140 lines
3.0 KiB
Markdown
140 lines
3.0 KiB
Markdown
# Lecture 25
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## Chapter VI Inner Product Spaces
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### Inner Products and Norms 6A
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#### Dot Product (Euclidean Inner Product)
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$$
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v\cdot w=v_1w_1+...+v_n w_n
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$$
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$$
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-\cdot -:\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}
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$$
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Some properties
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* $v\cdot v\geq 0$
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* $v\cdot v=0\iff v=0$
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* $(u+v)\cdot w=u\cdot w+v\cdot w$
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* $(c\cdot v)\cdot w=c\cdot(v\cdot w)$
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#### Definition 6.2
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An inner product $<,>:V\times V\to \mathbb{F}$
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Positivity: $<v,v>\geq 0$
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Definiteness: $<v,v>=0\iff v=0$
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Additivity: $<u+v,w>=<u,w>+<v,w>$
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Homogeneity: $<\lambda u, v>=\lambda<u,v>$
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Conjugate symmetry: $<u,v>=\overline{<v,u>}$
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Note: the dot product on $\mathbb{R}^n$ satisfies these properties
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Example:
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$V=C^0([-1,-])$
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$L_2$ - inner product.
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$<f,g>=\int^1_{-1} f\cdot g$
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$<f,f>=\int ^1_{-1}f^2\geq 0$
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$<f+g,h>=<f,h>+<g,h>$
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$<\lambda f,g>=\lambda<f,g>$
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$<f,g>=\int^1_{-1} f\cdot g=\int^1_{-1} g\cdot f=<g,f>$
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The result is in real vector space so no conjugate...
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#### Theorem 6.6
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For $<,>$ an inner product
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(a) Fix $V$, then the map given by $u\mapsto <u,v>$ is a linear map (Warning: if $\mathbb{F}=\mathbb{C}$, then $u\mapsto<u,v>$ is not linear).
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(b,c) $<0,v>=<v,0>=0$
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(d) $<u,v+w>=<u,v>+<u,w>$ (second terms are additive.)
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(e) $<u,\lambda v>=\bar{\lambda}<u,v>$
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#### Definition 6.4
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An **inner product space** is a pair of vector space and inner product on it. $(v,<,>)$. In practice, we will say "$V$ is an inner product space" and treat $V$ as the vector space.
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For the remainder of the chapter. $V,W$ are inner product vector spaces...
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#### Definition 6.7
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For $v\in V$ the **norm of $V$** is given by $||v||:=\sqrt{<v,v>}$
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#### Theorem 6.9
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Suppose $v\in V$.
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(a) $||v||=0\iff v=0$
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(b) $||\lambda v||=|\lambda|\ ||v||$
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Proof:
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$||\lambda v||^2=<\lambda v,\lambda v> =\lambda<v,\lambda v>=\lambda\bar{\lambda}<v,v>$
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So $|\lambda|^2 <v,v>=|\lambda|^2||v||^2$, $||\lambda v||=|\lambda|\ ||v||$
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#### Definition 6.10
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$v,u\in V$ are **orthogonal** if $<v,u>=0$.
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#### Theorem 6.12 (Pythagorean Theorem)
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If $u,v\in V$ are orthogonal, then $||u+v||^2=||u||^2+||v||$
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Proof:
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$$
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\begin{aligned}
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||u+v||^2&=<u+v,u+v>\\
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&=<u,u+v>+<v,u+v>\\
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&=<u,u>+<u,v>+<v,u>+<v,v>\\
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&=||u||^2+||v||^2
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\end{aligned}
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$$
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#### Theorem 6.13
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Suppose $u,v\in V$, $v\neq 0$, set $c=\frac{<u,v>}{||v||^2}$, then let $w=u-v\cdot v$, then $v$ and $w$ are orthogonal.
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#### Theorem 6.14 (Cauchy-Schwarz)
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Let $u,v\in V$, then $|<u,v>|\leq ||u||\ ||v||$ where equality occurs only $u,v$ are parallel...
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Proof:
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Take the square norm of $u=\frac{<u,v>}{||u||^2}v+w$.
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#### Theorem 6.17 Triangle Inequality
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If $u,v\in V$, then $||u+v||\leq ||u||+||v||$
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Proof:
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$$
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\begin{aligned}
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||u+v||^2&=<u+v,u+v>\\
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&=<u,u>+<u,v>+<v,u>+<v,v>\\
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&=||u||^2+||v||^2+2Re(<u,v>)\\
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&\leq ||u||^2+||v||^2+2|<u,v>|\\
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&\leq ||u||^2+||v||^2+2||u||\ ||v||\\
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&\leq (||u||+||v ||)^2
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\end{aligned}
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$$
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