NoteNextra-origin/content/Math429/Math429_L16.md

Lecture 16

Chapter IV Polynomials

\mathbb{F} denotes \mathbb{R} or $\mathbb{C}$

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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)

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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