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content/Math429/Math429_L26.md

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+# Lecture 26
+
+## Chapter VI Inner Product Spaces
+
+### Inner Products and Norms 6A
+
+---
+
+Review
+
+#### Dot products
+
+#### Inner product
+
+An inner product $\langle,\rangle:V\times V\to \mathbb{F}$
+
+Positivity: $\langle v,v\rangle\geq 0$
+
+Definiteness: $\langle v,v\rangle=0\iff v=0$
+
+Additivity: $<u+v,w>=<u,w>+<v,w>$
+
+Homogeneity: $<\lambda u, v>=\lambda<u,v>$
+
+Conjugate symmetry: $<u,v>=\overline{<v,u>}$
+
+#### Norm
+
+$||v||=\sqrt{<v,v>}$
+
+---
+
+New materials
+
+### Orthonormal basis 6B
+
+#### Definition 6.22
+
+A list of vectors is **orthonormal** if each vector has norm = 1, and is orthogonal to every other vectors in the list.
+
+if a list $e_1,...,e_m\in V$ is orthonormal if $<e_j,e_k>=1\begin{cases}
+    1 \textup{ if } j=k\\
+    0 \textup{ if }j\neq k
+\end{cases}$.
+
+Example:
+
+* Standard basis in $\mathbb{F}^n$ is orthonormal.  
+* $(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}),(\frac{-1}{\sqrt{2}},\frac{1}{\sqrt{2}},0),(\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}},\frac{-2}{\sqrt{6}})$ in $\mathbb{F}^3$ is orthonormal.
+* For $<p,q>=\int^1_{-1}pq$ on $\mathscr{P}_2(\mathbb{R})$. The standard basis $(1,x,x^2)$ is not orthonormal.
+
+#### Theorem 6.24
+
+Suppose $e_1,...,e_m$ is an orthonormal list, then $||a_1 e_1+...+a_m e_m||^2=|a_1|^2+...+|a_m|^2$
+
+Proof:
+
+Using induction of $m$.
+
+$m=1$, clear ($||e_1||^2=1$)
+$m>1$, $||a_1 e_1+...a_{m-1}e_{m-1}||^2=|a_1|^2+...+|a_{m-1}|^2$ and $<a_1 e_1+...+a_{m-1} e_{m-1},a_m e_m>=0$ by Pythagorean Theorem. $||(a_1 e_1+...a_{m-1}e_{m-1})+a_m e_m||^2=||a_1 e_1+...a_{m-1}e_{m-1}||^2+||a_m e_m||^2=|a_1|^2+...+|a_{m-1}|^2+|a_m|^2$
+
+#### Theorem 6.25
+
+Every orthonormal list is linearly independent.
+
+Proof:
+
+$||a_1 e_1+...+a_m e_m||^2=0$, then $|a_1|^2+...+|a_m|^2=0$, then $a_1=...=a_m=0$
+
+#### Theorem 6.28
+
+Every orthonormal list of length $dim\ V$ is a basis. 
+
+#### Definition 6.27
+
+An orthonormal basis is a basis that is an orthonormal list.
+
+#### Theorem 6.26 Bessel's Inequality
+
+Suppose $e_1,...,e_m$ is an orthonormal list $v\in V$
+
+$$
+|<v,e_1>|^2+...+|<v,e_m>|^2\leq ||v||^2
+$$
+
+Proof:
+
+Let $v\in V$, then let $n=<v,e_1>e_1+...+<v,e_m>e_m$,
+
+let $w=v-u$, Note that $<u,e_k>=<v,e_k>$, thus $<w,e_k>=0, <w,u>=0$, apply Pythagorean Theorem.
+
+$$
+||w+u||^2=||w||^2+||u||^2\\
+||v||^2\geq ||u||^2
+$$
+
+#### Theorem 6.30
+
+Suppose $e_1,...,e_n$ is an orthonormal basis, and $u,v\in V$, then 
+
+(a) $v=<v,e_1>e_1+...+<v,e_n>e_n$  
+(b) $||v||^2=|<v,e_1>|^2+...+|<v,e_n>|^2$  
+(c) $<u,v>=<u,e_1>\overline{<v,e_1>}+...+<u,e_n>\overline{<v,e_n>}$
+
+Proof:
+
+(a) let $a_1,...,a_n\in \mathbb{F}$ such that $v=a_1 e_1+...+a_n e_n$.
+
+$$
+\begin{aligned}
+<v,e_k>&=<a,e_1,e_k>+...+<a_k e_k,e_k>+...+<a_n e_n,e_n>\\
+&=<a_k e_k,e_k>\\
+&= a_k
+\end{aligned}
+$$
+
+---
+
+Note *6.30 (c)*  means up to change of basis, every inner product on a finite dimensional vector space "looks like" an euclidean inner products...
+
+#### Theorem 6.32 Gram-Schmidt
+
+Let $v_1,...,v_m$ be a linearly independent list. 
+
+Define $f_k\in V$ by $f_1=v_1,f_k=v_k-\sum_{j=1}^{k-1}\frac{<v_k,f_j>}{||f_j||^2}f_j$
+
+Define $e_k=\frac{f_k}{||f_k||}$, then $e_1,...,e_m$ is orthonormal $Span(v_1,...,v_m)=Span(f_1,...,f_m)$