diff --git a/content/CSE5313/CSE5313_L5.md b/content/CSE5313/CSE5313_L5.md new file mode 100644 index 0000000..147715e --- /dev/null +++ b/content/CSE5313/CSE5313_L5.md @@ -0,0 +1,272 @@ +# CSE5313 Coding and information theory for data science (Lecture 5) + +## Recap + +### Group + +1. Closure: $\forall a,b\in G, a\cdot b\in G$. +2. Associativity: $\forall a,b,c\in G, (a\cdot b)\cdot c=a\cdot (b\cdot c)$. +3. Identity: $\exists e\in G, \forall a\in G, a\cdot e=e\cdot a=a$. +4. Inverses: $\forall a\in G, \exists a^{-1}\in G, a\cdot a^{-1}=a^{-1}\cdot a=e$. + +May not be commutative (group of invertible matrices). + +### Order of element in group + +$a\in G$ is of order $k$ if $a^k=e$ and $k$ is the smallest positive integer such that $a^k=e$. + +If $a^n=e$, then $O(a)\mid n$. + +### Generator of group + +$a\in G$ is a generator of $G$ if $\mathcal{O}(a)=|G|$. (for finite groups) + +For infinite groups, $\langle a\rangle=G$. + +
+Example + +$(\mathbb{Z}_n,+)$ has generator $1$. + +$(\mathbb{Z}_8^*,\cdot)$ has generator $3$. (Recall $\mathbb{Z}_8^*=\{x\in\mathbb{Z}_8:gcd(x,8)=1\}$. for multiplicative inverse.) + +
+ +## New content + +### Subgroups + +A subgroup $H$ of $G$ is a nonempty subset of $G$ that is itself a group under the operation of $G$. + +Denoted as $H\leq G$. + +
+Example + +$(\mathbb{Z}_6,+)$ has subgroups $H=(\{0,2,4\},+)$. + +Only need to check three: + +- non-empty +- closure +- finite + +
+ +#### Theorem for finite subgroups + +If $H$ is finite, non-empty, and closed under the operation of $G$, then $H$ is a subgroup of $G$. + +### Equivalence relations + +An equivalence relation $\sim$ on a set $X$ is a relation that is + +- reflexive: $\forall x\in X, x\sim x$ +- symmetric: $\forall x,y\in X, x\sim y\implies y\sim x$ +- transitive: $\forall x,y,z\in X, x\sim y\text{ and } y\sim z\implies x\sim z$ + +
+Example + +Let $S$ be points on land, and $a\sim b$ if $a$ and $b$ are connected by land. + +
+ +#### Equivalence classes + +An equivalence relation on $S$ partitions $S$ into equivalence classes. + +Equivalence classes are: + +- Disjoint +- Cover $S$ + +### Cosets + +Let $G$ be a group and $H$ its subgroup. + +The coset of $H$ in $G$ is the equivalence class under congruence modulo $H$. + +Alternatively, (more convenient) + +$$ +\{h+a|h\in H,a\in G\} +$$ + +
+Example + +Let $G=(\mathbb{Z}_6,+)$ and $H=(\{0,2,4\},+)$. + +Define the equivalence relation on $G$ as: + +$a\sim b$ if $a+b^{-1}\in H$ (congruence modulo $H$) + +To find the equivalence classes of this relation for $G=(\mathbb{Z}_6,+)$ and $H=(\{0,2,4\},+)$, we have: + +- $0\sim 0, 0\sim 2, 0\sim 4$ +- $1\sim 1, 1\sim 3, 1\sim 5$ + +
+ +### Lagrange's theorem + +For every finite group $G$, the order of every subgroup $H$ of $G$ divides the order of $G$. + +#### Corollary of Lagrange's theorem + +If $|G|$ is prime, then $G$ is cyclic. + +
+Proof + +Let $A=\{a^0=e,a,a^2,\cdots,a^{|G|-1}\}$. + +$A$ is a cyclic subgroup of $G$. of order at least two. + +Then $|A|||G|$. + +So $|A|=|G|$. + +So $A=G$. + +So $G$ is cyclic. + +
+ +### Additive group of a field + +Any finite field has two types groups: + +- Additive group: $(\mathbb{F},+)$ +- Multiplicative group: $(\mathbb{F}^*,\cdot)$ + +The "integer" of $F$ is: + +$$ +\{a\in F|1^k=a,k\in\mathbb{N}\} +$$ + +The "characteristic" of $F$ is: + +- The order of $1$ in additive group +- Number of times that $1$ is added to itself to get $0$ +- Denoted by $\operatorname{c}(F)$. + +
+Example + +$\operatorname{c}(\mathbb{Z}_7)=7$. + +$\operatorname{c}(\mathbb{R})=0$. + +$\operatorname{c}(\mathbb{Z}_2[x] \mod x^2+x+1)=2$. + +
+ +#### Theorem field characteristic is prime + +If $\operatorname{c}(F)>0$, then $\operatorname{c}(F)$ is prime. + +
+Proof + +Suppose $\operatorname{c}(F)=mn$, then $0=\sum_{i=0}^{m-1}1\cdot\sum_{j=0}^{n-1}1=0$. + +So $m$ or $n$ must be $0$. + +So $\operatorname{c}(F)$ is prime. + +
+ +#### Theorem of linear power over additive group with prime characteristic + +Let $F$ be a field with characteristic $p>0$, then operation $^p$ is linear. + +That is, $(a+b)^p=a^p+b^p$. + +
+Proof + +$$ +(a+b)^p=\sum_{i=0}^p \binom{p}{i} a^i b^{p-i} +$$ + +$$ +\begin{aligned} +\binom{p}{i}&=\frac{p!}{i!(p-i)!}\\ +&=\frac{p(p-1)\cdots(p-i+1)}{i(i-1)\cdots 1} +\end{aligned} +$$ + +> Informally, $p$ divides the numerator but not the denominator. So the whole fraction is an integer. + +Since $\binom{p}{i}$ is an integer of $F$ except for $i=0$ and $i=p$, we have $\binom{p}{i}=0$ for $i=1,\cdots,p-1$. + +So $(a+b)^p=a^p+b^p$. + +
+ +### Multiplicative group of a field + +Every element in a multiplicative group of a field is cyclic. + +Corollary: + +- Every finite field has a generator, called a primitive element. +- This is an element $\gamma$ such that $\mathbb{F}^*=\langle \gamma\rangle$. +- Every element of $\mathbb{F}^*$ is a power of $\gamma$. + +
+Example + +Build $F_16=\mathbb{Z}_2[\zeta] \mod \zeta^4+\zeta+1$. + +The elements are: + +| Power of $\zeta$ | Element | As vector in $\mathbb{Z}_2^4$ | +|---|---|---| +| 0 | 0 | (0,0,0,0) | +| 1 | 1 | (1,0,0,0) | +| 2 | $\zeta$ | (0,1,0,0) | +| 3 | $\zeta^2$ | (0,0,1,0) | +| 4 | $\zeta^3$ | (0,0,0,1) | +| 5 | $\zeta+1$ | (1,1,0,0) | +| 6 | $\zeta^2+\zeta$ | (0,1,1,0) | +| 7 | $\zeta^3+\zeta^2$ | (0,0,1,1) | +| 8 | $\zeta^3+\zeta^2+1$ | (1,1,1,0) | + +The primitive element is $\zeta$. + +### Vector spaces and subspaces over finite fields + +$\mathbb{F}^n$ is a vector space over $\mathbb{F}$. + +With point-wise vector addition and scalar multiplication. + +
+Example + +$\mathbb{F}_2^4$ is a vector space over $\mathbb{F}_2$. + +Let $v=\begin{pmatrix} +1 & 1 & 1 & 1 +\end{pmatrix}$ + +Then $v$ is a vector in $\mathbb{F}_2^4$ that's "orthogonal" to itself. + +$v\cdot v=1+1+1+1=4=0$ in $\mathbb{F}_2$. + +In general field, the dual space and space may intersect non-trivially. + +
+ +Let $V$ be a subspace of $\mathbb{F}^n$. + +$V$ is a subgroup of $\mathbb{F}^n$ under vector addition. + +- Apply the theorem: If $H$ is finite, non-empty, and closed under the operation of $G$, then $H$ is a subgroup of $G$. + +> Is every subgroup of $\mathbb{F}^n$ a subspace? + +Cosets in this definition are called Affine subspaces.