LEC6

• Proof of Theorem for Real Polynomials.
• Vector Algebra

Proof of Theorem for Real Polynomials

• Take $p(x)$ and take conjugate of the entire expression, both LHS and RHS.
• $\overline{p(x)}=\overline{a_n{w}^n}+...+\overline{a_{1}w}+\overline{a_0}$
• Since conjugate of purely reals is same as the real number itself, we can get rid of the bars over $a_i$.
• $\overline{p(x)}=a_n\overline{w^n}+...+a_1\overline{w}+a_0$
• This is same as using $\overline{w}$ inside $p(x)$ instead of $w$.
• Hence $p(w)=p(\overline{w})=0$

Vector Algebra

• Instead of writing $(x,y)$ as a coordinate, we write it as a column vector.

$$\left[\begin{array}{c}x\\ y\end{array}\right]$$

$R^2$ is the set of all vectors with 2 components.

$$R^n=\{\left[\begin{array}{c}{x}_1\\ .\\ .\\ .\\ x_n\end{array}\right]|x_1,...,x_n\in \mathbb{R}\}$$

Basic operations

• Pretty obvios

Parallel Vectors

$$\vec{x}=c\vec{y}⟹\vec{x}\parallel \vec{y}$$

(assuming nothing is zero etc etc)