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content/Math429/Math429_L16.md

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+# 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 $b_k^2\leq 4c_k$