture 21 - -## Chapter V Eigenvalue and Eigenvectors - -### Minimal polynomial 5B - -#### Odd Dimensional Real Vector Spaces - -#### Theorem 5.34 - -Let $V$ be an odd dimensional real vector space and $T\in \mathscr{L}(V)$ a linear operator then $T$ has an eigenvalue. - -#### Theorem 5.33 - -Let $\mathbb{F}=\mathbb{R}$, $V$ be a finite dimensional vector space. $T\in\mathscr{L}(V)$ then $dim\ null\ (T^2+bT+cI)$ is even for $b^2\leq 4c$. - -Proof: - -$null\ (T^2+bT+cI)$ is invariant under $T$, so it suffices to consider $V=null\ (T^2+bT+cI)$. Thus $T^2+bT+cI=0$. - -Suppose $\lambda \in \mathbb{R}$ and $v\in V$ such that $Tv=\lambda v$, then if $v\neq 0$, then $z-\lambda$ must divide $z^2+bz+c$. but $z^2+bz+c$ does not factor over $\mathbb{R}$. Then we don't have eigenvalues. - -Let $U$ be the **largest invariant subspace** of even dimension. Suppose $w\in V$ and $w\cancel{\in} U$ consider $W=Span\ (w,Tw)$ note $dim\ (w)=2$. Consider $dim(U+W)=dim U+dim W-dim(U\cap W)$. - -So if $dim(U\cap W)=2$ then $w\in U$, which is a contradiction ($w\cancel{\in} U$). - -If $dim(U\cap W)=1$ then $U\cap W$ invariant and gives an eigenvalue, which is a contradiction (don't have eigenvalues). - -If $dim(U\cap W)=0$ $U+W$ is a larger even dimensional invariant subspace, which is a contradiction ($U$ be the **largest invariant subspace** of even dimension). - -So $U=V$, $dim\ V$ is even. - -### Upper Triangular Matrices 5C - -#### Definition 5.38 - -A square matrix is **upper triangular** if all entries below the diagonal are zero. - -Example: - -$$ -\begin{pmatrix} - 1& 2& 3\\ - 0& 3 &4\\ - 0& 0& 5 -\end{pmatrix} -$$ - -#### Theorem 5.39 - -Suppose $T\in \mathscr{L}(V)$ and $v_1,...,v_n$ is a basis, then the following are equal: - -a) $M(T,(v_1,...,v_n))$ is upper triangular -b) $Span\ (v_1,...,v_n)$ is invariant $\forall k=1,...,n$ -c) $Tv_k\in Span\ (v_1,...,v_n)$ $\forall k=1,...,n$ - -Sketch of Proof: - -a)$\implies$c) is clear... (probably) b)$\iff$ c), then do c)$\implies$a), go step by step and construct $M(T,(v_1,...,v_n))$. - -#### Theorem 5.41 - -Suppose $T\in\mathscr{L}(V)$ if there exists a basis where $M(T)$ is upper triangular with diagonal entries $\lambda_1,...,\lambda_n$, and $(T-\lambda _1 I)(T-\lambda_2 I)...(T-\lambda_n I)=0$, then $\lambda_1,...,\lambda_n$ are precisely the eigenvalues. - -Proof: - -Note that for $(T-\lambda_1 I)v_1=0$, consider $(T-\lambda_k I)v_k\in Span\ (v_1,...,v_{k-1})$, consider $w=Span\ (v_1,...,v_k)$ then $(T-\lambda_k I)\vert_w$ is not injective since $range\ (T-\lambda_k I)\vert_w=Span\ (v_1,...,v_{k-1})$, so $\lambda_k$ is an eigenvalue. - -but the minimal polynomial divides $(z-\lambda_1)...(z-\lambda_n)$, so every eigenvalue is in. - -#### Theorem 5.40 - -Suppose $T\in\mathscr{L}(V)$ if there exists a basis where $M(T)$ is upper triangular with diagonal entries $\lambda_1,...,\lambda_n$, then $(T-\lambda _1 I)(T-\lambda_2 I)...(T-\lambda_n I)=0$. - -Proof: - +# Lecture 21 + +## Chapter V Eigenvalue and Eigenvectors + +### Minimal polynomial 5B + +#### Odd Dimensional Real Vector Spaces + +#### Theorem 5.34 + +Let $V$ be an odd dimensional real vector space and $T\in \mathscr{L}(V)$ a linear operator then $T$ has an eigenvalue. + +#### Theorem 5.33 + +Let $\mathbb{F}=\mathbb{R}$, $V$ be a finite dimensional vector space. $T\in\mathscr{L}(V)$ then $dim\ null\ (T^2+bT+cI)$ is even for $b^2\leq 4c$. + +Proof: + +$null\ (T^2+bT+cI)$ is invariant under $T$, so it suffices to consider $V=null\ (T^2+bT+cI)$. Thus $T^2+bT+cI=0$. + +Suppose $\lambda \in \mathbb{R}$ and $v\in V$ such that $Tv=\lambda v$, then if $v\neq 0$, then $z-\lambda$ must divide $z^2+bz+c$. but $z^2+bz+c$ does not factor over $\mathbb{R}$. Then we don't have eigenvalues. + +Let $U$ be the **largest invariant subspace** of even dimension. Suppose $w\in V$ and $w\cancel{\in} U$ consider $W=Span\ (w,Tw)$ note $dim\ (w)=2$. Consider $dim(U+W)=dim U+dim W-dim(U\cap W)$. + +So if $dim(U\cap W)=2$ then $w\in U$, which is a contradiction ($w\cancel{\in} U$). + +If $dim(U\cap W)=1$ then $U\cap W$ invariant and gives an eigenvalue, which is a contradiction (don't have eigenvalues). + +If $dim(U\cap W)=0$ $U+W$ is a larger even dimensional invariant subspace, which is a contradiction ($U$ be the **largest invariant subspace** of even dimension). + +So $U=V$, $dim\ V$ is even. + +### Upper Triangular Matrices 5C + +#### Definition 5.38 + +A square matrix is **upper triangular** if all entries below the diagonal are zero. + +Example: + +$$ +\begin{pmatrix} + 1& 2& 3\\ + 0& 3 &4\\ + 0& 0& 5 +\end{pmatrix} +$$ + +#### Theorem 5.39 + +Suppose $T\in \mathscr{L}(V)$ and $v_1,...,v_n$ is a basis, then the following are equal: + +a) $M(T,(v_1,...,v_n))$ is upper triangular +b) $Span\ (v_1,...,v_n)$ is invariant $\forall k=1,...,n$ +c) $Tv_k\in Span\ (v_1,...,v_n)$ $\forall k=1,...,n$ + +Sketch of Proof: + +a)$\implies$c) is clear... (probably) b)$\iff$ c), then do c)$\implies$a), go step by step and construct $M(T,(v_1,...,v_n))$. + +#### Theorem 5.41 + +Suppose $T\in\mathscr{L}(V)$ if there exists a basis where $M(T)$ is upper triangular with diagonal entries $\lambda_1,...,\lambda_n$, and $(T-\lambda _1 I)(T-\lambda_2 I)...(T-\lambda_n I)=0$, then $\lambda_1,...,\lambda_n$ are precisely the eigenvalues. + +Proof: + +Note that for $(T-\lambda_1 I)v_1=0$, consider $(T-\lambda_k I)v_k\in Span\ (v_1,...,v_{k-1})$, consider $w=Span\ (v_1,...,v_k)$ then $(T-\lambda_k I)\vert_w$ is not injective since $range\ (T-\lambda_k I)\vert_w=Span\ (v_1,...,v_{k-1})$, so $\lambda_k$ is an eigenvalue. + +but the minimal polynomial divides $(z-\lambda_1)...(z-\lambda_n)$, so every eigenvalue is in. + +#### Theorem 5.40 + +Suppose $T\in\mathscr{L}(V)$ if there exists a basis where $M(T)$ is upper triangular with diagonal entries $\lambda_1,...,\lambda_n$, then $(T-\lambda _1 I)(T-\lambda_2 I)...(T-\lambda_n I)=0$. + +Proof: + Note that for $(T-\lambda_1 I)v_1=0$ and $Tv_k\in Span\ (v_1,...,v_k)$, and $Tv_k=\lambda_k v_k+...+\lambda_1 v_1$, $(T-\lambda_k I)\in Span\ (v_1,...,v_k)$ \ No newline at end of file diff --git a/pages/Math429/Math429_L22.md b/content/Math429/Math429_L22.md similarity index 100% rename from pages/Math429/Math429_L22.md rename to content/Math429/Math429_L22.md diff --git a/pages/Math429/Math429_L23.md b/content/Math429/Math429_L23.md similarity index 100% rename from pages/Math429/Math429_L23.md rename to content/Math429/Math429_L23.md diff --git a/pages/Math429/Math429_L24.md b/content/Math429/Math429_L24.md similarity index 97% rename from pages/Math429/Math429_L24.md rename to content/Math429/Math429_L24.md index 4d24d5f..cc9d937 100644 --- a/pages/Math429/Math429_L24.md +++ b/content/Math429/Math429_L24.md @@ -1,89 +1,89 @@ -# Lecture 24 - -## Chapter V Eigenvalue and Eigenvectors - -### 5E Commuting Operators - -#### Definition 5.71 - -* For $T,S\in\mathscr{L}(V)$, $T$ and $S$ commute if $ST=TS$. -* For $A,B\in \mathbb{F}^{n,n}$, $A$ and $B$ commute if $AB=BA$. - -Example: -For $p,q\in \mathscr{P}(\mathbb{F})$, $p(T)$ and $q(T)$ commute - -* Partial Derivatives $\frac{d}{dx},\frac{d}{dy}:\mathscr{P}_m(\mathbb{R}^2)\to \mathscr{P}_m(\mathbb{R}^2)$, $\frac{d}{dy}\frac{d}{dd}=\frac{d}{dx}\frac{d}{dy}$ -* Diagonal matrices commute with each other - -#### Proposition 5.74 - -Given $S,T\in \mathscr{L}(V)$, $S,T$ commute if and only if $M(S), M(T)$ commute. - -Proof: $ST=TS\iff M(ST)=M(TS)\iff M(S)M(T)=M(T)M(S)$ - -#### Proposition 5.75 - -Suppose $S,T\in \mathscr{L}(V)$ commute and $\lambda\in \mathbb{F}$, then the eigenspace $E(\lambda, S)$ is invariant under $T$. - -Proof: -Suppose $V\in E(\lambda, S), S(Tv)=(ST)v=(TS)v=T(Sv)=T\lambda v=\lambda Tv$ - -$Tv$ is an eigenvector with eigenvalue $\lambda$ (or $Tv=0$) $Tv\in E(\lambda, S)$ - -#### Theorem 5.76 - -Let $S,T\in \mathscr{L}(V)$ be diagonalizable operators. Then $S,T$ are **diagonalizable with respect to the same basis** (simultaneously diagonalizable) if and only if $S,T$ commute. - -Proof: - -$\Rightarrow$ - -diagonal matrices commute - -$\Leftarrow$ - -Since $S$ is diagonalizable, $V=E(\lambda, S)\oplus...\oplus E(\lambda_m,S)$, where $\lambda_1,...,\lambda_m$ are the (distinct) eigenvalues of $S$. consider $T\vert_{E(\lambda_k,S)}$ (**Theorem 5.65**) this operator is diagonalizable because $T$ is diagonalizable. We can chose a basis of $E(\lambda_k,S)$ such that $T\vert_{E(\lambda_k,S)}$ gives a diagonal matrix. Take the basis of $V$ given by concatenating the bases of $E(\lambda_k,S)$ the elements of this basis eigenvectors of $S$ and $T$ so $S$ and $T$ are diagonalizable with respect to this basis. - -#### Proposition 5.78 - -Every pair of commuting operators on a finite dimensional complex nonzero vector spaces has at least one common eigenvectors. - -Proof: apply (5.75) and the fact that operator on complex vector spaces has at least one eigenvector. - -#### Theorem 5.80 - -If $S,T$ are commuting operators, then there is a basis where both have upper triangular matrices. - -Proof: - -Induction on $n=dim\ V$. - -$n=1$, clear - -$n>1$, use (5.78) to find $v_1$ and eigenvector for $S$ and $T$. Decompose $V$ as $V=Span(v_1)\oplus W$ then defined a map $P:V\to W,P(av_1+w)=w$ define $\hat{S}:W\to W$ as $\hat{S}(w)=P(S(w))$ similarly $\hat{T}(w)=P(T(w))$ now apply the inductive hypothesis to $\hat{S}$ and $\hat{T}$. get a basis $v_2,...,v_n$ where they are both upper triangular and then exercise: $S,T$ are upper triangular with respect to the basis $v_1,...,v_n$. - -#### Theorem 5.81 - -For $V$ finite dimensional $S,T\in \mathscr{L}(V)$ commuting operators then every eigenvalue of $S+T$ is a sum of an eigenvector of $S$ and an eigenvalue of $T$; every eigenvalue of $S\cdot T$ is a product of an eigenvector of $S$ and an eigenvalue of $T$. - -Proof: - -For upper triangular matrices - -$$ -\begin{pmatrix} - \lambda_1 & & *\\ - & \ddots & \\ - 0 & & \lambda_m -\end{pmatrix}+ -\begin{pmatrix} - \mu_1 & & *\\ - & \ddots & \\ - 0 & & \mu_m -\end{pmatrix}= -\begin{pmatrix} - \lambda_1+\mu_1 & & *\\ - & \ddots & \\ - 0 & & \lambda_m+\mu_m -\end{pmatrix} +# Lecture 24 + +## Chapter V Eigenvalue and Eigenvectors + +### 5E Commuting Operators + +#### Definition 5.71 + +* For $T,S\in\mathscr{L}(V)$, $T$ and $S$ commute if $ST=TS$. +* For $A,B\in \mathbb{F}^{n,n}$, $A$ and $B$ commute if $AB=BA$. + +Example: +For $p,q\in \mathscr{P}(\mathbb{F})$, $p(T)$ and $q(T)$ commute + +* Partial Derivatives $\frac{d}{dx},\frac{d}{dy}:\mathscr{P}_m(\mathbb{R}^2)\to \mathscr{P}_m(\mathbb{R}^2)$, $\frac{d}{dy}\frac{d}{dd}=\frac{d}{dx}\frac{d}{dy}$ +* Diagonal matrices commute with each other + +#### Proposition 5.74 + +Given $S,T\in \mathscr{L}(V)$, $S,T$ commute if and only if $M(S), M(T)$ commute. + +Proof: $ST=TS\iff M(ST)=M(TS)\iff M(S)M(T)=M(T)M(S)$ + +#### Proposition 5.75 + +Suppose $S,T\in \mathscr{L}(V)$ commute and $\lambda\in \mathbb{F}$, then the eigenspace $E(\lambda, S)$ is invariant under $T$. + +Proof: +Suppose $V\in E(\lambda, S), S(Tv)=(ST)v=(TS)v=T(Sv)=T\lambda v=\lambda Tv$ + +$Tv$ is an eigenvector with eigenvalue $\lambda$ (or $Tv=0$) $Tv\in E(\lambda, S)$ + +#### Theorem 5.76 + +Let $S,T\in \mathscr{L}(V)$ be diagonalizable operators. Then $S,T$ are **diagonalizable with respect to the same basis** (simultaneously diagonalizable) if and only if $S,T$ commute. + +Proof: + +$\Rightarrow$ + +diagonal matrices commute + +$\Leftarrow$ + +Since $S$ is diagonalizable, $V=E(\lambda, S)\oplus...\oplus E(\lambda_m,S)$, where $\lambda_1,...,\lambda_m$ are the (distinct) eigenvalues of $S$. consider $T\vert_{E(\lambda_k,S)}$ (**Theorem 5.65**) this operator is diagonalizable because $T$ is diagonalizable. We can chose a basis of $E(\lambda_k,S)$ such that $T\vert_{E(\lambda_k,S)}$ gives a diagonal matrix. Take the basis of $V$ given by concatenating the bases of $E(\lambda_k,S)$ the elements of this basis eigenvectors of $S$ and $T$ so $S$ and $T$ are diagonalizable with respect to this basis. + +#### Proposition 5.78 + +Every pair of commuting operators on a finite dimensional complex nonzero vector spaces has at least one common eigenvectors. + +Proof: apply (5.75) and the fact that operator on complex vector spaces has at least one eigenvector. + +#### Theorem 5.80 + +If $S,T$ are commuting operators, then there is a basis where both have upper triangular matrices. + +Proof: + +Induction on $n=dim\ V$. + +$n=1$, clear + +$n>1$, use (5.78) to find $v_1$ and eigenvector for $S$ and $T$. Decompose $V$ as $V=Span(v_1)\oplus W$ then defined a map $P:V\to W,P(av_1+w)=w$ define $\hat{S}:W\to W$ as $\hat{S}(w)=P(S(w))$ similarly $\hat{T}(w)=P(T(w))$ now apply the inductive hypothesis to $\hat{S}$ and $\hat{T}$. get a basis $v_2,...,v_n$ where they are both upper triangular and then exercise: $S,T$ are upper triangular with respect to the basis $v_1,...,v_n$. + +#### Theorem 5.81 + +For $V$ finite dimensional $S,T\in \mathscr{L}(V)$ commuting operators then every eigenvalue of $S+T$ is a sum of an eigenvector of $S$ and an eigenvalue of $T$; every eigenvalue of $S\cdot T$ is a product of an eigenvector of $S$ and an eigenvalue of $T$. + +Proof: + +For upper triangular matrices + +$$ +\begin{pmatrix} + \lambda_1 & & *\\ + & \ddots & \\ + 0 & & \lambda_m +\end{pmatrix}+ +\begin{pmatrix} + \mu_1 & & *\\ + & \ddots & \\ + 0 & & \mu_m +\end{pmatrix}= +\begin{pmatrix} + \lambda_1+\mu_1 & & *\\ + & \ddots & \\ + 0 & & \lambda_m+\mu_m +\end{pmatrix} $$ \ No newline at end of file diff --git a/pages/Math429/Math429_L25.md b/content/Math429/Math429_L25.md similarity index 95% rename from pages/Math429/Math429_L25.md rename to content/Math429/Math429_L25.md index 31254fc..953f3e0 100644 --- a/pages/Math429/Math429_L25.md +++ b/content/Math429/Math429_L25.md @@ -1,139 +1,139 @@ -# Lecture 25 - -## Chapter VI Inner Product Spaces - -### Inner Products and Norms 6A - -#### Dot Product (Euclidean Inner Product) - -$$ -v\cdot w=v_1w_1+...+v_n w_n -$$ - -$$ --\cdot -:\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R} -$$ - -Some properties - -* $v\cdot v\geq 0$ -* $v\cdot v=0\iff v=0$ -* $(u+v)\cdot w=u\cdot w+v\cdot w$ -* $(c\cdot v)\cdot w=c\cdot(v\cdot w)$ - -#### Definition 6.2 - -An inner product $<,>:V\times V\to \mathbb{F}$ - -Positivity: $\geq 0$ - -Definiteness: $=0\iff v=0$ - -Additivity: $=+$ - -Homogeneity: $<\lambda u, v>=\lambda$ - -Conjugate symmetry: $=\overline{}$ - -Note: the dot product on $\mathbb{R}^n$ satisfies these properties - -Example: - -$V=C^0([-1,-])$ - -$L_2$ - inner product. - -$=\int^1_{-1} f\cdot g$ - -$=\int ^1_{-1}f^2\geq 0$ - -$=+$ - -$<\lambda f,g>=\lambda$ - -$=\int^1_{-1} f\cdot g=\int^1_{-1} g\cdot f=$ - -The result is in real vector space so no conjugate... - -#### Theorem 6.6 - -For $<,>$ an inner product - -(a) Fix $V$, then the map given by $u\mapsto $ is a linear map (Warning: if $\mathbb{F}=\mathbb{C}$, then $u\mapsto$ is not linear). - -(b,c) $<0,v>==0$ - -(d) $=+$ (second terms are additive.) - -(e) $=\bar{\lambda}$ - -#### Definition 6.4 - -An **inner product space** is a pair of vector space and inner product on it. $(v,<,>)$. In practice, we will say "$V$ is an inner product space" and treat $V$ as the vector space. - -For the remainder of the chapter. $V,W$ are inner product vector spaces... - -#### Definition 6.7 - -For $v\in V$ the **norm of $V$** is given by $||v||:=\sqrt{}$ - -#### Theorem 6.9 - -Suppose $v\in V$. - -(a) $||v||=0\iff v=0$ -(b) $||\lambda v||=|\lambda|\ ||v||$ - -Proof: - -$||\lambda v||^2=<\lambda v,\lambda v> =\lambda=\lambda\bar{\lambda}$ - -So $|\lambda|^2 =|\lambda|^2||v||^2$, $||\lambda v||=|\lambda|\ ||v||$ - -#### Definition 6.10 - -$v,u\in V$ are **orthogonal** if $=0$. - -#### Theorem 6.12 (Pythagorean Theorem) - -If $u,v\in V$ are orthogonal, then $||u+v||^2=||u||^2+||v||$ - -Proof: - -$$ -\begin{aligned} - ||u+v||^2&=\\ - &=+\\ - &=+++\\ - &=||u||^2+||v||^2 -\end{aligned} -$$ - -#### Theorem 6.13 - -Suppose $u,v\in V$, $v\neq 0$, set $c=\frac{}{||v||^2}$, then let $w=u-v\cdot v$, then $v$ and $w$ are orthogonal. - -#### Theorem 6.14 (Cauchy-Schwarz) - -Let $u,v\in V$, then $||\leq ||u||\ ||v||$ where equality occurs only $u,v$ are parallel... - -Proof: - -Take the square norm of $u=\frac{}{||u||^2}v+w$. - -#### Theorem 6.17 Triangle Inequality - -If $u,v\in V$, then $||u+v||\leq ||u||+||v||$ - -Proof: - -$$ -\begin{aligned} - ||u+v||^2&=\\ - &=+++\\ - &=||u||^2+||v||^2+2Re()\\ - &\leq ||u||^2+||v||^2+2||\\ - &\leq ||u||^2+||v||^2+2||u||\ ||v||\\ - &\leq (||u||+||v ||)^2 -\end{aligned} -$$ +# Lecture 25 + +## Chapter VI Inner Product Spaces + +### Inner Products and Norms 6A + +#### Dot Product (Euclidean Inner Product) + +$$ +v\cdot w=v_1w_1+...+v_n w_n +$$ + +$$ +-\cdot -:\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R} +$$ + +Some properties + +* $v\cdot v\geq 0$ +* $v\cdot v=0\iff v=0$ +* $(u+v)\cdot w=u\cdot w+v\cdot w$ +* $(c\cdot v)\cdot w=c\cdot(v\cdot w)$ + +#### Definition 6.2 + +An inner product $<,>:V\times V\to \mathbb{F}$ + +Positivity: $\geq 0$ + +Definiteness: $=0\iff v=0$ + +Additivity: $=+$ + +Homogeneity: $<\lambda u, v>=\lambda$ + +Conjugate symmetry: $=\overline{}$ + +Note: the dot product on $\mathbb{R}^n$ satisfies these properties + +Example: + +$V=C^0([-1,-])$ + +$L_2$ - inner product. + +$=\int^1_{-1} f\cdot g$ + +$=\int ^1_{-1}f^2\geq 0$ + +$=+$ + +$<\lambda f,g>=\lambda$ + +$=\int^1_{-1} f\cdot g=\int^1_{-1} g\cdot f=$ + +The result is in real vector space so no conjugate... + +#### Theorem 6.6 + +For $<,>$ an inner product + +(a) Fix $V$, then the map given by $u\mapsto $ is a linear map (Warning: if $\mathbb{F}=\mathbb{C}$, then $u\mapsto$ is not linear). + +(b,c) $<0,v>==0$ + +(d) $=+$ (second terms are additive.) + +(e) $=\bar{\lambda}$ + +#### Definition 6.4 + +An **inner product space** is a pair of vector space and inner product on it. $(v,<,>)$. In practice, we will say "$V$ is an inner product space" and treat $V$ as the vector space. + +For the remainder of the chapter. $V,W$ are inner product vector spaces... + +#### Definition 6.7 + +For $v\in V$ the **norm of $V$** is given by $||v||:=\sqrt{}$ + +#### Theorem 6.9 + +Suppose $v\in V$. + +(a) $||v||=0\iff v=0$ +(b) $||\lambda v||=|\lambda|\ ||v||$ + +Proof: + +$||\lambda v||^2=<\lambda v,\lambda v> =\lambda=\lambda\bar{\lambda}$ + +So $|\lambda|^2 =|\lambda|^2||v||^2$, $||\lambda v||=|\lambda|\ ||v||$ + +#### Definition 6.10 + +$v,u\in V$ are **orthogonal** if $=0$. + +#### Theorem 6.12 (Pythagorean Theorem) + +If $u,v\in V$ are orthogonal, then $||u+v||^2=||u||^2+||v||$ + +Proof: + +$$ +\begin{aligned} + ||u+v||^2&=\\ + &=+\\ + &=+++\\ + &=||u||^2+||v||^2 +\end{aligned} +$$ + +#### Theorem 6.13 + +Suppose $u,v\in V$, $v\neq 0$, set $c=\frac{}{||v||^2}$, then let $w=u-v\cdot v$, then $v$ and $w$ are orthogonal. + +#### Theorem 6.14 (Cauchy-Schwarz) + +Let $u,v\in V$, then $||\leq ||u||\ ||v||$ where equality occurs only $u,v$ are parallel... + +Proof: + +Take the square norm of $u=\frac{}{||u||^2}v+w$. + +#### Theorem 6.17 Triangle Inequality + +If $u,v\in V$, then $||u+v||\leq ||u||+||v||$ + +Proof: + +$$ +\begin{aligned} + ||u+v||^2&=\\ + &=+++\\ + &=||u||^2+||v||^2+2Re()\\ + &\leq ||u||^2+||v||^2+2||\\ + &\leq ||u||^2+||v||^2+2||u||\ ||v||\\ + &\leq (||u||+||v ||)^2 +\end{aligned} +$$ diff --git a/pages/Math429/Math429_L26.md b/content/Math429/Math429_L26.md similarity index 100% rename from pages/Math429/Math429_L26.md rename to content/Math429/Math429_L26.md diff --git a/pages/Math429/Math429_L27.md b/content/Math429/Math429_L27.md similarity index 100% rename from pages/Math429/Math429_L27.md rename to content/Math429/Math429_L27.md diff --git a/pages/Math429/Math429_L28.md b/content/Math429/Math429_L28.md similarity index 96% rename from pages/Math429/Math429_L28.md rename to content/Math429/Math429_L28.md index e6d9c58..2dd39c1 100644 --- a/pages/Math429/Math429_L28.md +++ b/content/Math429/Math429_L28.md @@ -1,132 +1,132 @@ -# Lecture 28 - -## Chapter VI Inner Product Spaces - -### Orthonormal basis 6B - -Example: - -Find a polynomial $q\in \mathscr{P}_2(\mathbb{R})$ such that - -$$ -\int^1_{-1}p(t)cos(\pi t)dt=\int^1_{-1}p(t)q(t)dt -$$ - -for $p\in \mathscr{P}_2(\mathbb{R})$ - -note that $\varphi(p)=\int^1_{-1}p(t)cos(\pi t)cos(\pi t)dt$ is a linear functional. Thus by **Riesz Representation Theorem**, $\exists$ unique $q$ such that $\varphi (p)=\langle p,q \rangle=\int^1_{-1}pq$ - -$$ -q=\overline{\varphi(e_0)}e_0+\overline{\varphi(e_z)}e_z -$$ - -where $e_0,e_1,e_z$ is an orthonormal basis. - -and $q=\frac{15}{2\pi^2}(1-3x^2)$ - -#### Orthogonal Projection and Minimization - -#### Definition 6.46 - -If $U$ is a subset of $V$, then the **orthogonal complement** of $U$ denoted $U^\perp$ - -$$ -U^\perp=\{v\in V\vert \langle u,v\rangle =0,\forall u\in U\} -$$ - -The set of vectors orthogonal to every vector in $U$. - -#### Theorem 6.48 - -Let $U$ be a subset of $V$. - -(a) $U^\perp$ is a subspace of $V$. -(b) $\{0\}^\perp=V$ -(c) $V^\perp =\{0\}$ -(d) $U\cap U^\perp\subseteq\{0\}$ -(e) If $G,H$ subsets of $V$ with $G\subseteq H$, then $H^\perp\subseteq G^\perp$ - -Example: - -Two perpendicular line in 2D plane. - -Let $e_1,...,e_m$ be an orthonormal list, let $u=Span(e_1,...,e_m)$ How do I find $U^\perp$? - -Extend to an orthonormal basis $e_1,...,e_m,f_1...,f_n$. $U^\perp=Span(f_1,...,f_n)$ - -#### Theorem 6.40 - -Suppose $U$ is finite dimensional subspace of $V$, then $V=U\oplus U^\perp$ - -Proof: - -Note $U\cap U^\perp=\{0\}$, so it suffices to show $U+U^\perp=V$. Fix an orthonormal basis $e_1,...,e_m$ of $U$. Let $v\in V$, let $u=\langle v,e_1\rangle e_1+...+\langle v,e_m\rangle e_m$ let $w=v-u$, then $v=u+W$ we need to check that $w\in U^\perp$ - -$\langle w,e_k \rangle=\langle v,e_k\rangle-\langle u,e_k\rangle=\langle v,e_k\rangle-\langle v,e_k\rangle=0$ - -So $w\in U^\perp$ - -#### Corollary 6.51 - -$$ -dim\ U^\perp=dim\ V-dim\ U -$$ - -#### Theorem 6.52 - -Let $U$ be a finite dimensional of a vector space $V$. Then $(U^\perp)^\perp=U$ - -Proof: - -First let $u\in U$ we want to show $u\in (U^\perp)^\perp$, then $\langle u,w \rangle=0$ for all $w\in U^\perp$ but then $u\in (U^\perp)^\perp$ - -Exxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxercise on the other directiononoooonononon. - -#### Corollary 6.54 - -$U^\vert=\{0\}\iff U=V$ - -Proof: - -$(U^\perp)^\perp=\{0\}^\perp\implies U=V$ - -#### Definition 6.55 - -Given $U$ a finite dimensional subspace of $V$. The **orthogonal projection of $V$ onto $U$** is the operator $P_u\in \mathscr{L}(V)$ defined by: For each $v$ write $v=u+w$ where $u\in U$ and $w\in U^\perp$ then $P_u v=u$ - -Formula: - -Let $e_1,...,e_n$ an orthonormal basis of $U$. - -$P_u v=\langle v,e_1\rangle e_1+...+\langle v,e_m\rangle e_m$ - -#### Theorem 6.57 - -(a) $P_u$ is linear. -(b) $P_u u=U,\forall u\in U$ -(c) $P_u w=0,\forall w\in U^\perp$ -(d) $range\ P_u=U$ -(e) $null\ P_u=U^\vert$ -(f) $v-P_u v\in U^\perp$ -(g) $P_u^2=P_u$ -(h) $||P_u v||\leq ||v||$ - -Proof: - -(a) Let $v,v'\in V$ and suppose $v=u+w,v'=u'+w'$, then $v+u'=(u+u')+(w+w')$ this implies that $P_u(v+v')=u+u'=P_u v+ P_u v'$ - -... - -#### Theorem 6.58 Riesz Representation Theorem - -Let $V$ be a finite dimensional vector space for $v\in V$ define $\varphi_v\in V'$ by $\varphi_v(u)=\langle u,v \rangle$. Then the map $v\to \varphi_v$ is a bijection. - -Proof: - -Surjectivity Ideal is let $w\in (null\ \varphi)^\perp$ - -$$ -v=\frac{\varphi(w)}{||w||^2}w,\varphi(v)=||v||^2 -$$ - -make sense $\varphi_v=\varphi$ +# Lecture 28 + +## Chapter VI Inner Product Spaces + +### Orthonormal basis 6B + +Example: + +Find a polynomial $q\in \mathscr{P}_2(\mathbb{R})$ such that + +$$ +\int^1_{-1}p(t)cos(\pi t)dt=\int^1_{-1}p(t)q(t)dt +$$ + +for $p\in \mathscr{P}_2(\mathbb{R})$ + +note that $\varphi(p)=\int^1_{-1}p(t)cos(\pi t)cos(\pi t)dt$ is a linear functional. Thus by **Riesz Representation Theorem**, $\exists$ unique $q$ such that $\varphi (p)=\langle p,q \rangle=\int^1_{-1}pq$ + +$$ +q=\overline{\varphi(e_0)}e_0+\overline{\varphi(e_z)}e_z +$$ + +where $e_0,e_1,e_z$ is an orthonormal basis. + +and $q=\frac{15}{2\pi^2}(1-3x^2)$ + +#### Orthogonal Projection and Minimization + +#### Definition 6.46 + +If $U$ is a subset of $V$, then the **orthogonal complement** of $U$ denoted $U^\perp$ + +$$ +U^\perp=\{v\in V\vert \langle u,v\rangle =0,\forall u\in U\} +$$ + +The set of vectors orthogonal to every vector in $U$. + +#### Theorem 6.48 + +Let $U$ be a subset of $V$. + +(a) $U^\perp$ is a subspace of $V$. +(b) $\{0\}^\perp=V$ +(c) $V^\perp =\{0\}$ +(d) $U\cap U^\perp\subseteq\{0\}$ +(e) If $G,H$ subsets of $V$ with $G\subseteq H$, then $H^\perp\subseteq G^\perp$ + +Example: + +Two perpendicular line in 2D plane. + +Let $e_1,...,e_m$ be an orthonormal list, let $u=Span(e_1,...,e_m)$ How do I find $U^\perp$? + +Extend to an orthonormal basis $e_1,...,e_m,f_1...,f_n$. $U^\perp=Span(f_1,...,f_n)$ + +#### Theorem 6.40 + +Suppose $U$ is finite dimensional subspace of $V$, then $V=U\oplus U^\perp$ + +Proof: + +Note $U\cap U^\perp=\{0\}$, so it suffices to show $U+U^\perp=V$. Fix an orthonormal basis $e_1,...,e_m$ of $U$. Let $v\in V$, let $u=\langle v,e_1\rangle e_1+...+\langle v,e_m\rangle e_m$ let $w=v-u$, then $v=u+W$ we need to check that $w\in U^\perp$ + +$\langle w,e_k \rangle=\langle v,e_k\rangle-\langle u,e_k\rangle=\langle v,e_k\rangle-\langle v,e_k\rangle=0$ + +So $w\in U^\perp$ + +#### Corollary 6.51 + +$$ +dim\ U^\perp=dim\ V-dim\ U +$$ + +#### Theorem 6.52 + +Let $U$ be a finite dimensional of a vector space $V$. Then $(U^\perp)^\perp=U$ + +Proof: + +First let $u\in U$ we want to show $u\in (U^\perp)^\perp$, then $\langle u,w \rangle=0$ for all $w\in U^\perp$ but then $u\in (U^\perp)^\perp$ + +Exxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxercise on the other directiononoooonononon. + +#### Corollary 6.54 + +$U^\vert=\{0\}\iff U=V$ + +Proof: + +$(U^\perp)^\perp=\{0\}^\perp\implies U=V$ + +#### Definition 6.55 + +Given $U$ a finite dimensional subspace of $V$. The **orthogonal projection of $V$ onto $U$** is the operator $P_u\in \mathscr{L}(V)$ defined by: For each $v$ write $v=u+w$ where $u\in U$ and $w\in U^\perp$ then $P_u v=u$ + +Formula: + +Let $e_1,...,e_n$ an orthonormal basis of $U$. + +$P_u v=\langle v,e_1\rangle e_1+...+\langle v,e_m\rangle e_m$ + +#### Theorem 6.57 + +(a) $P_u$ is linear. +(b) $P_u u=U,\forall u\in U$ +(c) $P_u w=0,\forall w\in U^\perp$ +(d) $range\ P_u=U$ +(e) $null\ P_u=U^\vert$ +(f) $v-P_u v\in U^\perp$ +(g) $P_u^2=P_u$ +(h) $||P_u v||\leq ||v||$ + +Proof: + +(a) Let $v,v'\in V$ and suppose $v=u+w,v'=u'+w'$, then $v+u'=(u+u')+(w+w')$ this implies that $P_u(v+v')=u+u'=P_u v+ P_u v'$ + +... + +#### Theorem 6.58 Riesz Representation Theorem + +Let $V$ be a finite dimensional vector space for $v\in V$ define $\varphi_v\in V'$ by $\varphi_v(u)=\langle u,v \rangle$. Then the map $v\to \varphi_v$ is a bijection. + +Proof: + +Surjectivity Ideal is let $w\in (null\ \varphi)^\perp$ + +$$ +v=\frac{\varphi(w)}{||w||^2}w,\varphi(v)=||v||^2 +$$ + +make sense $\varphi_v=\varphi$ diff --git a/pages/Math429/Math429_L29.md b/content/Math429/Math429_L29.md similarity index 96% rename from pages/Math429/Math429_L29.md rename to content/Math429/Math429_L29.md index 441dd0e..6e6a91b 100644 --- a/pages/Math429/Math429_L29.md +++ b/content/Math429/Math429_L29.md @@ -1,110 +1,110 @@ -# Lecture 29 - -## Chapter VI Inner Product Spaces - -### Orthogonal Complements and Minimization Problems 6C - -#### Minimization Problems - -#### Theorem 6.61 - -Suppose $U$ is a finite dimensional subspace of $V$. Let $v\in V$, $u\in U$. Then $||v-P_u v||\leq|| v-u||$. with equality if and only if $u=P_u v$ - -Proof: - -Using triangle inequality - -$$ -\begin{aligned} -||v-P_u v||^2 &\leq ||v-P_u v||^2+||P_u v-u||^2\\ -&=||(v-P_u v)+(P_u v-u)||^2\\ -&=||v-u||^2 -\end{aligned} -$$ - -Example: - -Find $u(x)\in \mathscr{P}_5(\mathbb{R}) minimizing - -$$ -\int^{\pi}_{-\pi}|sin(x)-u(x)|^2 dx -$$ - -$V=C([-\pi,\pi])=$ continuous (real valued) function on $[-\pi,\pi]$ - -$u=\mathscr{P}_5(\mathbb{R})$. Note $U\subseteq V$ and $u$ is finite dimensional. - -$\langle f,g \rangle=\int^{\pi}_{-\pi}fg$ gives an inner product on $V$. - -Minimize $||sin-u||^2$, choose an orthonormal basis $e_0,...,e_5$ of $\mathscr{P}_5(\mathbb{R})$, so $u=P_u(sin)=\langle e_0,sin\rangle e_0+...+\langle e_5,sin \rangle e_5$ - -#### Pseudo inverses - -Idea: Want to (approximately) solve $Tx=b$. - -- If $T$ is invertible $x=T^{-1}b$ -- If $T$ is not invertible, want $T^{T}$ such that $y=T^{T}b$ is the "best solution" - -#### Lemma 6.67 - -If $V$ is a finite dimensional vector space, $T\in \mathscr{L}(V,W)$ then $T\vert_{{null\ T}^\perp}$ is one to one onto $range\ T$. - -Proof: - -Note $(null\ T)^\perp \simeq V/(null\ T)$ - -Exercise, prove this... - -If $v\in null(T\vert_{{null}^\perp})\implies v\in null\ T,$ and $v\in (null\ T)^\perp\implies v=0$ - -If $w\in range\ T$ so $\exists v\in V$ such that $Tv=w$ write $v$ as $v=u+x$ sor $u\in null\ T,x\in (null\ T)^\perp$. - -#### Definition 6.68 - -V is a finite dimensional space $T\in \mathscr{L}(V,W)$. The **pseudo-inverse** denoted $T^\dag\in \mathscr{L}(W,V)$ is given by - -$$ -T^\dag w=(T\vert_{{null\ T}^\perp})^{-1}P_{range\ T}w -$$ - -Some explanation: - -_Let $T\in \mathscr{L}(V,W)$.Since there exists isomorphism between $(null\ T)^\perp\subseteq V$ and $range\ T\subseteq W$.We can always map $W$ to $V$ using $T^\dag\in \mathscr{L}(W,V)$. $P_{range\ T}$ is the map that $W\mapsto range\ T$ and $(T\vert_{{null\ T}^\perp})^{-1}$ is a linear map that map $w\in W$_ - -#### Proposition 6.69 - -$V$ is a finite dimensional vector space. $T\in\mathscr{L}(V,W)$, then - -(a) If $T$ is invertible, then $T^\dag=T^{-1}$. -(b) $TT^\dag=P_{range\ T}$. -(c) $T^\dag T=P_{(null\ T)^\perp}$. - -#### Theorem 6.70 - -$V$ is a finite dimensional vector space. $T\in\mathscr{L}(V,W)$, for $b\in W$, then - -(a) If $x\in V$, then $||T(T^* b)-b||\leq ||Tx-b||$ with equality if and only if $x\in T^\dag b+null\ T$ (_$T^\dag$ is the best solution we can have as "inverse" for non-invertible linear map_) - -(b) If $x\in T^\dag b+null\ T$ then - -$$ -||T^\dag b ||\leq ||x|| -$$ - -Proof: - -(a) $Tx-b=(Tx-TT^\dag b)+(TT^\dag b-b)$ - -Using pythagorean theorem, we have - -$||Tx-b||\geq ||TT^\dag b-b||$ - -## Chapter VII Operators on Inner Product Spaces - -### Self adjoint and Normal Operators 7A - -#### Definition 7.1 - -Let $T\in \mathscr{L}(V,W)$, then the **adjoint** of $T$ denoted $T^*$ is the function $T^*:W\to V$ such that $\langle Tv,w \rangle =\langle v,T^* w \rangle$ - -For euclidean inner product $T^*$ is given by the conjugate transpose. +# Lecture 29 + +## Chapter VI Inner Product Spaces + +### Orthogonal Complements and Minimization Problems 6C + +#### Minimization Problems + +#### Theorem 6.61 + +Suppose $U$ is a finite dimensional subspace of $V$. Let $v\in V$, $u\in U$. Then $||v-P_u v||\leq|| v-u||$. with equality if and only if $u=P_u v$ + +Proof: + +Using triangle inequality + +$$ +\begin{aligned} +||v-P_u v||^2 &\leq ||v-P_u v||^2+||P_u v-u||^2\\ +&=||(v-P_u v)+(P_u v-u)||^2\\ +&=||v-u||^2 +\end{aligned} +$$ + +Example: + +Find $u(x)\in \mathscr{P}_5(\mathbb{R}) minimizing + +$$ +\int^{\pi}_{-\pi}|sin(x)-u(x)|^2 dx +$$ + +$V=C([-\pi,\pi])=$ continuous (real valued) function on $[-\pi,\pi]$ + +$u=\mathscr{P}_5(\mathbb{R})$. Note $U\subseteq V$ and $u$ is finite dimensional. + +$\langle f,g \rangle=\int^{\pi}_{-\pi}fg$ gives an inner product on $V$. + +Minimize $||sin-u||^2$, choose an orthonormal basis $e_0,...,e_5$ of $\mathscr{P}_5(\mathbb{R})$, so $u=P_u(sin)=\langle e_0,sin\rangle e_0+...+\langle e_5,sin \rangle e_5$ + +#### Pseudo inverses + +Idea: Want to (approximately) solve $Tx=b$. + +- If $T$ is invertible $x=T^{-1}b$ +- If $T$ is not invertible, want $T^{T}$ such that $y=T^{T}b$ is the "best solution" + +#### Lemma 6.67 + +If $V$ is a finite dimensional vector space, $T\in \mathscr{L}(V,W)$ then $T\vert_{{null\ T}^\perp}$ is one to one onto $range\ T$. + +Proof: + +Note $(null\ T)^\perp \simeq V/(null\ T)$ + +Exercise, prove this... + +If $v\in null(T\vert_{{null}^\perp})\implies v\in null\ T,$ and $v\in (null\ T)^\perp\implies v=0$ + +If $w\in range\ T$ so $\exists v\in V$ such that $Tv=w$ write $v$ as $v=u+x$ sor $u\in null\ T,x\in (null\ T)^\perp$. + +#### Definition 6.68 + +V is a finite dimensional space $T\in \mathscr{L}(V,W)$. The **pseudo-inverse** denoted $T^\dag\in \mathscr{L}(W,V)$ is given by + +$$ +T^\dag w=(T\vert_{{null\ T}^\perp})^{-1}P_{range\ T}w +$$ + +Some explanation: + +_Let $T\in \mathscr{L}(V,W)$.Since there exists isomorphism between $(null\ T)^\perp\subseteq V$ and $range\ T\subseteq W$.We can always map $W$ to $V$ using $T^\dag\in \mathscr{L}(W,V)$. $P_{range\ T}$ is the map that $W\mapsto range\ T$ and $(T\vert_{{null\ T}^\perp})^{-1}$ is a linear map that map $w\in W$_ + +#### Proposition 6.69 + +$V$ is a finite dimensional vector space. $T\in\mathscr{L}(V,W)$, then + +(a) If $T$ is invertible, then $T^\dag=T^{-1}$. +(b) $TT^\dag=P_{range\ T}$. +(c) $T^\dag T=P_{(null\ T)^\perp}$. + +#### Theorem 6.70 + +$V$ is a finite dimensional vector space. $T\in\mathscr{L}(V,W)$, for $b\in W$, then + +(a) If $x\in V$, then $||T(T^* b)-b||\leq ||Tx-b||$ with equality if and only if $x\in T^\dag b+null\ T$ (_$T^\dag$ is the best solution we can have as "inverse" for non-invertible linear map_) + +(b) If $x\in T^\dag b+null\ T$ then + +$$ +||T^\dag b ||\leq ||x|| +$$ + +Proof: + +(a) $Tx-b=(Tx-TT^\dag b)+(TT^\dag b-b)$ + +Using pythagorean theorem, we have + +$||Tx-b||\geq ||TT^\dag b-b||$ + +## Chapter VII Operators on Inner Product Spaces + +### Self adjoint and Normal Operators 7A + +#### Definition 7.1 + +Let $T\in \mathscr{L}(V,W)$, then the **adjoint** of $T$ denoted $T^*$ is the function $T^*:W\to V$ such that $\langle Tv,w \rangle =\langle v,T^* w \rangle$ + +For euclidean inner product $T^*$ is given by the conjugate transpose. diff --git a/pages/Math429/Math429_L3.md b/content/Math429/Math429_L3.md similarity index 100% rename from pages/Math429/Math429_L3.md rename to content/Math429/Math429_L3.md diff --git a/pages/Math429/Math429_L30.md b/content/Math429/Math429_L30.md similarity index 96% rename from pages/Math429/Math429_L30.md rename to content/Math429/Math429_L30.md index 49b4074..001ac3c 100644 --- a/pages/Math429/Math429_L30.md +++ b/content/Math429/Math429_L30.md @@ -1,104 +1,104 @@ -# Lecture 30 - -## Chapter VII Operators on Inner Product Spaces - -**Assumption: $V,W$ are finite dimensional inner product spaces.** - -### Self adjoint and Normal Operators 7A - -#### Definition 7.1 - -Suppose $T\in \mathscr{L}(V,W)$. The adjoint is the function $T^*:W\to V$ such that - -$$ -\langle Tv,w \rangle=\langle v,T^*w \rangle, \forall v\in V, w\in W -$$ - -#### Theorem 7.4 - -Suppose $T\in \mathscr{L}(V,W)$ then $T^*\in \mathscr{L}(W,V)$ - -Proof: - -Additivity, let $w_1,w_2\in W$. We want to show $T^*(w_1+w_2)T^*w_1+T^*w_2$ - -Let $v\in V$, then - -$$ -\begin{aligned} - \langle v,T^*(w_1+w_2) \rangle &=\langle v,T^*(w_1+w_2) \rangle\\ - &=\langle Tv,w_1+w_2 \rangle\\ - &=\langle Tv,w_1 \rangle+\langle Tv,w_2 \rangle\\ - &=\langle v,T^*w_1 \rangle+\langle v,T^* w_2 \rangle\\ - &=\langle v,T^*w_1 +T^* w_2 \rangle\\ -\end{aligned} -$$ - -Note: If $\langle v,u \rangle=\langle v,u'\rangle$, forall $v\in V$ then $u=u'$ - -Homogeneity: same as idea above. - -#### Theorem 7.5 - -Suppose $S,T\in \mathscr{L}(V,W)$, and $\lambda\in \mathbb{F}$, then - -(a) $(S+T)^*=S^*+T^*$ -(b) $(ST)^*=T^* S^*$ -(c) $(\lambda T)^*=\bar{\lambda}S^*$ -(d) $I^*=I$ -(e) $(T^*)^*=T$ -(f) If $T$ is invertible, then $(T^*)^{-1}=(T^{-1})^*$ - -Proof: - -(d) $\langle (ST)v,u \rangle=\langle S(Tv),u \rangle=\langle Tv,S^*u \rangle=\langle v,T^*S^*u \rangle$ - -#### Theorem 7.6 - -Suppose $T\in\mathscr{L}(V,W)$, then - -(a) $null\ T^*=(range\ T)^\perp$ -(b) $null\ T=(range\ T^*)^\perp$ -(c) $range\ T^*=(null\ T)^\perp$ -(d) $range\ T=(null\ T^*)^\perp$ - -Proof: - -$(a)\iff (c)$ since we can use **Theorem 7.5** (c) while replacing $T$ with $T^*$. Same idea give $(b)\iff (d)$. Also $(a)\iff (d)$ Since $(V^\perp)^\perp=V$ - -Now we prove (a). Suppose $w\in null\ T^*\iff T^*w=0$ - -$T^*w=0\iff \langle v,T^* w\rangle=0\forall v\in V\iff \langle Tv,w\rangle =0\forall v\in V\iff w\in (range\ T)^\perp$ - -#### Definition 7.7 - -The **conjugate transpose** of a $m\times n$ matrix $A$ is the $n\times m$ matrix denoted $A^*$ given the conjugate of the transpose. - -ie. $(A^*)_{j,k}=A_{j,k}$ - -#### Theorem 7.9 - -Let $T\in \mathscr{L}(V,W)$ and $e_1,..,e_n$ an orthonormal basis of $V$, $f_1,...,f_m$ be an orthonormal basis of $W$. Then $M(T^*,(f_1,...,f_m),(e_1,..,e_n))=M(T,(f_1,...,f_m),(e_1,..,e_n))^*$ - -Proof: - -The k-th column of $T$ is given by writing $Te_k$. in the basis $f_1,...,f_m$. ie. the $k,j$ entry of $M(T)$ is $\langle Te_k,f_j \rangle$, but then the $j,k$ entry of $M(T^*)$ is $\langle T^*f,e_k \rangle$. But $\langle Te_k,f_j\rangle=\langle e_k,T^*f_j\rangle=\overline{\langle T^*f_j,e_k\rangle}$ - -Example: - -Suppose $T(x_1,x_2,x_3)=(x_2+3x_3,2x_1)$ - -$$ -\begin{aligned} - \langle T(x_1,x_2,x_3),(y_1,y_2)\rangle&=\langle (x_2+3x_3,2x_1),(y_1,y_2)\rangle\\ - &=(x_2+3x_3,2x_1)\bar{y_1},(x_2+3x_3,2x_1)\bar{y_2}\\ - &=\bar{y_1}x_2+3\bar{y_1}x_3+2\bar{y_2}x_1\\ - &=\langle (x_1,x_2,x_3),(2y_2,y_1,3y_1)\rangle -\end{aligned} -$$ - -So $T^*(y_1,y_2)=(2y_2,y_1,3y_1)$ - -Idea: Reisz Representation gives a function from $V$ to $V'$ (3.118) tells us given $T\in \mathscr{L}(V,W)$, we have - - +# Lecture 30 + +## Chapter VII Operators on Inner Product Spaces + +**Assumption: $V,W$ are finite dimensional inner product spaces.** + +### Self adjoint and Normal Operators 7A + +#### Definition 7.1 + +Suppose $T\in \mathscr{L}(V,W)$. The adjoint is the function $T^*:W\to V$ such that + +$$ +\langle Tv,w \rangle=\langle v,T^*w \rangle, \forall v\in V, w\in W +$$ + +#### Theorem 7.4 + +Suppose $T\in \mathscr{L}(V,W)$ then $T^*\in \mathscr{L}(W,V)$ + +Proof: + +Additivity, let $w_1,w_2\in W$. We want to show $T^*(w_1+w_2)T^*w_1+T^*w_2$ + +Let $v\in V$, then + +$$ +\begin{aligned} + \langle v,T^*(w_1+w_2) \rangle &=\langle v,T^*(w_1+w_2) \rangle\\ + &=\langle Tv,w_1+w_2 \rangle\\ + &=\langle Tv,w_1 \rangle+\langle Tv,w_2 \rangle\\ + &=\langle v,T^*w_1 \rangle+\langle v,T^* w_2 \rangle\\ + &=\langle v,T^*w_1 +T^* w_2 \rangle\\ +\end{aligned} +$$ + +Note: If $\langle v,u \rangle=\langle v,u'\rangle$, forall $v\in V$ then $u=u'$ + +Homogeneity: same as idea above. + +#### Theorem 7.5 + +Suppose $S,T\in \mathscr{L}(V,W)$, and $\lambda\in \mathbb{F}$, then + +(a) $(S+T)^*=S^*+T^*$ +(b) $(ST)^*=T^* S^*$ +(c) $(\lambda T)^*=\bar{\lambda}S^*$ +(d) $I^*=I$ +(e) $(T^*)^*=T$ +(f) If $T$ is invertible, then $(T^*)^{-1}=(T^{-1})^*$ + +Proof: + +(d) $\langle (ST)v,u \rangle=\langle S(Tv),u \rangle=\langle Tv,S^*u \rangle=\langle v,T^*S^*u \rangle$ + +#### Theorem 7.6 + +Suppose $T\in\mathscr{L}(V,W)$, then + +(a) $null\ T^*=(range\ T)^\perp$ +(b) $null\ T=(range\ T^*)^\perp$ +(c) $range\ T^*=(null\ T)^\perp$ +(d) $range\ T=(null\ T^*)^\perp$ + +Proof: + +$(a)\iff (c)$ since we can use **Theorem 7.5** (c) while replacing $T$ with $T^*$. Same idea give $(b)\iff (d)$. Also $(a)\iff (d)$ Since $(V^\perp)^\perp=V$ + +Now we prove (a). Suppose $w\in null\ T^*\iff T^*w=0$ + +$T^*w=0\iff \langle v,T^* w\rangle=0\forall v\in V\iff \langle Tv,w\rangle =0\forall v\in V\iff w\in (range\ T)^\perp$ + +#### Definition 7.7 + +The **conjugate transpose** of a $m\times n$ matrix $A$ is the $n\times m$ matrix denoted $A^*$ given the conjugate of the transpose. + +ie. $(A^*)_{j,k}=A_{j,k}$ + +#### Theorem 7.9 + +Let $T\in \mathscr{L}(V,W)$ and $e_1,..,e_n$ an orthonormal basis of $V$, $f_1,...,f_m$ be an orthonormal basis of $W$. Then $M(T^*,(f_1,...,f_m),(e_1,..,e_n))=M(T,(f_1,...,f_m),(e_1,..,e_n))^*$ + +Proof: + +The k-th column of $T$ is given by writing $Te_k$. in the basis $f_1,...,f_m$. ie. the $k,j$ entry of $M(T)$ is $\langle Te_k,f_j \rangle$, but then the $j,k$ entry of $M(T^*)$ is $\langle T^*f,e_k \rangle$. But $\langle Te_k,f_j\rangle=\langle e_k,T^*f_j\rangle=\overline{\langle T^*f_j,e_k\rangle}$ + +Example: + +Suppose $T(x_1,x_2,x_3)=(x_2+3x_3,2x_1)$ + +$$ +\begin{aligned} + \langle T(x_1,x_2,x_3),(y_1,y_2)\rangle&=\langle (x_2+3x_3,2x_1),(y_1,y_2)\rangle\\ + &=(x_2+3x_3,2x_1)\bar{y_1},(x_2+3x_3,2x_1)\bar{y_2}\\ + &=\bar{y_1}x_2+3\bar{y_1}x_3+2\bar{y_2}x_1\\ + &=\langle (x_1,x_2,x_3),(2y_2,y_1,3y_1)\rangle +\end{aligned} +$$ + +So $T^*(y_1,y_2)=(2y_2,y_1,3y_1)$ + +Idea: Reisz Representation gives a function from $V$ to $V'$ (3.118) tells us given $T\in \mathscr{L}(V,W)$, we have + + diff --git a/pages/Math429/Math429_L31.md b/content/Math429/Math429_L31.md similarity index 95% rename from pages/Math429/Math429_L31.md rename to content/Math429/Math429_L31.md index f255a96..41f0095 100644 --- a/pages/Math429/Math429_L31.md +++ b/content/Math429/Math429_L31.md @@ -1,142 +1,142 @@ -# Lecture 31 - -## Chapter VII Operators on Inner Product Spaces - -**Assumption: $V,W$ are finite dimensional inner product spaces.** - -### Self adjoint and Normal Operators 7A - -#### Definition 7.10 - -An operator $T\in\mathscr{L}(V)$ is **self adjoint** if $T=T^*$. ie. $\langle Tv,u\rangle=\langle v,Tu \rangle$ for $u,v\in V$. - -Example: - -Consider $M(T)=\begin{pmatrix} - 2 & i\\ - -i& 3 -\end{pmatrix}$, Then - -$$ -M(T^*)=M(T)^*=\begin{pmatrix} - \bar{2},\bar{-i}\\ - \bar{i},\bar{3} -\end{pmatrix}=\begin{pmatrix} - 2 &i\\ - -i& 3 -\end{pmatrix}=M(T) -$$ - -So $T=T^*$ so $T$ is self adjoint - -#### Theorem 7.12 - -Every eigenvalue of a self adjoint operator $T$ is real. - -Proof: - -Suppose $T$ is self adjoint and $\lambda$ is an eigenvalue of $T$, and $v$ is an eigenvector with eigenvalue $\lambda$. - -Consider $\langle Tv,v\rangle$ - -$$ -\langle Tv, v\rangle= -\langle v, Tv\rangle= -\langle v,\lambda v\rangle= -\bar{\lambda}\langle v,v\rangle=\bar{\lambda}||v||^2 -$$ -$$ -\langle Tv, v\rangle= -\langle \lambda v, v\rangle= -\langle v, v\rangle= -\lambda\langle v,v\rangle=\lambda||v||^2\\ -$$ - -So $\lambda=\bar{\lambda}$, so $\lambda$ is real. - -NoteL (7.12) is only interesting for complex vector spaces. - -#### Theorem 7.13 - -Suppose $V$ is a complex inner product space and $T\in\mathscr{L}(V)$, then - -$$ -\langle Tv, v\rangle =0 \textup{ for every }v\in V\iff T=0 -$$ - -Note: (7.13) is **False** over real vector spaces. The counterexample is $T$ the rotation by $90\degree$ operator. ie. $M(T)=\begin{pmatrix} - 0&-1\\ - 1&0 -\end{pmatrix}$ - -Proof: - -$\Rightarrow$ Suppose $u,w\in V$ - -$$ -\begin{aligned} -\langle Tu,w \rangle&=\frac{\langle T(u+w),u+w\rangle -\langle T(u-w),u-w\rangle}{4}+\frac{\langle T(u+iw),u+iw\rangle -\langle T(u-iw),u-iw\rangle}{4}i\\ -&=0 -\end{aligned} -$$ - -Since $w$ is arbitrary $\implies Tu=0, \forall u\in V\implies T=0$. - -#### Theorem 7.14 - -Suppose $V$ is a complex inner product space and $T\in \mathscr{L}(V)$ thne - -$$ -T \textup{ is self adjoint }\iff \langle Tv, v\rangle \in \mathbb{R} \textup{ for every} v \in V -$$ - -Proof: - -$$ -\begin{aligned} - T\textup{ is self adjoint}&\iff T-T^*=0\\ - &\iff \langle (T-T^*)v,v\rangle =0 (\textup{ by \textbf{7.13}})\\ - &\iff \langle Tv, v\rangle -\langle T^*v,v \rangle =0\\ - &\iff \langle Tv, v\rangle -\overline{\langle T,v \rangle} =0\\ - &\iff\langle Tv,v\rangle \in \mathbb{R} -\end{aligned} -$$ - -#### Theorem 7.16 - -Suppose $T$ is a self adjoint operator, then $\langle Tv, v\rangle =0,\forall v\in V\iff T=0$ - -Proof: - -Note the complex case is **Theorem 7.13**, so assume $V$ is a real vector space. Let $u,w\in V$ consider - -$\Rightarrow$ - -$$ -\langle Tu,w\rangle=\frac{\langle T(u+w),u+w\rangle -\langle T(u-w),u-w\rangle}{4}=0 -$$ - -We set $\langle Tw,u\rangle=\langle w,Tu\rangle =\langle Tu,w\rangle$ - -#### Normal Operators - -#### Definition 7.18 - -An operator $T\in \mathscr{L}(V)$ on an inner product space is **normal** if $TT^*=T^*T$ ie. $T$ commutes with its adjoint - -#### Theorem (7.20) - -An operator $T$ is normal if and only if - -$$ -||Tv||=||T^*v||,\forall v\in V -$$ - -Proof: - -The key idea is that $T^*T-TT^*$ is self adjoint. - -$$ -(T^*T-TT^*)^*=(T^*T)^*-(TT^*)^*=T^*T-TT^* -$$ - +# Lecture 31 + +## Chapter VII Operators on Inner Product Spaces + +**Assumption: $V,W$ are finite dimensional inner product spaces.** + +### Self adjoint and Normal Operators 7A + +#### Definition 7.10 + +An operator $T\in\mathscr{L}(V)$ is **self adjoint** if $T=T^*$. ie. $\langle Tv,u\rangle=\langle v,Tu \rangle$ for $u,v\in V$. + +Example: + +Consider $M(T)=\begin{pmatrix} + 2 & i\\ + -i& 3 +\end{pmatrix}$, Then + +$$ +M(T^*)=M(T)^*=\begin{pmatrix} + \bar{2},\bar{-i}\\ + \bar{i},\bar{3} +\end{pmatrix}=\begin{pmatrix} + 2 &i\\ + -i& 3 +\end{pmatrix}=M(T) +$$ + +So $T=T^*$ so $T$ is self adjoint + +#### Theorem 7.12 + +Every eigenvalue of a self adjoint operator $T$ is real. + +Proof: + +Suppose $T$ is self adjoint and $\lambda$ is an eigenvalue of $T$, and $v$ is an eigenvector with eigenvalue $\lambda$. + +Consider $\langle Tv,v\rangle$ + +$$ +\langle Tv, v\rangle= +\langle v, Tv\rangle= +\langle v,\lambda v\rangle= +\bar{\lambda}\langle v,v\rangle=\bar{\lambda}||v||^2 +$$ +$$ +\langle Tv, v\rangle= +\langle \lambda v, v\rangle= +\langle v, v\rangle= +\lambda\langle v,v\rangle=\lambda||v||^2\\ +$$ + +So $\lambda=\bar{\lambda}$, so $\lambda$ is real. + +NoteL (7.12) is only interesting for complex vector spaces. + +#### Theorem 7.13 + +Suppose $V$ is a complex inner product space and $T\in\mathscr{L}(V)$, then + +$$ +\langle Tv, v\rangle =0 \textup{ for every }v\in V\iff T=0 +$$ + +Note: (7.13) is **False** over real vector spaces. The counterexample is $T$ the rotation by $90\degree$ operator. ie. $M(T)=\begin{pmatrix} + 0&-1\\ + 1&0 +\end{pmatrix}$ + +Proof: + +$\Rightarrow$ Suppose $u,w\in V$ + +$$ +\begin{aligned} +\langle Tu,w \rangle&=\frac{\langle T(u+w),u+w\rangle -\langle T(u-w),u-w\rangle}{4}+\frac{\langle T(u+iw),u+iw\rangle -\langle T(u-iw),u-iw\rangle}{4}i\\ +&=0 +\end{aligned} +$$ + +Since $w$ is arbitrary $\implies Tu=0, \forall u\in V\implies T=0$. + +#### Theorem 7.14 + +Suppose $V$ is a complex inner product space and $T\in \mathscr{L}(V)$ thne + +$$ +T \textup{ is self adjoint }\iff \langle Tv, v\rangle \in \mathbb{R} \textup{ for every} v \in V +$$ + +Proof: + +$$ +\begin{aligned} + T\textup{ is self adjoint}&\iff T-T^*=0\\ + &\iff \langle (T-T^*)v,v\rangle =0 (\textup{ by \textbf{7.13}})\\ + &\iff \langle Tv, v\rangle -\langle T^*v,v \rangle =0\\ + &\iff \langle Tv, v\rangle -\overline{\langle T,v \rangle} =0\\ + &\iff\langle Tv,v\rangle \in \mathbb{R} +\end{aligned} +$$ + +#### Theorem 7.16 + +Suppose $T$ is a self adjoint operator, then $\langle Tv, v\rangle =0,\forall v\in V\iff T=0$ + +Proof: + +Note the complex case is **Theorem 7.13**, so assume $V$ is a real vector space. Let $u,w\in V$ consider + +$\Rightarrow$ + +$$ +\langle Tu,w\rangle=\frac{\langle T(u+w),u+w\rangle -\langle T(u-w),u-w\rangle}{4}=0 +$$ + +We set $\langle Tw,u\rangle=\langle w,Tu\rangle =\langle Tu,w\rangle$ + +#### Normal Operators + +#### Definition 7.18 + +An operator $T\in \mathscr{L}(V)$ on an inner product space is **normal** if $TT^*=T^*T$ ie. $T$ commutes with its adjoint + +#### Theorem (7.20) + +An operator $T$ is normal if and only if + +$$ +||Tv||=||T^*v||,\forall v\in V +$$ + +Proof: + +The key idea is that $T^*T-TT^*$ is self adjoint. + +$$ +(T^*T-TT^*)^*=(T^*T)^*-(TT^*)^*=T^*T-TT^* +$$ + diff --git a/pages/Math429/Math429_L32.md b/content/Math429/Math429_L32.md similarity index 96% rename from pages/Math429/Math429_L32.md rename to content/Math429/Math429_L32.md index bba6f66..3c4953c 100644 --- a/pages/Math429/Math429_L32.md +++ b/content/Math429/Math429_L32.md @@ -1,108 +1,108 @@ -# Lecture 32 - -## Chapter VII Operators on Inner Product Spaces - -**Assumption: $V,W$ are finite dimensional inner product spaces.** - -### Spectral Theorem 7B - -Recall - -#### Definition 7.10 - -An operator $T\in \mathscr{L}(V)$ is self adjoint if $T=T^*$ - -#### Definition 7.18 - -AN operator $T\in\mathscr{L}(V)$ is normal if $TT^*=T^*T$ - -#### Theorem 7.20 - -Suppose $T\in \mathscr{L}(V)$, $T$ is normal $\iff ||Tv||=||T^*v||$ - -#### Lemma 7,26 - -Suppose $T\in\mathscr{L}(V)$ is self adjoint operator and $b,c\in \mathbb{R}$ such that $b^2<4c$, then - -$$ -T^2+bT+cI -$$ - -is invertible. - -Proof: - -Prove $T^2+bT+cI$ is injective by showing $\langle(T^2+bT+cI),v\rangle\neq 0$ (for $v\neq 0$) - -$$ -\begin{aligned} -\langle(T^2+bT+cI),v\rangle&=\langle T^2v,v\rangle+\langle bTv,v\rangle+c\langle v,v\rangle\\ -&=\langle Tv,Tv\rangle+b\langle Tv,v\rangle +c||v||^2\\ -&\geq ||Tv||^2-|b|\ ||Tv||\ ||v||+c||v||^2 \textup{ by cauchy schuarz}\\ -&=\left(||Tv||-\frac{b||v||}{2}\right)^2+\left(c-\frac{b^2}{4}\right)||v||^2>0 -\end{aligned} -$$ - -#### Theorem 7.27 - -Suppose $T\in \mathscr{L}(V)$ is self adjoint. Then the minimal polynomial is of the form $(z-\lambda_1)...(z-\lambda_m)$ for some $\lambda_1,...,\lambda_m\in\mathbb{R}$ - -Proof: - -$\mathbb{F}=\mathbb{C}$ clear from previous results - -$\mathbb{F}=\mathbb{R}$ assume for contradiction $q(z)$, where $b^2\leq 4c$. Then $P(T)=0$ but $q(T)\neq 0$. So let $v\in V$ such that $q(T)v\neq 0$. - -then $(T^2+bT+cI)(q(T)v)=0$ but $T^2+bT+cI$ is invertible so $q(T)v=0$ this is a contradiction so $p(z)=(z-\lambda_1)...(z-\lambda_m)$ - -#### Theorem 7.29 Real Spectral theorem - -Suppose $V$ is a finite dimensional real inner product space and $T\in \mathscr{L}(V)$ then the following are equivalent. - -(a) $T$ is self adjoint. -(b) $T$ has a diagonal matrix with respect to same orthonormal basis. -(c) $V$ has an orthonormal basis of eigenvectors of $T$ - -Proof: - -$b\iff c$ clear by definition - -$b\implies a$ because the transpose of a diagonal matrix is itself. - -$a\implies b$ by (**Theorem 7.27**) there exists an orthonormal basis such that $M(T)$ is upper triangular. But $M(T^*)=M(T)$ and $M(T^*)=(M(T))^*$ - -but this $M(T)$ is both upper and lower triangular, so $M(T)$ is diagonal. - -#### Theorem 7.31 Complete Spectral Theorem - -Suppose $V$ is a complex finite dimensional inner product space. $T\in \mathscr{L}(V)$, then the following are equivalent. - -(a) $T$ is normal -(b) $T$ has a diagonal matrix with respect to an orthonormal basis -(c) $V$ has an orthonormal basis of eigenvectors of $T$. - -$a\implies b$ - -$$ -M(T)=\begin{pmatrix} - a_{1,1}&\dots&a_{1,n}\\ - &\ddots &\vdots\\ - 0& & a_{n,n} -\end{pmatrix} -$$ - -with respect to an appropriate basis $e_1,...,e_n$ - -Then $||Te_1||^2=|a_{1,1}|^2$, $||Te_1||^2=||T^*e_1||^2=|a_{1,1}|^2+|a_{1,2}|^2+...+|a_{1,n}|^2$. So $a_{1,2}=...=a_{1,n}=0$, without loss of generality, $||Te_2||^2=0$. Repeating this procedure we have $M(T)$ is diagonal. - -Example: - -$T\in \mathscr{L}(\mathbb{C}^2)$ $M(T)=\begin{pmatrix} - 2&-3\\ - 3&2 -\end{pmatrix}$ - -$M(T,(f_1,f_2))=\begin{pmatrix} - 2+3c&0\\ - 0&2-3c -\end{pmatrix}$ +# Lecture 32 + +## Chapter VII Operators on Inner Product Spaces + +**Assumption: $V,W$ are finite dimensional inner product spaces.** + +### Spectral Theorem 7B + +Recall + +#### Definition 7.10 + +An operator $T\in \mathscr{L}(V)$ is self adjoint if $T=T^*$ + +#### Definition 7.18 + +AN operator $T\in\mathscr{L}(V)$ is normal if $TT^*=T^*T$ + +#### Theorem 7.20 + +Suppose $T\in \mathscr{L}(V)$, $T$ is normal $\iff ||Tv||=||T^*v||$ + +#### Lemma 7,26 + +Suppose $T\in\mathscr{L}(V)$ is self adjoint operator and $b,c\in \mathbb{R}$ such that $b^2<4c$, then + +$$ +T^2+bT+cI +$$ + +is invertible. + +Proof: + +Prove $T^2+bT+cI$ is injective by showing $\langle(T^2+bT+cI),v\rangle\neq 0$ (for $v\neq 0$) + +$$ +\begin{aligned} +\langle(T^2+bT+cI),v\rangle&=\langle T^2v,v\rangle+\langle bTv,v\rangle+c\langle v,v\rangle\\ +&=\langle Tv,Tv\rangle+b\langle Tv,v\rangle +c||v||^2\\ +&\geq ||Tv||^2-|b|\ ||Tv||\ ||v||+c||v||^2 \textup{ by cauchy schuarz}\\ +&=\left(||Tv||-\frac{b||v||}{2}\right)^2+\left(c-\frac{b^2}{4}\right)||v||^2>0 +\end{aligned} +$$ + +#### Theorem 7.27 + +Suppose $T\in \mathscr{L}(V)$ is self adjoint. Then the minimal polynomial is of the form $(z-\lambda_1)...(z-\lambda_m)$ for some $\lambda_1,...,\lambda_m\in\mathbb{R}$ + +Proof: + +$\mathbb{F}=\mathbb{C}$ clear from previous results + +$\mathbb{F}=\mathbb{R}$ assume for contradiction $q(z)$, where $b^2\leq 4c$. Then $P(T)=0$ but $q(T)\neq 0$. So let $v\in V$ such that $q(T)v\neq 0$. + +then $(T^2+bT+cI)(q(T)v)=0$ but $T^2+bT+cI$ is invertible so $q(T)v=0$ this is a contradiction so $p(z)=(z-\lambda_1)...(z-\lambda_m)$ + +#### Theorem 7.29 Real Spectral theorem + +Suppose $V$ is a finite dimensional real inner product space and $T\in \mathscr{L}(V)$ then the following are equivalent. + +(a) $T$ is self adjoint. +(b) $T$ has a diagonal matrix with respect to same orthonormal basis. +(c) $V$ has an orthonormal basis of eigenvectors of $T$ + +Proof: + +$b\iff c$ clear by definition + +$b\implies a$ because the transpose of a diagonal matrix is itself. + +$a\implies b$ by (**Theorem 7.27**) there exists an orthonormal basis such that $M(T)$ is upper triangular. But $M(T^*)=M(T)$ and $M(T^*)=(M(T))^*$ + +but this $M(T)$ is both upper and lower triangular, so $M(T)$ is diagonal. + +#### Theorem 7.31 Complete Spectral Theorem + +Suppose $V$ is a complex finite dimensional inner product space. $T\in \mathscr{L}(V)$, then the following are equivalent. + +(a) $T$ is normal +(b) $T$ has a diagonal matrix with respect to an orthonormal basis +(c) $V$ has an orthonormal basis of eigenvectors of $T$. + +$a\implies b$ + +$$ +M(T)=\begin{pmatrix} + a_{1,1}&\dots&a_{1,n}\\ + &\ddots &\vdots\\ + 0& & a_{n,n} +\end{pmatrix} +$$ + +with respect to an appropriate basis $e_1,...,e_n$ + +Then $||Te_1||^2=|a_{1,1}|^2$, $||Te_1||^2=||T^*e_1||^2=|a_{1,1}|^2+|a_{1,2}|^2+...+|a_{1,n}|^2$. So $a_{1,2}=...=a_{1,n}=0$, without loss of generality, $||Te_2||^2=0$. Repeating this procedure we have $M(T)$ is diagonal. + +Example: + +$T\in \mathscr{L}(\mathbb{C}^2)$ $M(T)=\begin{pmatrix} + 2&-3\\ + 3&2 +\end{pmatrix}$ + +$M(T,(f_1,f_2))=\begin{pmatrix} + 2+3c&0\\ + 0&2-3c +\end{pmatrix}$ diff --git a/pages/Math429/Math429_L33.md b/content/Math429/Math429_L33.md similarity index 100% rename from pages/Math429/Math429_L33.md rename to content/Math429/Math429_L33.md diff --git a/pages/Math429/Math429_L34.md b/content/Math429/Math429_L34.md similarity index 100% rename from pages/Math429/Math429_L34.md rename to content/Math429/Math429_L34.md diff --git a/pages/Math429/Math429_L35.md b/content/Math429/Math429_L35.md similarity index 97% rename from pages/Math429/Math429_L35.md rename to content/Math429/Math429_L35.md index 0523ea7..9c5fc51 100644 --- a/pages/Math429/Math429_L35.md +++ b/content/Math429/Math429_L35.md @@ -1,114 +1,114 @@ -# Lecture 35 - -## Chapter VIII Operators on complex vector spaces - -### Generalized Eigenvectors and Nilpotent Operators 8A - -Recall: Definition 8.8 - -Suppose $T\in \mathscr{L}(V)$ and $\lambda$ is an eigenvalue of $T$. A vector $v\in V$ is called a **generalized eigenvector** of $T$ corresponding to $\lambda$ if $v\neq 0$ and - -$$ -(T-\lambda I)^k v=0 -$$ - -for some positive integer $k$. - -Example: - -For $T\in\mathscr{L}(\mathbb{F})$ - -The matrix for $T$ is $\begin{pmatrix} 0&1\\0&0 \end{pmatrix}$ - -When $\lambda=0$, $\begin{pmatrix} 1 & 0 \end{pmatrix}$ is an eigenvector $\begin{pmatrix} 0&1 \end{pmatrix}$ is not and eigenvector but it is a generalized eigenvector. - -In fact $\begin{pmatrix} 0&1\\0&0 \end{pmatrix}^2=\begin{pmatrix} 0&0\\0&0 \end{pmatrix}$, so any nonzero vector is a generalized eigenvector. is a generalized eigenvector of $T$ corresponding to eigenvalue $0$. - -Fact: $v\in V$ is a generalized eigenvector of $T$ corresponding to $\lambda\iff (T-\lambda I)^{dim\ V}v=0$ - -#### Theorem 8.9 - -Suppose $\mathbb{F}=\mathbb{C}$ and $T\in \mathscr{L}(V)$ Then $\exists$ basis of $V$ consisting of generalized eigenvector of $T$. - -Proof: Let $n=dim\ V$ we will induct on $n$. - -Base case $n=1$, Every nonzero vector in $V$ is an eigenvector of $T$. - -Inductive step: Let $n=dim\ V$, assume the theorem is tru for all vector spaces with $dim - -$$ -\begin{aligned} - 0&=(T-\lambda I)^n v\\ - &=(B+A)^n v\\ - &=\sum^n_{k=0} \begin{pmatrix} - n\\k - \end{pmatrix} A^{n-k}B^kv -\end{aligned} -$$ - -Then we apply $(T-\alpha I)^{m-1}$, which is $B^{m-1}$ to both sides - -$$ -\begin{aligned} - 0&=A^nB^{m-1}v -\end{aligned} -$$ - +# Lecture 35 + +## Chapter VIII Operators on complex vector spaces + +### Generalized Eigenvectors and Nilpotent Operators 8A + +Recall: Definition 8.8 + +Suppose $T\in \mathscr{L}(V)$ and $\lambda$ is an eigenvalue of $T$. A vector $v\in V$ is called a **generalized eigenvector** of $T$ corresponding to $\lambda$ if $v\neq 0$ and + +$$ +(T-\lambda I)^k v=0 +$$ + +for some positive integer $k$. + +Example: + +For $T\in\mathscr{L}(\mathbb{F})$ + +The matrix for $T$ is $\begin{pmatrix} 0&1\\0&0 \end{pmatrix}$ + +When $\lambda=0$, $\begin{pmatrix} 1 & 0 \end{pmatrix}$ is an eigenvector $\begin{pmatrix} 0&1 \end{pmatrix}$ is not and eigenvector but it is a generalized eigenvector. + +In fact $\begin{pmatrix} 0&1\\0&0 \end{pmatrix}^2=\begin{pmatrix} 0&0\\0&0 \end{pmatrix}$, so any nonzero vector is a generalized eigenvector. is a generalized eigenvector of $T$ corresponding to eigenvalue $0$. + +Fact: $v\in V$ is a generalized eigenvector of $T$ corresponding to $\lambda\iff (T-\lambda I)^{dim\ V}v=0$ + +#### Theorem 8.9 + +Suppose $\mathbb{F}=\mathbb{C}$ and $T\in \mathscr{L}(V)$ Then $\exists$ basis of $V$ consisting of generalized eigenvector of $T$. + +Proof: Let $n=dim\ V$ we will induct on $n$. + +Base case $n=1$, Every nonzero vector in $V$ is an eigenvector of $T$. + +Inductive step: Let $n=dim\ V$, assume the theorem is tru for all vector spaces with $dim + +$$ +\begin{aligned} + 0&=(T-\lambda I)^n v\\ + &=(B+A)^n v\\ + &=\sum^n_{k=0} \begin{pmatrix} + n\\k + \end{pmatrix} A^{n-k}B^kv +\end{aligned} +$$ + +Then we apply $(T-\alpha I)^{m-1}$, which is $B^{m-1}$ to both sides + +$$ +\begin{aligned} + 0&=A^nB^{m-1}v +\end{aligned} +$$ + Since $(T-\alpha I)^{m-1}\neq 0$, $A=0$, then $\alpha I-\lambda I=0$, $\alpha=\lambda$ \ No newline at end of file diff --git a/pages/Math429/Math429_L36.md b/content/Math429/Math429_L36.md similarity index 100% rename from pages/Math429/Math429_L36.md rename to content/Math429/Math429_L36.md diff --git a/pages/Math429/Math429_L37.md b/content/Math429/Math429_L37.md similarity index 96% rename from pages/Math429/Math429_L37.md rename to content/Math429/Math429_L37.md index 681bce7..9fee1aa 100644 --- a/pages/Math429/Math429_L37.md +++ b/content/Math429/Math429_L37.md @@ -1,126 +1,126 @@ -# Lecture 37 - -## Chapter VIII Operators on complex vector spaces - -### Generalized Eigenspace Decomposition 8B - ---- -Review - - -#### Definition 8.19 - -The generalized eigenspace of $T$ for $\lambda \in \mathbb{F}$ is $G(\lambda,T)=\{v\in V\vert (T-\lambda I)^k v=0\textup{ for some k>0}\}$ - -#### Theorem 8.20 - -$G(\lambda, T)=null((T-\lambda I)^{dim\ V})$ - ---- -New materials - -#### Theorem 8.31 - -Suppose $v_1,...,v_n$ is a basis where $M(T,(v_1,...,v_k))$ is upper triangular. Then the number of times $\lambda$ appears on the diagonal is the multiplicity of $\lambda$ as an eigenvalue of $T$. - -Proof: - -Let $\lambda_1,...,\lambda_n$ be the diagonal entries, $S$ be such that $M(S,(v_1,...,v_n))$ is upper triangular. Note that if $\mu_1,...,\mu_n$ are the diagonal entires of $M(S)$, then the diagonal entires of $M(S^n)$ are $\mu_1^n,...,\mu_n^n$ - -$$ -\begin{aligned} -dim(null\ S^n)&=n-dim\ range\ (S^n)\leq n-\textup{ number of non-zero diagonal entries on } S^n\\ -&=\textup{ number of zero diagonal entries of }S^n -\end{aligned} -$$ - -plus in $S=T-\lambda I$, then - -$$ -\begin{aligned} -dim G(\lambda, T)&=dim(null\ (T-\lambda I)^n)\\ -&\leq \textup{number times where }\lambda \textup{ appears on the diagonal of }M(T)\\ -\end{aligned} -$$ - -Note: - -$V=G(\lambda_1, T)\oplus \dots \oplus G(\lambda_k, T)$ - -for distinct $\lambda_1,...,\lambda_k$ thus $n=dim\ G(\lambda_1,T)+\dots +dim\ (\lambda_k, T)$ - -on the other hand $n=\textup{ number of times }\lambda_1 \textup{ appears as a diagonal entry}+\dots +\textup{ number of times }\lambda_k \textup{ appears as a diagonal entry}+\dots $ - -So $dim\ G(\lambda_i, T)=$ number of times where $\lambda_i$ appears oas a diagonal entry. - -#### Definition 8.35 - -A **block diagonal matrix** is a matrix of the form $\begin{pmatrix} - A_1& & 0\\ - & \ddots &\\ - 0& & A_m -\end{pmatrix}$ where $A_k$ is a **square matrix**. - -Example: - -$ -\begin{pmatrix} - 1&0&0 & 0&0\\ - 0 & 2 &1&0&0\\ - 0 & 0 &2&0&0\\ - 0& 0&0& 4&1\\ - 0& 0&0& 0&4\\ -\end{pmatrix}$ - -#### Theorem - -Let $V$ be a complex vector space and let $\lambda_1,...,\lambda_m$ be the distinct eigenvalue of $T$ with multiplicity $d_1,...,d_m$, then there exists a basis where $\begin{pmatrix} - A_1& & 0\\ - & \ddots &\\ - 0& & A_m -\end{pmatrix}$ where $A_k$ is a $d_k\times d_k$ matrix upper triangular with only $\lambda_k$ on the diagonal. - -Proof: - -Note that $(T-\lambda_k I)\vert_{G(\lambda_k,T)}$ is nilpotent. So there is a basis of $G(\lambda_k,T)$ where $(T-\lambda_k I)\vert_{G(\lambda_k,T)}$ is upper triangular with zeros on the diagonal. Then $(T-\lambda_k I)\vert_{G(\lambda_k,T)}$ is upper triangular with $\lambda_k$ on the diagonal. - -### Jordan Normal Form 8C - -Nilpotent operators - -Example: $T(x,y,z)=(0,x,y), M(T)=\begin{pmatrix} - 0&1&0\\ - 0&0&1\\ - 0&0&0 -\end{pmatrix}$ - -#### Definition 8.44 - -Let $T\in \mathscr{L}(V)$ a basis of $V$ is a **Jordan basis** of $T$ if in that basis $\begin{pmatrix} - A_1& & 0\\ - & \ddots &\\ - 0& & A_p -\end{pmatrix}$ where each $A_k=\begin{pmatrix} - \lambda_1& 1& & 0\\ - & \ddots& \ddots &\\ - &&\ddots& 1\\ - 0&&&\lambda_k\\ -\end{pmatrix}$ - -#### Theorem 8.45 - -Suppose $T\in \mathscr{L}(V)$ is nilpotent, then there exists a basis of $V$ that is a Jordan basis of $T$. - -Sketch of Proof: - -Induct on $dim\ V$, if $dim\ V=1$, clear. - -if $dim\ V>1$, then let $m$ be such that $T^m=0$ and $T^{m-1}\neq 0$. Then $\exists u\in V$ such that $T^{m-1}u\neq 0$, then $Span (u,Tu, ...,T^{m-1}u)$ is $m$ dimensional. - -#### Theorem 8.46 - -Suppose $V$ is a complex vector space $T\in \mathscr{L}(V)$ then $T$ has a Jordan basis. - -Proof: - +# Lecture 37 + +## Chapter VIII Operators on complex vector spaces + +### Generalized Eigenspace Decomposition 8B + +--- +Review + + +#### Definition 8.19 + +The generalized eigenspace of $T$ for $\lambda \in \mathbb{F}$ is $G(\lambda,T)=\{v\in V\vert (T-\lambda I)^k v=0\textup{ for some k>0}\}$ + +#### Theorem 8.20 + +$G(\lambda, T)=null((T-\lambda I)^{dim\ V})$ + +--- +New materials + +#### Theorem 8.31 + +Suppose $v_1,...,v_n$ is a basis where $M(T,(v_1,...,v_k))$ is upper triangular. Then the number of times $\lambda$ appears on the diagonal is the multiplicity of $\lambda$ as an eigenvalue of $T$. + +Proof: + +Let $\lambda_1,...,\lambda_n$ be the diagonal entries, $S$ be such that $M(S,(v_1,...,v_n))$ is upper triangular. Note that if $\mu_1,...,\mu_n$ are the diagonal entires of $M(S)$, then the diagonal entires of $M(S^n)$ are $\mu_1^n,...,\mu_n^n$ + +$$ +\begin{aligned} +dim(null\ S^n)&=n-dim\ range\ (S^n)\leq n-\textup{ number of non-zero diagonal entries on } S^n\\ +&=\textup{ number of zero diagonal entries of }S^n +\end{aligned} +$$ + +plus in $S=T-\lambda I$, then + +$$ +\begin{aligned} +dim G(\lambda, T)&=dim(null\ (T-\lambda I)^n)\\ +&\leq \textup{number times where }\lambda \textup{ appears on the diagonal of }M(T)\\ +\end{aligned} +$$ + +Note: + +$V=G(\lambda_1, T)\oplus \dots \oplus G(\lambda_k, T)$ + +for distinct $\lambda_1,...,\lambda_k$ thus $n=dim\ G(\lambda_1,T)+\dots +dim\ (\lambda_k, T)$ + +on the other hand $n=\textup{ number of times }\lambda_1 \textup{ appears as a diagonal entry}+\dots +\textup{ number of times }\lambda_k \textup{ appears as a diagonal entry}+\dots $ + +So $dim\ G(\lambda_i, T)=$ number of times where $\lambda_i$ appears oas a diagonal entry. + +#### Definition 8.35 + +A **block diagonal matrix** is a matrix of the form $\begin{pmatrix} + A_1& & 0\\ + & \ddots &\\ + 0& & A_m +\end{pmatrix}$ where $A_k$ is a **square matrix**. + +Example: + +$ +\begin{pmatrix} + 1&0&0 & 0&0\\ + 0 & 2 &1&0&0\\ + 0 & 0 &2&0&0\\ + 0& 0&0& 4&1\\ + 0& 0&0& 0&4\\ +\end{pmatrix}$ + +#### Theorem + +Let $V$ be a complex vector space and let $\lambda_1,...,\lambda_m$ be the distinct eigenvalue of $T$ with multiplicity $d_1,...,d_m$, then there exists a basis where $\begin{pmatrix} + A_1& & 0\\ + & \ddots &\\ + 0& & A_m +\end{pmatrix}$ where $A_k$ is a $d_k\times d_k$ matrix upper triangular with only $\lambda_k$ on the diagonal. + +Proof: + +Note that $(T-\lambda_k I)\vert_{G(\lambda_k,T)}$ is nilpotent. So there is a basis of $G(\lambda_k,T)$ where $(T-\lambda_k I)\vert_{G(\lambda_k,T)}$ is upper triangular with zeros on the diagonal. Then $(T-\lambda_k I)\vert_{G(\lambda_k,T)}$ is upper triangular with $\lambda_k$ on the diagonal. + +### Jordan Normal Form 8C + +Nilpotent operators + +Example: $T(x,y,z)=(0,x,y), M(T)=\begin{pmatrix} + 0&1&0\\ + 0&0&1\\ + 0&0&0 +\end{pmatrix}$ + +#### Definition 8.44 + +Let $T\in \mathscr{L}(V)$ a basis of $V$ is a **Jordan basis** of $T$ if in that basis $\begin{pmatrix} + A_1& & 0\\ + & \ddots &\\ + 0& & A_p +\end{pmatrix}$ where each $A_k=\begin{pmatrix} + \lambda_1& 1& & 0\\ + & \ddots& \ddots &\\ + &&\ddots& 1\\ + 0&&&\lambda_k\\ +\end{pmatrix}$ + +#### Theorem 8.45 + +Suppose $T\in \mathscr{L}(V)$ is nilpotent, then there exists a basis of $V$ that is a Jordan basis of $T$. + +Sketch of Proof: + +Induct on $dim\ V$, if $dim\ V=1$, clear. + +if $dim\ V>1$, then let $m$ be such that $T^m=0$ and $T^{m-1}\neq 0$. Then $\exists u\in V$ such that $T^{m-1}u\neq 0$, then $Span (u,Tu, ...,T^{m-1}u)$ is $m$ dimensional. + +#### Theorem 8.46 + +Suppose $V$ is a complex vector space $T\in \mathscr{L}(V)$ then $T$ has a Jordan basis. + +Proof: + take $V=G(\lambda_1, T)\oplus \dots \oplus G(\lambda_m, T)$, then look at $(T-\lambda_k I)\vert_{G(\lambda_k,T)}$ \ No newline at end of file diff --git a/pages/Math429/Math429_L38.md b/content/Math429/Math429_L38.md similarity index 96% rename from pages/Math429/Math429_L38.md rename to content/Math429/Math429_L38.md index 5273cef..8129515 100644 --- a/pages/Math429/Math429_L38.md +++ b/content/Math429/Math429_L38.md @@ -1,119 +1,119 @@ -# Lecture 38 - -## Chapter VIII Operators on complex vector spaces - -### Trace 8D - -#### Definition 8.47 - -For a square matrix $A$, the **trace of** $A$ is the sum of the diagonal entries denoted $tr(A)$. - -#### Theorem 8.49 - -Suppose $A$ is $m\times n$, $B$ is $n\times m$ matrices, then $tr(AB)=tr(BA)$. - -Proof: - -By pure computation. - -#### Theorem 8.50 - -Suppose $T\in \mathscr{L}(V)$ and $u_1,...,u_n$ and $v_1,...,v_n$ are bases of $V$. - -$$ -tr(M(T,(u_1,...,u_n)))=tr(M(T,(v_1,...,v_n))) -$$ - -Proof: - -Let $A=tr(M(T,(u_1,...,u_n)))$ and $B=tr(M(T,(v_1,...,v_n)))$, then there exists $C$, invertible such that $A=CBC^{-1}$, - -$$ -tr(A)=tr((CB)C^{-1})=tr(C^{-1}(CB))=tr(B) -$$ - -#### Definition 8.51 - -Given $T\in \mathscr{L}(V)$ the trace of $T$ denoted $tr(T)$ is given by $tr(T)=tr(M(T))$. - -Note: For an upper triangular matrix, the diagonal entries are the eigenvalues with multiplicity - -#### Theorem 8.52 - -Suppose $V$ is a complex vector space such that $T\in \mathscr{L}(V)$, then $tr(T)$ is the sum of the eigenvalues counted with multiplicity. - -Proof: - -Over $\mathbb{C}$, there is a basis where $M(T)$ is upper triangular. - -#### Theorem 8.54 - -Suppose $V$ is a complex vector space, $n=dim\ V$.$T\in \mathscr{L}(V)$. Then the coefficient on $z^{n-1}$ in the characteristic polynomial is $tr(T)$. - -Proof: - -$(z-\lambda_1)\dots(z-\lambda_n)=z^{n}-(\lambda_1+...+\lambda_n)z^{n-1}+\dots$ - -#### Theorem 8.56 - -Trance is linear - -Proof: - -- Additivity - $tr(T+S)=tr(M(T)+M(S))=tr(T)+tr(S)$ -- Homogeneity - $tr(cT)=ctr(M(T))=ctr(T)$ - -#### Theorem/Example 8.10 - -Trace is the unique linear functional $\mathscr{L}\to \mathbb{F}$ such that $tr(ST)=tr(TS)$ and $tr(I)=dim\ V$ - -Proof: - -Let $\varphi:\mathscr{L}(V)\to \mathbb{F}$ be a linear functional such that $\varphi(ST)=\varphi(TS)$ and $\varphi(I)=n$ where $n=dim\ V$. Let $v_1,...,v_n$ be a basis for $V$ define $P_{j,k}$ to be the operator $M(P_{j,k})=\begin{pmatrix} - 0&0&0\\ - 0&1&0\\ - 0&0&0 -\end{pmatrix}$. Note $P_{j,k}$ form a basis of $L(V)$, now we must show $\varphi(P_{j,k})=tr(P_{j,k})=\begin{cases}1\textup{ if }j=k\\0\textup{ if }j \neq k\end{cases}$ - -- For $j\neq k$ - $\varphi(P_{j,j}P_{j,k})=\varphi(P_{j,k})=0$ - - $\varphi(P_{j,k}P_{j,j})=\varphi(P_{j,k})=0$ -- For $j=k$ - $\varphi(P_{k,j},P_{j,k})=\varphi(P_{k,k})=1$ - - $\varphi(P_{j,k},P_{k,j})=\varphi(P_{j,j})=1$ - -So $\varphi(I)=\varphi(P_{1,1}+...+P_{n,n})=\varphi(P_{1,1})+...+\varphi(P_{n,n})=n$ - -#### Theorem 8.57 - -Suppose $V$ is finite dimensional vector space, then there does not exists $S,T\in \mathscr{L}(V)$ such that $ST-TS=I$. ($ST-TS$ is called communicator) - -Proof: - -$tr(ST-TS)=tr(ST)-tr(TS)=tr(ST)-tr(ST)=0$, since $tr(I)=dim\ V$, so $ST-TS\neq I$ - -Note: **requires finite dimensional.** - -## Chapter ? Multilinear Algebra and Determinants - -### Determinants ?A - -#### Definition ?.1 - -The determinant of $T\in \mathscr{L}(V)$ is the product of eigenvalues counted with multiplicity. - -#### Definition ?.2 - -The determinant of a matrix is given by - -$$ -det(A)=\sum_{\sigma\in perm(n)}A_{\sigma(1),1}\cdot ...\cdot A_{\sigma(n),n}\cdot sign(\sigma) -$$ - -$perm(\sigma)=$ all recordings of $1,...,n$, number of swaps needed to write $\sigma$ - -$$ +# Lecture 38 + +## Chapter VIII Operators on complex vector spaces + +### Trace 8D + +#### Definition 8.47 + +For a square matrix $A$, the **trace of** $A$ is the sum of the diagonal entries denoted $tr(A)$. + +#### Theorem 8.49 + +Suppose $A$ is $m\times n$, $B$ is $n\times m$ matrices, then $tr(AB)=tr(BA)$. + +Proof: + +By pure computation. + +#### Theorem 8.50 + +Suppose $T\in \mathscr{L}(V)$ and $u_1,...,u_n$ and $v_1,...,v_n$ are bases of $V$. + +$$ +tr(M(T,(u_1,...,u_n)))=tr(M(T,(v_1,...,v_n))) +$$ + +Proof: + +Let $A=tr(M(T,(u_1,...,u_n)))$ and $B=tr(M(T,(v_1,...,v_n)))$, then there exists $C$, invertible such that $A=CBC^{-1}$, + +$$ +tr(A)=tr((CB)C^{-1})=tr(C^{-1}(CB))=tr(B) +$$ + +#### Definition 8.51 + +Given $T\in \mathscr{L}(V)$ the trace of $T$ denoted $tr(T)$ is given by $tr(T)=tr(M(T))$. + +Note: For an upper triangular matrix, the diagonal entries are the eigenvalues with multiplicity + +#### Theorem 8.52 + +Suppose $V$ is a complex vector space such that $T\in \mathscr{L}(V)$, then $tr(T)$ is the sum of the eigenvalues counted with multiplicity. + +Proof: + +Over $\mathbb{C}$, there is a basis where $M(T)$ is upper triangular. + +#### Theorem 8.54 + +Suppose $V$ is a complex vector space, $n=dim\ V$.$T\in \mathscr{L}(V)$. Then the coefficient on $z^{n-1}$ in the characteristic polynomial is $tr(T)$. + +Proof: + +$(z-\lambda_1)\dots(z-\lambda_n)=z^{n}-(\lambda_1+...+\lambda_n)z^{n-1}+\dots$ + +#### Theorem 8.56 + +Trance is linear + +Proof: + +- Additivity + $tr(T+S)=tr(M(T)+M(S))=tr(T)+tr(S)$ +- Homogeneity + $tr(cT)=ctr(M(T))=ctr(T)$ + +#### Theorem/Example 8.10 + +Trace is the unique linear functional $\mathscr{L}\to \mathbb{F}$ such that $tr(ST)=tr(TS)$ and $tr(I)=dim\ V$ + +Proof: + +Let $\varphi:\mathscr{L}(V)\to \mathbb{F}$ be a linear functional such that $\varphi(ST)=\varphi(TS)$ and $\varphi(I)=n$ where $n=dim\ V$. Let $v_1,...,v_n$ be a basis for $V$ define $P_{j,k}$ to be the operator $M(P_{j,k})=\begin{pmatrix} + 0&0&0\\ + 0&1&0\\ + 0&0&0 +\end{pmatrix}$. Note $P_{j,k}$ form a basis of $L(V)$, now we must show $\varphi(P_{j,k})=tr(P_{j,k})=\begin{cases}1\textup{ if }j=k\\0\textup{ if }j \neq k\end{cases}$ + +- For $j\neq k$ + $\varphi(P_{j,j}P_{j,k})=\varphi(P_{j,k})=0$ + + $\varphi(P_{j,k}P_{j,j})=\varphi(P_{j,k})=0$ +- For $j=k$ + $\varphi(P_{k,j},P_{j,k})=\varphi(P_{k,k})=1$ + + $\varphi(P_{j,k},P_{k,j})=\varphi(P_{j,j})=1$ + +So $\varphi(I)=\varphi(P_{1,1}+...+P_{n,n})=\varphi(P_{1,1})+...+\varphi(P_{n,n})=n$ + +#### Theorem 8.57 + +Suppose $V$ is finite dimensional vector space, then there does not exists $S,T\in \mathscr{L}(V)$ such that $ST-TS=I$. ($ST-TS$ is called communicator) + +Proof: + +$tr(ST-TS)=tr(ST)-tr(TS)=tr(ST)-tr(ST)=0$, since $tr(I)=dim\ V$, so $ST-TS\neq I$ + +Note: **requires finite dimensional.** + +## Chapter ? Multilinear Algebra and Determinants + +### Determinants ?A + +#### Definition ?.1 + +The determinant of $T\in \mathscr{L}(V)$ is the product of eigenvalues counted with multiplicity. + +#### Definition ?.2 + +The determinant of a matrix is given by + +$$ +det(A)=\sum_{\sigma\in perm(n)}A_{\sigma(1),1}\cdot ...\cdot A_{\sigma(n),n}\cdot sign(\sigma) +$$ + +$perm(\sigma)=$ all recordings of $1,...,n$, number of swaps needed to write $\sigma$ + +$$ diff --git a/pages/Math429/Math429_L39.md b/content/Math429/Math429_L39.md similarity index 96% rename from pages/Math429/Math429_L39.md rename to content/Math429/Math429_L39.md index bb173a6..43ec17e 100644 --- a/pages/Math429/Math429_L39.md +++ b/content/Math429/Math429_L39.md @@ -1,107 +1,107 @@ -# Lecture 39 - -## Chapter IX Multilinear Algebra and Determinants - -### Exterior Powers ?A - -#### Definitions ?.1 - -Let $V$ be a vector space, the **n-th** exterior power of $V$ denoted $\wedge^m V$ is a vector space formed by finite linear combination of expression of the form $v_1\wedge v_2\wedge\dots \wedge v_m$. subject to relations: - -1. $c(v_1\wedge v_2\wedge\dots \wedge v_m)=(cv_1)\wedge v_2\wedge\dots \wedge v_m$ -2. $(v_1+w_1)\wedge v_2\wedge\dots \wedge v_m=(v_1\wedge v_2\wedge\dots \wedge v_m)+(w_1\wedge v_2\wedge\dots \wedge v_m)$ -3. Swapping two entires in ($v_1\wedge v_2\wedge\dots \wedge v_m$) gives a negative sign. - -Example: - -$\wedge^2\mathbb{R}^3$ - -$$ -\begin{aligned} - &(1,0,0)\wedge(0,1,0)+(1,0,1)\wedge(1,1,1)\in \wedge^2\mathbb{R}^3\\ - &=(1,0,0)\wedge(0,1,0)+((1,0,0)+(0,0,1))\wedge(1,1,1)\\ - &=(1,0,0)\wedge(0,1,0)+(1,0,0)\wedge(1,1,1)+(0,0,1)\wedge(1,1,1)\\ - &=(1,0,0)\wedge(1,2,1)+(0,0,1)\wedge(1,1,1) -\end{aligned} -$$ - -#### Theorem ?.2 - -$0\wedge v_1\wedge\dots\wedge v_m=0$ - -Proof: - -$$ -\begin{aligned} -\vec{0}\wedge v_2\wedge\dots \wedge v_m &=(0\cdot \vec{0})\wedge v_2\wedge \dots\wedge v_m\\ -&=0(\vec{0}\wedge v_2\wedge \dots\wedge v_m)\\ -&=0 -\end{aligned} -$$ - -#### Theorem ?.3 - -$v_1\wedge v_1\wedge\dots\wedge v_m=0$ - -Proof: - -swap $v_1$ and $v_1$. - -$$ -\begin{aligned} -v_1\wedge v_1 \wedge v_2\wedge\dots \wedge v_m &=-(v_1\wedge v_1 \wedge v_2\wedge\dots \wedge v_m) \\ -v_1\wedge v_1 \wedge v_2\wedge\dots \wedge v_m&=0 -\end{aligned} -$$ - -#### Theorem ?.4 - -$v_1\wedge v_2\wedge\dots\wedge v_m\neq 0$ if and only if $v_1,\dots ,v_m$ are linearly independent. - -Proof: - -We first prove forward direction, - -Suppose $v_1,\dots, v_m$ are linearly dependent then let $a_1v_1+\dots +a_nv_m=0$ be a linear dependence. Without loss of generality. $a\neq 0$ then consider - -$$ -\begin{aligned} -0&=0\wedge v_2\wedge\dots\wedge v_m\\ -&=(a_1,v_1+...+a_m v_m)\wedge v_2\wedge \dots \wedge v_m\\ -&=a_1(v_1\wedge \dots v_m)+a_2(v_2\wedge v_2\wedge \dots \wedge v_m)+a_m(v_m\wedge v_2\wedge\dots\wedge v_m)\\ -&=a_1(v_1\wedge \dots v_m) -\end{aligned} -$$ - -reverse is the similar. - -#### Theorem ?.5 - -If $v_1,\dots v_n$ forms a basis for $V$, then expressions of the form $v_{i_1}\wedge\dots \wedge v_{i_m}$ for $1\leq i_1\leq i_m\leq n$ forms a basis of $\wedge^m V$ - -Proof: - -Spanning: Let $u_1\wedge\dots \wedge u_m\in \wedge^m V$ where $u_1=a_{1,1}v_1+\dots+a_{1,n}v_n,u_m=a_{m,1}v_1+\dots+a_{m,n}v_n$ - -Expand: then we set expressions of the form $\plusmn c(v_{i_1}\wedge \dots \wedge v_{i_m})$. Let $A=(a_{i,j})$ , $c$ is the $m\times m$ minor for the columns $i_1,..,i_m$. - -#### Corollary ?.6 - -Let $n=dim\ V$ then $dim\ \wedge^n v=1$ - -Note $dim\ \wedge^m V=\begin{pmatrix} - n\\m -\end{pmatrix}$ - -Proof: Chose a basis $v_1,...,v_n$ of $V$ then $v_1\wedge \dots \wedge v_n$ generates $\wedge^n v$. - -#### Definition ?.7 - -Let $T\in\mathscr{L}(V)$, $n=dim\ V$ define $det\ T$ to be the unique number such that for $v_1\wedge\dots\wedge v_n\in \wedge^n V$. $(Tv_1\wedge\dots\wedge Tv_n)=(det\ T)(v_1\wedge \dots \wedge v_n)$ - -#### Theorem ?.8 - -1. Swapping columns negates the determinants -2. $T$ is invertible if and only if $det\ T\neq 0$ -3. $det(ST)=det(S)det(T)$ +# Lecture 39 + +## Chapter IX Multilinear Algebra and Determinants + +### Exterior Powers ?A + +#### Definitions ?.1 + +Let $V$ be a vector space, the **n-th** exterior power of $V$ denoted $\wedge^m V$ is a vector space formed by finite linear combination of expression of the form $v_1\wedge v_2\wedge\dots \wedge v_m$. subject to relations: + +1. $c(v_1\wedge v_2\wedge\dots \wedge v_m)=(cv_1)\wedge v_2\wedge\dots \wedge v_m$ +2. $(v_1+w_1)\wedge v_2\wedge\dots \wedge v_m=(v_1\wedge v_2\wedge\dots \wedge v_m)+(w_1\wedge v_2\wedge\dots \wedge v_m)$ +3. Swapping two entires in ($v_1\wedge v_2\wedge\dots \wedge v_m$) gives a negative sign. + +Example: + +$\wedge^2\mathbb{R}^3$ + +$$ +\begin{aligned} + &(1,0,0)\wedge(0,1,0)+(1,0,1)\wedge(1,1,1)\in \wedge^2\mathbb{R}^3\\ + &=(1,0,0)\wedge(0,1,0)+((1,0,0)+(0,0,1))\wedge(1,1,1)\\ + &=(1,0,0)\wedge(0,1,0)+(1,0,0)\wedge(1,1,1)+(0,0,1)\wedge(1,1,1)\\ + &=(1,0,0)\wedge(1,2,1)+(0,0,1)\wedge(1,1,1) +\end{aligned} +$$ + +#### Theorem ?.2 + +$0\wedge v_1\wedge\dots\wedge v_m=0$ + +Proof: + +$$ +\begin{aligned} +\vec{0}\wedge v_2\wedge\dots \wedge v_m &=(0\cdot \vec{0})\wedge v_2\wedge \dots\wedge v_m\\ +&=0(\vec{0}\wedge v_2\wedge \dots\wedge v_m)\\ +&=0 +\end{aligned} +$$ + +#### Theorem ?.3 + +$v_1\wedge v_1\wedge\dots\wedge v_m=0$ + +Proof: + +swap $v_1$ and $v_1$. + +$$ +\begin{aligned} +v_1\wedge v_1 \wedge v_2\wedge\dots \wedge v_m &=-(v_1\wedge v_1 \wedge v_2\wedge\dots \wedge v_m) \\ +v_1\wedge v_1 \wedge v_2\wedge\dots \wedge v_m&=0 +\end{aligned} +$$ + +#### Theorem ?.4 + +$v_1\wedge v_2\wedge\dots\wedge v_m\neq 0$ if and only if $v_1,\dots ,v_m$ are linearly independent. + +Proof: + +We first prove forward direction, + +Suppose $v_1,\dots, v_m$ are linearly dependent then let $a_1v_1+\dots +a_nv_m=0$ be a linear dependence. Without loss of generality. $a\neq 0$ then consider + +$$ +\begin{aligned} +0&=0\wedge v_2\wedge\dots\wedge v_m\\ +&=(a_1,v_1+...+a_m v_m)\wedge v_2\wedge \dots \wedge v_m\\ +&=a_1(v_1\wedge \dots v_m)+a_2(v_2\wedge v_2\wedge \dots \wedge v_m)+a_m(v_m\wedge v_2\wedge\dots\wedge v_m)\\ +&=a_1(v_1\wedge \dots v_m) +\end{aligned} +$$ + +reverse is the similar. + +#### Theorem ?.5 + +If $v_1,\dots v_n$ forms a basis for $V$, then expressions of the form $v_{i_1}\wedge\dots \wedge v_{i_m}$ for $1\leq i_1\leq i_m\leq n$ forms a basis of $\wedge^m V$ + +Proof: + +Spanning: Let $u_1\wedge\dots \wedge u_m\in \wedge^m V$ where $u_1=a_{1,1}v_1+\dots+a_{1,n}v_n,u_m=a_{m,1}v_1+\dots+a_{m,n}v_n$ + +Expand: then we set expressions of the form $\plusmn c(v_{i_1}\wedge \dots \wedge v_{i_m})$. Let $A=(a_{i,j})$ , $c$ is the $m\times m$ minor for the columns $i_1,..,i_m$. + +#### Corollary ?.6 + +Let $n=dim\ V$ then $dim\ \wedge^n v=1$ + +Note $dim\ \wedge^m V=\begin{pmatrix} + n\\m +\end{pmatrix}$ + +Proof: Chose a basis $v_1,...,v_n$ of $V$ then $v_1\wedge \dots \wedge v_n$ generates $\wedge^n v$. + +#### Definition ?.7 + +Let $T\in\mathscr{L}(V)$, $n=dim\ V$ define $det\ T$ to be the unique number such that for $v_1\wedge\dots\wedge v_n\in \wedge^n V$. $(Tv_1\wedge\dots\wedge Tv_n)=(det\ T)(v_1\wedge \dots \wedge v_n)$ + +#### Theorem ?.8 + +1. Swapping columns negates the determinants +2. $T$ is invertible if and only if $det\ T\neq 0$ +3. $det(ST)=det(S)det(T)$ 4. $det(cT)=c^n det(T)$ \ No newline at end of file diff --git a/pages/Math429/Math429_L4.md b/content/Math429/Math429_L4.md similarity index 100% rename from pages/Math429/Math429_L4.md rename to content/Math429/Math429_L4.md diff --git a/pages/Math429/Math429_L5.md b/content/Math429/Math429_L5.md similarity index 100% rename from pages/Math429/Math429_L5.md rename to content/Math429/Math429_L5.md diff --git a/pages/Math429/Math429_L6.md b/content/Math429/Math429_L6.md similarity index 100% rename from pages/Math429/Math429_L6.md rename to content/Math429/Math429_L6.md diff --git a/pages/Math429/Math429_L7.md b/content/Math429/Math429_L7.md similarity index 100% rename from pages/Math429/Math429_L7.md rename to content/Math429/Math429_L7.md diff --git a/pages/Math429/Math429_L8.md b/content/Math429/Math429_L8.md similarity index 96% rename from pages/Math429/Math429_L8.md rename to content/Math429/Math429_L8.md index ea25a8b..9079afd 100644 --- a/pages/Math429/Math429_L8.md +++ b/content/Math429/Math429_L8.md @@ -1,92 +1,92 @@ -# Lecture 8 - -## Chapter III Linear maps - -**Assumption: $U,V,W$ are vector spaces (over $\mathbb{F}$)** - -### Vector Space of Linear Maps 3A - -#### Definition 3.1 - -A **linear map** from $V$ to $W$ is a function from $T:V\to W$ with the following properties: - -1. Additivity: $T(u+v)=T(u)+T(v),\forall u,v\in V$ -2. Homogeneity: $T(\lambda v)=\lambda T(v),\forall \lambda \in \mathbb{F},v\in V$ - -#### Notation - -* $Tv=T(v)$ -* $\mathscr{L}(V,W)$ denotes the set of linear maps from $V$ to $W$. (homomorphism, $Hom(V,W)$) -* $\mathscr{L}(V)$ denotes the set of linear maps from $V$ to $V$. (endomorphism, $End(V)$) - -#### Example - -* zero map $0(v)\in \mathscr{L}(V,W)$ $0(v)=0$ -* identity map $I\in \mathscr{L}(V,W)$, $I(v)=v$ -* scaling map $T\in \mathscr{L}(V,W)$, $T(v)=av,a\in \mathbb{F}$ -* differentiation map $D\in \mathscr{L}(\mathscr{P}_m(\mathbb{F}),\mathscr{P}_{m-1}(\mathbb{F}))$, $D(f)=f'$ - -#### Lemma 3.10 - -$T0=0$ for $T\in \mathscr{L}(V,W)$ - -Proof: - -$T(0+0)=T(0)+T(0)$ - - -#### Theorem 3.4 Linear map lemma - -Suppose $v_1,...,v_n$ is a basis for $V$, and suppose $w_1,...,w_n\in W$ are arbitrary vector. Then, there exists a unique linear map. $T:V\to W$ such that $T_{v_i}=w_i$ for $i=1,...,n$ - -Proof: - -First we show existence. - -by constrains, - -$T(c_1 v_1,...+c_n v_n)=c_1w_1+...+c_n w_n$ - -T is well defined because $v_1,....v_n$ are a basis. - -Need to show that $T$ is a linear map. - -* Additivity: let $u,v\in V$ and suppose $a_1,...,a_n,b_1,...,b_n\in \mathbb{F}$ -with $u=a_1v_1+....+a_n v_n ,v=b_1v_1+...+b_2v_n$, then $T(u+v)=T((a_1+b_1)v_1+...+(a_n+b_n)v_n)=Tu+Tv$ - -Proof for homogeneity used for exercise. - -Need to show $T$ is unique. Let $S\in\mathscr{L}(V,W)$ such that $Sv_i=w_i,i=1,...,n$ - -$$ -S(c_1 v_1+...+c_n v_n)=S(c_1v_1)+S(...)+S(c_n v_n)=c_1S(v_1)+...+c_nS(v_n) -+c_1w_1+...+c_nw_n -$$ - -Then $S=T$ - -#### Definition 3.5 - -Let $S,T\in \mathscr{L}(V,W)$, then define - -* $(S+T)\in\mathscr{L}(V,W)$ by $(S+T)(v)=Sv+Tv$ -* for $\lambda \in \mathbb{F}$, $(\lambda T)\in \mathscr{L}(V,W)$, $(\lambda T)(v)=\lambda T(v)$ - -Exercises: Show that $S+T$ and $\lambda T$ are linear maps. - -#### Theorem 3.6 - -$\mathscr{L}(V,W)$ is a vector space. - -Sketch of proof: - -* additive identity: $0(v)=0$ -* associativity: -* commutativity: -* additive inverse: $T\to (-1)T=-T$ -* scalar multiplication $1T=T$ -* distributive - -#### Definition 3.7 - -Multiplication for linear map: $(ST)v=S(T(v))=(S\circ T)(v)$ **Not commutative but associative**. +# Lecture 8 + +## Chapter III Linear maps + +**Assumption: $U,V,W$ are vector spaces (over $\mathbb{F}$)** + +### Vector Space of Linear Maps 3A + +#### Definition 3.1 + +A **linear map** from $V$ to $W$ is a function from $T:V\to W$ with the following properties: + +1. Additivity: $T(u+v)=T(u)+T(v),\forall u,v\in V$ +2. Homogeneity: $T(\lambda v)=\lambda T(v),\forall \lambda \in \mathbb{F},v\in V$ + +#### Notation + +* $Tv=T(v)$ +* $\mathscr{L}(V,W)$ denotes the set of linear maps from $V$ to $W$. (homomorphism, $Hom(V,W)$) +* $\mathscr{L}(V)$ denotes the set of linear maps from $V$ to $V$. (endomorphism, $End(V)$) + +#### Example + +* zero map $0(v)\in \mathscr{L}(V,W)$ $0(v)=0$ +* identity map $I\in \mathscr{L}(V,W)$, $I(v)=v$ +* scaling map $T\in \mathscr{L}(V,W)$, $T(v)=av,a\in \mathbb{F}$ +* differentiation map $D\in \mathscr{L}(\mathscr{P}_m(\mathbb{F}),\mathscr{P}_{m-1}(\mathbb{F}))$, $D(f)=f'$ + +#### Lemma 3.10 + +$T0=0$ for $T\in \mathscr{L}(V,W)$ + +Proof: + +$T(0+0)=T(0)+T(0)$ + + +#### Theorem 3.4 Linear map lemma + +Suppose $v_1,...,v_n$ is a basis for $V$, and suppose $w_1,...,w_n\in W$ are arbitrary vector. Then, there exists a unique linear map. $T:V\to W$ such that $T_{v_i}=w_i$ for $i=1,...,n$ + +Proof: + +First we show existence. + +by constrains, + +$T(c_1 v_1,...+c_n v_n)=c_1w_1+...+c_n w_n$ + +T is well defined because $v_1,....v_n$ are a basis. + +Need to show that $T$ is a linear map. + +* Additivity: let $u,v\in V$ and suppose $a_1,...,a_n,b_1,...,b_n\in \mathbb{F}$ +with $u=a_1v_1+....+a_n v_n ,v=b_1v_1+...+b_2v_n$, then $T(u+v)=T((a_1+b_1)v_1+...+(a_n+b_n)v_n)=Tu+Tv$ + +Proof for homogeneity used for exercise. + +Need to show $T$ is unique. Let $S\in\mathscr{L}(V,W)$ such that $Sv_i=w_i,i=1,...,n$ + +$$ +S(c_1 v_1+...+c_n v_n)=S(c_1v_1)+S(...)+S(c_n v_n)=c_1S(v_1)+...+c_nS(v_n) ++c_1w_1+...+c_nw_n +$$ + +Then $S=T$ + +#### Definition 3.5 + +Let $S,T\in \mathscr{L}(V,W)$, then define + +* $(S+T)\in\mathscr{L}(V,W)$ by $(S+T)(v)=Sv+Tv$ +* for $\lambda \in \mathbb{F}$, $(\lambda T)\in \mathscr{L}(V,W)$, $(\lambda T)(v)=\lambda T(v)$ + +Exercises: Show that $S+T$ and $\lambda T$ are linear maps. + +#### Theorem 3.6 + +$\mathscr{L}(V,W)$ is a vector space. + +Sketch of proof: + +* additive identity: $0(v)=0$ +* associativity: +* commutativity: +* additive inverse: $T\to (-1)T=-T$ +* scalar multiplication $1T=T$ +* distributive + +#### Definition 3.7 + +Multiplication for linear map: $(ST)v=S(T(v))=(S\circ T)(v)$ **Not commutative but associative**. diff --git a/pages/Math429/Math429_L9.md b/content/Math429/Math429_L9.md similarity index 96% rename from pages/Math429/Math429_L9.md rename to content/Math429/Math429_L9.md index e48cec4..694bfd6 100644 --- a/pages/Math429/Math429_L9.md +++ b/content/Math429/Math429_L9.md @@ -1,114 +1,114 @@ -# Lecture 9 - -## Chapter III Linear maps - -**Assumption: $U,V,W$ are vector spaces (over $\mathbb{F}$)** - -### Vector Space of Linear Maps 3A - -Review - -$\mathscr{L}(V,W) =$ space of linear maps form $V$ to $W$. - -$\mathscr{L}(V)=\mathscr{L}(V,V)$ - -Key facts: - -* $\mathscr{L}(V,W)$ is a vector space -* given $T\in\mathscr{L}(U,V),S\in \mathscr{L}(V,W)$, we have $TS\in \mathscr{L}(V,W)$ -* not commutative - -### Null spaces and Range 3B - -#### Definition 3.11 - -Null space and injectivity - -For $T\in \mathscr{L}(V,W)$, the **null space** of $T$, denoted as $null(T)$ (sometime also noted as $ker\ T$), is a subset of $V$ given by - -$$ -ker\ T=null(T)=\{v\in V \vert Tv=0\} -$$ - -Examples: - -* $0\in \mathscr{L}(V,W)$, then $null\ 0=V$, $null(I)=\{0\}$ -* $T\in \mathscr{L}(\mathbb{R}^3,\mathbb{R}^2),T(x,y,z)=(x+y,y+z)$, to find the null space, we set $T(x,y,z)=0$, then $x+y=0,y+z=0$, $x=-y,x=z$. So $null(T)=\{(x,-x,x)\in \mathbb{R}^2\vert x\in \mathbb{R}\}$ -* Let $D\in \mathscr{L}(\mathscr{P}(\mathbb{R})),D(f)=f'$, $null (D)=$ the set of constant functions. (because the derivatives of them are zero.) - -#### Theorem 3.13 - -Given $T\in \mathscr{L}(V,W)$, $null(T)$ is a subspace of $V$. - -Proof: - -We check the conditions for the subspace. -* $T0=0$, so $0\in null(T)$ -* $u,v\in null(T)$, then consider $T(u+v)=Tu+Tv=0+0=0$, so $u+v\in null(T)$ -* Let $v\in null (T),\lambda \in \mathbb{F}$, then $T(\lambda v)=\lambda (Tv)=\lambda 0=0$, so $\lambda v \in null (T)$ - -So $null(T)$ is a subspace. - -#### Definition 3.14 - -A function $f:V\to W$ is **injective** (also called one-to-one, 1-1) if for all $u,v\in V$, if $Tv=Tu$, then $T=U$. - -#### Lemma 3.15 - -Let $T\in \mathscr{L}(V,W)$ then $T$ is injective if and only if $null(T)=\{0\}$ - -Proof: - -$\Rightarrow$ - -Let $T\in \mathscr{L}(V,W)$ be injective, and let $v\in null (T)$. Then $Tv=0=T0$ so because $T$ is injective $v=0\implies null (T)=\{0\}$ - -$\Leftarrow$ - -Suppose $T\in \mathscr{L}(V,W)$ with $null (T)=\{0\}$. Let $u,v\in V$ with $Tu=Tv$, $Tu-Tv=0,T(u-v)=0,u-v=0,u=v$, so $T$ is injective - -#### Definition 3.16 - -Range and surjectivity - -For $T\in \mathscr{L}(V,W)$ the range of $T$ denoted $range(T)$, is given by - -$$ -range(T)=\{Tv\vert v\in V\} -$$ - -Example: - -* $0\in \mathscr{L}(V,W)$, $range(0)=\{0\}$ -* $I\in \mathscr{L}(V,W)$, $range(I)=V$ -* Let $T:\mathbb{R}\to \mathbb{R}^2$ given by $T(x)=(x,2x)$, $range(T)=\{(x,2x)\vert x\in \mathbb{R}\}$ - -#### Theorem 3.18 - -Given $T\in \mathscr{L}(V,W)$, $range(T)$ is a subspace of $W$ - -Proof: - -Exercise, not interesting. - -#### Definition 3.19 - -A function $T:V\to W$ is surjective (also called onto) if $range(T)=W$ - -#### Theorem 3.21 (The Fundamental Theorem of Linear Maps, Rank-nullity Theorem) - -Suppose $V$ is finite dimensional, and $T\in \mathscr{L}(V,W)$, then $range(T)$ is finite dimensional ($W$ don't need to be finite dimensional). and - -$$ -dim(V)=dim(null (T))+dim(range(T)) -$$ - -#### Theorem 3.22 - -Let $T\in \mathscr{L}(V,W)$ and suppose $dim(V)>dim(W)$. Then $T$ is not injective. - -Proof: - -By **Theorem 3.21**, $dim(V)=dim(null (T))+dim(range(T))$, $dim(V)=dim(null (T))+dim(W)$, $00\implies null (T)\neq \{0\}$ - -So $T$ is not injective. +# Lecture 9 + +## Chapter III Linear maps + +**Assumption: $U,V,W$ are vector spaces (over $\mathbb{F}$)** + +### Vector Space of Linear Maps 3A + +Review + +$\mathscr{L}(V,W) =$ space of linear maps form $V$ to $W$. + +$\mathscr{L}(V)=\mathscr{L}(V,V)$ + +Key facts: + +* $\mathscr{L}(V,W)$ is a vector space +* given $T\in\mathscr{L}(U,V),S\in \mathscr{L}(V,W)$, we have $TS\in \mathscr{L}(V,W)$ +* not commutative + +### Null spaces and Range 3B + +#### Definition 3.11 + +Null space and injectivity + +For $T\in \mathscr{L}(V,W)$, the **null space** of $T$, denoted as $null(T)$ (sometime also noted as $ker\ T$), is a subset of $V$ given by + +$$ +ker\ T=null(T)=\{v\in V \vert Tv=0\} +$$ + +Examples: + +* $0\in \mathscr{L}(V,W)$, then $null\ 0=V$, $null(I)=\{0\}$ +* $T\in \mathscr{L}(\mathbb{R}^3,\mathbb{R}^2),T(x,y,z)=(x+y,y+z)$, to find the null space, we set $T(x,y,z)=0$, then $x+y=0,y+z=0$, $x=-y,x=z$. So $null(T)=\{(x,-x,x)\in \mathbb{R}^2\vert x\in \mathbb{R}\}$ +* Let $D\in \mathscr{L}(\mathscr{P}(\mathbb{R})),D(f)=f'$, $null (D)=$ the set of constant functions. (because the derivatives of them are zero.) + +#### Theorem 3.13 + +Given $T\in \mathscr{L}(V,W)$, $null(T)$ is a subspace of $V$. + +Proof: + +We check the conditions for the subspace. +* $T0=0$, so $0\in null(T)$ +* $u,v\in null(T)$, then consider $T(u+v)=Tu+Tv=0+0=0$, so $u+v\in null(T)$ +* Let $v\in null (T),\lambda \in \mathbb{F}$, then $T(\lambda v)=\lambda (Tv)=\lambda 0=0$, so $\lambda v \in null (T)$ + +So $null(T)$ is a subspace. + +#### Definition 3.14 + +A function $f:V\to W$ is **injective** (also called one-to-one, 1-1) if for all $u,v\in V$, if $Tv=Tu$, then $T=U$. + +#### Lemma 3.15 + +Let $T\in \mathscr{L}(V,W)$ then $T$ is injective if and only if $null(T)=\{0\}$ + +Proof: + +$\Rightarrow$ + +Let $T\in \mathscr{L}(V,W)$ be injective, and let $v\in null (T)$. Then $Tv=0=T0$ so because $T$ is injective $v=0\implies null (T)=\{0\}$ + +$\Leftarrow$ + +Suppose $T\in \mathscr{L}(V,W)$ with $null (T)=\{0\}$. Let $u,v\in V$ with $Tu=Tv$, $Tu-Tv=0,T(u-v)=0,u-v=0,u=v$, so $T$ is injective + +#### Definition 3.16 + +Range and surjectivity + +For $T\in \mathscr{L}(V,W)$ the range of $T$ denoted $range(T)$, is given by + +$$ +range(T)=\{Tv\vert v\in V\} +$$ + +Example: + +* $0\in \mathscr{L}(V,W)$, $range(0)=\{0\}$ +* $I\in \mathscr{L}(V,W)$, $range(I)=V$ +* Let $T:\mathbb{R}\to \mathbb{R}^2$ given by $T(x)=(x,2x)$, $range(T)=\{(x,2x)\vert x\in \mathbb{R}\}$ + +#### Theorem 3.18 + +Given $T\in \mathscr{L}(V,W)$, $range(T)$ is a subspace of $W$ + +Proof: + +Exercise, not interesting. + +#### Definition 3.19 + +A function $T:V\to W$ is surjective (also called onto) if $range(T)=W$ + +#### Theorem 3.21 (The Fundamental Theorem of Linear Maps, Rank-nullity Theorem) + +Suppose $V$ is finite dimensional, and $T\in \mathscr{L}(V,W)$, then $range(T)$ is finite dimensional ($W$ don't need to be finite dimensional). and + +$$ +dim(V)=dim(null (T))+dim(range(T)) +$$ + +#### Theorem 3.22 + +Let $T\in \mathscr{L}(V,W)$ and suppose $dim(V)>dim(W)$. Then $T$ is not injective. + +Proof: + +By **Theorem 3.21**, $dim(V)=dim(null (T))+dim(range(T))$, $dim(V)=dim(null (T))+dim(W)$, $00\implies null (T)\neq \{0\}$ + +So $T$ is not injective. diff --git a/pages/Math429/_meta.js b/content/Math429/_meta.js similarity index 98% rename from pages/Math429/_meta.js rename to content/Math429/_meta.js index c238059..391b275 100644 --- a/pages/Math429/_meta.js +++ b/content/Math429/_meta.js @@ -1,5 +1,5 @@ export default { - index: "Course Description", + //index: "Course Description", "---":{ type: 'separator' }, diff --git a/pages/Math429/index.md b/content/Math429/index.md similarity index 100% rename from pages/Math429/index.md rename to content/Math429/index.md diff --git a/pages/Swap/CSE361S_L2.md b/content/Swap/CSE361S_L2.md similarity index 100% rename from pages/Swap/CSE361S_L2.md rename to content/Swap/CSE361S_L2.md diff --git a/pages/Swap/CSE361S_W1.md b/content/Swap/CSE361S_W1.md similarity index 100% rename from pages/Swap/CSE361S_W1.md rename to content/Swap/CSE361S_W1.md diff --git a/pages/Swap/Math4351_L1.md b/content/Swap/Math4351_L1.md similarity index 100% rename from pages/Swap/Math4351_L1.md rename to content/Swap/Math4351_L1.md diff --git a/pages/_meta.js b/content/_meta.js similarity index 86% rename from pages/_meta.js rename to content/_meta.js index 9967089..e1b7520 100644 --- a/pages/_meta.js +++ b/content/_meta.js @@ -1,5 +1,5 @@ export default { - index: { + menu: { title: 'Home', type: 'menu', items: { @@ -16,10 +16,6 @@ export default { href: '/contact' } }, - theme:{ - sidebar: false, - timestamp: true, - } }, Math3200: { type: 'page', @@ -58,24 +54,28 @@ export default { } }, CSE332S: { + title: 'CSE332S', type: 'page', theme:{ timestamp: true, } }, CSE347: { + title: 'CSE347', type: 'page', theme:{ timestamp: true, } }, CSE442T: { + title: 'CSE442T', type: 'page', theme:{ timestamp: true, } }, CSE559A: { + title: 'CSE559A', type: 'page', theme:{ timestamp: true, @@ -87,16 +87,25 @@ export default { timestamp: true, } }, + index: { + display: 'hidden', + theme:{ + sidebar: false, + timestamp: true, + } + }, about: { display: 'hidden', theme:{ - timestamp: true, + sidebar: false, + timestamp: true, } }, contact: { display: 'hidden', theme:{ - timestamp: true, + sidebar: false, + timestamp: true, } } } \ No newline at end of file diff --git a/pages/about.md b/content/about.md similarity index 100% rename from pages/about.md rename to content/about.md diff --git a/pages/contact.md b/content/contact.md similarity index 100% rename from pages/contact.md rename to content/contact.md diff --git a/pages/index.md b/content/index.md similarity index 100% rename from pages/index.md rename to content/index.md diff --git a/content/layout.tsx b/content/layout.tsx new file mode 100644 index 0000000..dad73d2 --- /dev/null +++ b/content/layout.tsx @@ -0,0 +1,90 @@ +/* eslint-env node */ +import { Footer, Layout, Navbar } from 'nextra-theme-docs' +import { Banner, Head } from 'nextra/components' +import { getPageMap } from 'nextra/page-map' +import 'nextra-theme-docs/style.css' +import { SpeedInsights } from "@vercel/speed-insights/next" +import { Analytics } from "@vercel/analytics/react" + +export const metadata = { + metadataBase: new URL('https://notenextra.trance-0.com'), + title: { + template: '%s - NoteNextra' + }, + description: 'A static note sharing site for minimum care', + applicationName: 'NoteNextra', + generator: 'Next.js', + appleWebApp: { + title: 'NoteNextra' + }, + other: { + 'msapplication-TileImage': '/ms-icon-144x144.png', + 'msapplication-TileColor': '#fff' + }, + twitter: { + site: 'https://notenextra.trance-0.com' + } +} + +export default async function RootLayout({ children }) { + const navbar = ( + + + + + + NoteNextra + + + } + projectLink="https://github.com/Trance-0/NoteNextra" + /> + ) + const pageMap = await getPageMap() + return ( + + + + + + MIT {new Date().getFullYear()} ©{' '} + + Trance-0 + + . + + + } + editLink="Edit this page on GitHub" + docsRepositoryBase="https://github.com/Trance-0/NoteNextra/tree/main" + sidebar={{ defaultMenuCollapseLevel: 1 }} + pageMap={pageMap} + > + {children} + {/* SpeedInsights in vercel */} + + {/* Analytics in vercel */} + + + + + ) +} \ No newline at end of file diff --git a/mdx-components.js b/mdx-components.js new file mode 100644 index 0000000..b015544 --- /dev/null +++ b/mdx-components.js @@ -0,0 +1,11 @@ +import { useMDXComponents as getDocsMDXComponents } from 'nextra-theme-docs' + +const docsComponents = getDocsMDXComponents() + +// Merge components +export const useMDXComponents = (components) => { + return { + ...docsComponents, + ...components + } +} \ No newline at end of file diff --git a/next-env.d.ts b/next-env.d.ts index 52e831b..1b3be08 100644 --- a/next-env.d.ts +++ b/next-env.d.ts @@ -2,4 +2,4 @@ /// // NOTE: This file should not be edited -// see https://nextjs.org/docs/pages/api-reference/config/typescript for more information. +// see https://nextjs.org/docs/app/api-reference/config/typescript for more information. diff --git a/next.config.mjs b/next.config.mjs index 0e155d9..71fce0a 100644 --- a/next.config.mjs +++ b/next.config.mjs @@ -1,29 +1,33 @@ import nextra from 'nextra' +import withBundleAnalyzer from '@next/bundle-analyzer' const withNextra = nextra({ - theme: 'nextra-theme-docs', - themeConfig: './theme.config.jsx', latex: { renderer: 'katex', options: { // suppress warnings from katex for `\\` - // strict: false, - // macros: { - // '\\RR': '\\mathbb{R}' - // } + strict: false, } - } + }, + mdxOptions: { + format: 'detect' + }, + contentDirBasePath: '/', }) -export default withNextra({ +const bundleAnalyzer = withBundleAnalyzer({ + enabled: process.env.ANALYZE === 'true' +}) + +export default bundleAnalyzer(withNextra({ output: 'standalone', - // eslint: { - // ignoreDuringBuilds: true, - // }, + eslint: { + ignoreDuringBuilds: true, + }, experimental: { - // optimize memory usage: https://nextjs.org/docs/app/building-your-application/optimizing/memory-usage webpackMemoryOptimizations: true, - },}) + }, +})) // If you have other Next.js configurations, you can pass them as the parameter: // export default withNextra({ /* other next.js config */ }) \ No newline at end of file diff --git a/package-lock.json b/package-lock.json index d8a239a..1066e56 100644 --- a/package-lock.json +++ b/package-lock.json @@ -1,11720 +1,6936 @@ { - 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"type": "github", + "url": "https://github.com/sponsors/wooorm" + } } + } } diff --git a/package.json b/package.json index b6c4934..daf8bb0 100644 --- a/package.json +++ b/package.json @@ -1,22 +1,24 @@ { - "scripts": { - "dev": "next", - "build": "next build", - "start": "next start" - }, - "dependencies": { - "@cloudflare/next-on-pages": "^1.13.7", - "@napi-rs/simple-git": "^0.1.19", - "@vercel/analytics": "^1.4.1", - "@vercel/speed-insights": "^1.1.0", - "next": "^15.3.5", - "nextra": "^3.3.1", - "nextra-theme-docs": "^3.3.1", - "react": "^18.3.1", - "react-dom": "^18.3.1", - "vercel": "^32.3.0" - }, - "devDependencies": { - "@types/node": "22.10.7" - } + "scripts": { + "dev": "next dev", + "build": "cross-env NODE_OPTIONS='--max-old-space-size=16384' next build", + "build:test": "cross-env ANALYZE=true NODE_OPTIONS='--inspect --max-old-space-size=4096' next build", + "build:analyze": "cross-env ANALYZE=true NODE_OPTIONS='--max-old-space-size=16384' next build", + "start": "next start" + }, + "dependencies": { + "@next/bundle-analyzer": "^15.3.5", + "@vercel/analytics": "^1.5.0", + "@vercel/speed-insights": "^1.2.0", + "cross-env": "^7.0.3", + "next": "^15.3.5", + "nextra": "^4.2.17", + "nextra-theme-docs": "^4.2.17", + "react": "^19.1.0", + "react-dom": "^19.1.0" + }, + "devDependencies": { + "@types/node": "24.0.10", + "@types/react": "19.1.8" + } } diff --git a/pages/_app.jsx b/pages/_app.jsx deleted file mode 100644 index aaf16ef..0000000 --- a/pages/_app.jsx +++ /dev/null @@ -1,15 +0,0 @@ -import { ThemeProvider } from 'next-themes' -import { SpeedInsights } from "@vercel/speed-insights/next" -import { Analytics } from "@vercel/analytics/react" - -export default function App({ Component, pageProps }) { - return ( - - {/* SpeedInsights in vercel */} - - {/* Analytics in vercel */} - - - - ) - } \ No newline at end of file diff --git a/theme.config.jsx b/theme.config.jsx deleted file mode 100644 index 0371389..0000000 --- a/theme.config.jsx +++ /dev/null @@ -1,80 +0,0 @@ -import { useRouter } from 'next/router' -import { useConfig } from 'nextra-theme-docs' - -export default { - color: { - hue: { - dark: 336, - light: 164 - }, - saturation: { - dark: 72, - light: 49 - }, - lightness: { - dark: 59, - light: 35 - } - }, - footer: { - content: ( - - MIT {new Date().getFullYear()} ©{' '} - - Trance-0 - - . - - ) - }, - head() { - const { asPath, defaultLocale, locale } = useRouter() - const { frontMatter } = useConfig() - const url = - 'https://notenextra.trance-0.com' + - (defaultLocale === locale ? asPath : `/${locale}${asPath}`) - - return ( - <> - {frontMatter.title || 'NoteNextra'} - - - - - ) - }, - - logo: ( - <> - - - - - NoteNextra - - - ), - readMore: 'Read More →', - postFooter: null, - darkMode: false, - navs: [ - { - url: 'https://github.com/Trance-0/NoteNextra', - name: 'Source' - } - ], - darkMode: true, - themeSwitch: { - useOptions() { - return { - light: 'Light', - dark: 'Dark', - system: 'System' - } - } - }, - docsRepositoryBase: 'https://github.com/Trance-0/NoteNextra/tree/main' -} \ No newline at end of file diff --git a/pages/CSE332S/LNG_v1.0.py b/toolboxes/LNG_v1.0.py similarity index 100% rename from pages/CSE332S/LNG_v1.0.py rename to toolboxes/LNG_v1.0.py diff --git a/pages/CSE347/code_test.py b/toolboxes/code_test.py similarity index 95% rename from pages/CSE347/code_test.py rename to toolboxes/code_test.py index 645571f..1a5b520 100644 --- a/pages/CSE347/code_test.py +++ b/toolboxes/code_test.py @@ -1,108 +1,108 @@ -import random -import time - -def partition(A,p,r): - x=A[r] - lo,hi=p,r-1 - for i in range(p,r): - if A[i]0: - for i in range(len(A)): - digit=(A[i]//exp)%b - buckets[digit].append(A[i]) - A=[] - for bucket in buckets: - A.extend(bucket) - exp*=b - return A - -if __name__=="__main__": - C=[random.randint(0,10000000) for _ in range(100000)] - A=C.copy() - start=time.time() - Ao=sorted(A) - end=time.time() - print(f"Time taken: for built-in sort {end-start} seconds") - A=C.copy() - start=time.time() - randomized_quicksort(A,0,len(A)-1) - end=time.time() - print(A==Ao) - print(f"Time taken: for randomized quicksort {end-start} seconds") - A=C.copy() - start=time.time() - quicksort(A,0,len(A)-1) - end=time.time() - print(A==Ao) - print(f"Time taken: for quicksort {end-start} seconds") - A=C.copy() - start=time.time() - merge_sort(A,0,len(A)-1) - end=time.time() - print(A==Ao) - print(f"Time taken: for merge sort {end-start} seconds") - A=C.copy() - start=time.time() - radix_sort(A) - end=time.time() - print(A==Ao) - print(f"Time taken: for radix sort {end-start} seconds") - +import random +import time + +def partition(A,p,r): + x=A[r] + lo,hi=p,r-1 + for i in range(p,r): + if A[i]0: + for i in range(len(A)): + digit=(A[i]//exp)%b + buckets[digit].append(A[i]) + A=[] + for bucket in buckets: + A.extend(bucket) + exp*=b + return A + +if __name__=="__main__": + C=[random.randint(0,10000000) for _ in range(100000)] + A=C.copy() + start=time.time() + Ao=sorted(A) + end=time.time() + print(f"Time taken: for built-in sort {end-start} seconds") + A=C.copy() + start=time.time() + randomized_quicksort(A,0,len(A)-1) + end=time.time() + print(A==Ao) + print(f"Time taken: for randomized quicksort {end-start} seconds") + A=C.copy() + start=time.time() + quicksort(A,0,len(A)-1) + end=time.time() + print(A==Ao) + print(f"Time taken: for quicksort {end-start} seconds") + A=C.copy() + start=time.time() + merge_sort(A,0,len(A)-1) + end=time.time() + print(A==Ao) + print(f"Time taken: for merge sort {end-start} seconds") + A=C.copy() + start=time.time() + radix_sort(A) + end=time.time() + print(A==Ao) + print(f"Time taken: for radix sort {end-start} seconds") + diff --git a/pages/CSE442T/fun.py b/toolboxes/fun.py similarity index 95% rename from pages/CSE442T/fun.py rename to toolboxes/fun.py index 56df8bb..7cf39e5 100644 --- a/pages/CSE442T/fun.py +++ b/toolboxes/fun.py @@ -1,65 +1,65 @@ -from math import gcd - -def euclidean_algorithm(a,b): - if a0: - if k==0: - break - # raise ValueError(f"Damn, {i} generates 0 for group {p}") - sg.append(k) - k=(k**f)%p - step-=1 - sg.append(1) - # if len(sg)!=(p-1): continue - g.append((i,[j for j in sg])) - return g - -def __list_print(arr): - for i in arr:print(i) - -def factorization(n): - # Pollard's rho integer factorization algorithm - # https://stackoverflow.com/questions/32871539/integer-factorization-in-python - factors = [] - - def get_factor(n): - x_fixed = 2 - cycle_size = 2 - x = 2 - factor = 1 - - while factor == 1: - for count in range(cycle_size): - if factor > 1: break - x = (x * x + 1) % n - factor = gcd(x - x_fixed, n) - - cycle_size *= 2 - x_fixed = x - - return factor - - while n > 1: - next = get_factor(n) - factors.append(next) - n //= next - - return factors - -if __name__=='__main__': - print(euclidean_algorithm(285,(10**9+7)*5)) - __list_print(get_generator(23)) +from math import gcd + +def euclidean_algorithm(a,b): + if a0: + if k==0: + break + # raise ValueError(f"Damn, {i} generates 0 for group {p}") + sg.append(k) + k=(k**f)%p + step-=1 + sg.append(1) + # if len(sg)!=(p-1): continue + g.append((i,[j for j in sg])) + return g + +def __list_print(arr): + for i in arr:print(i) + +def factorization(n): + # Pollard's rho integer factorization algorithm + # https://stackoverflow.com/questions/32871539/integer-factorization-in-python + factors = [] + + def get_factor(n): + x_fixed = 2 + cycle_size = 2 + x = 2 + factor = 1 + + while factor == 1: + for count in range(cycle_size): + if factor > 1: break + x = (x * x + 1) % n + factor = gcd(x - x_fixed, n) + + cycle_size *= 2 + x_fixed = x + + return factor + + while n > 1: + next = get_factor(n) + factors.append(next) + n //= next + + return factors + +if __name__=='__main__': + print(euclidean_algorithm(285,(10**9+7)*5)) + __list_print(get_generator(23)) print(factorization(162000)) \ No newline at end of file diff --git a/pages/CSE559A/mlp_image_reconstruction.py b/toolboxes/mlp_image_reconstruction.py similarity index 100% rename from pages/CSE559A/mlp_image_reconstruction.py rename to toolboxes/mlp_image_reconstruction.py diff --git a/pages/Math3200/public_html_ifram_gen.py b/toolboxes/public_html_ifram_gen.py similarity index 100% rename from pages/Math3200/public_html_ifram_gen.py rename to toolboxes/public_html_ifram_gen.py diff --git a/tsconfig.json b/tsconfig.json index c3208e2..1fda10e 100644 --- a/tsconfig.json +++ b/tsconfig.json @@ -13,15 +13,22 @@ "incremental": true, "module": "esnext", "esModuleInterop": true, - "moduleResolution": "node", + "moduleResolution": "bundler", "resolveJsonModule": true, "isolatedModules": true, - "jsx": "preserve" + "jsx": "preserve", + "plugins": [ + { + "name": "next" + } + ], + "strictNullChecks": true }, "include": [ - "next-env.d.ts", "**/*.ts", - "**/*.tsx" + "**/*.tsx", + "next-env.d.ts", + ".next/types/**/*.ts" ], "exclude": [ "node_modules"