# MATH 323 Lecture 22

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## Inner Product and Norm

### Cauchy-Schwarz Inequality

${\displaystyle |\langle u,v\rangle |\leq \|u\|\|v\|}$

Can be rewritten as

${\displaystyle -1\leq \underbrace {\frac {\langle u,v\rangle }{\|u\|\|v\|}} _{\cos {\theta }}\leq 1}$

${\displaystyle V}$ is normed linear space if with each ${\displaystyle v\in V}$, the number \|v\| is associated:

1. ${\displaystyle \|v\|\geq 0}$ and ${\displaystyle \|v\|=0\iff v={\vec {0}}}$
2. ${\displaystyle \|\alpha v\|=|\alpha |\|v\|\quad \forall \alpha \in \mathbb {R} }$
3. ${\displaystyle \|v+w\|\leq \|v\|+\|w\|}$ (triangle inequality)

### Theorem 5.4.3

If ${\displaystyle V}$ is an inner product space, then ${\displaystyle \|v\|={\sqrt {\langle v,v\rangle }}}$ is a norm.

### Euclidean Norm

${\displaystyle \mathbb {R} ^{n}}$, ${\displaystyle \langle x,y\rangle =x\cdot y}$, ${\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}}$ is a norm

Euclidean Norm given by ${\displaystyle \|x\|_{p}}$ is defined for ${\displaystyle p\geq 1}$ as

${\displaystyle \|x\|_{p}=\left(\sum _{i=1}^{n}|x_{i}|^{p}\right)^{\frac {1}{p}}}$

Supreme norm ${\displaystyle \|x\|_{\infty }=\max _{1\leq i\leq n}|x_{i}|}$

#### Example

For ${\displaystyle {\vec {x}}=\langle 4,-5,3\rangle \in \mathbb {R} ^{3}}$

• ${\displaystyle \|x\|_{1}=4+5+3=12}$
• ${\displaystyle \|x\|_{2}={\sqrt {16+25+9}}=5{\sqrt {2}}}$
• ${\displaystyle \|x\|_{\infty }=5}$

### Applications in Orthogonality

Inner product space ${\displaystyle V}$, ${\displaystyle \langle \cdot ,\cdot \langle }$

For vectors ${\displaystyle \{v_{1},\ldots ,v_{n}\}\subset V}$ such that ${\displaystyle \langle v_{i},v_{j}\rangle =0}$ when ${\displaystyle i\neq j}$, ${\displaystyle \{v_{1},\ldots ,v_{n}\}}$ is an orthogonal set of vectors.

If we transform ${\displaystyle \{v_{1},\ldots ,v_{n}\}}$ into a set of unit vectors ${\displaystyle \{u_{1},\ldots ,u_{n}\}}$, then

${\displaystyle \langle u_{i},u_{j}\rangle =\delta _{ij}={\begin{cases}0&i\neq j\\1i=j\end{cases}}}$

This is called an orthonormal set

#### Theorem 5.5.1

If ${\displaystyle \{v_{1},\ldots ,v_{n}\}}$ is an orthogonal set of nonzero vectors, then the vectors are linearly independent.

{\displaystyle {\begin{aligned}\langle v_{j},c_{1}\,v_{1}+\dots +c_{n}\,v_{n}\rangle &=\langle v_{j},0\rangle =0\\&=c_{1}\langle v_{j},v_{1}\rangle +\dots +c_{j}\langle v_{j},v_{j}\rangle +\dots +c_{n}\langle v_{j},v_{n}\rangle \\&=c_{j}\langle v_{j},v_{j}\rangle =0\iff c_{j}=0\end{aligned}}}

#### Theorem 5.5.2

An orthonormal set ${\displaystyle B=\{u_{1},\ldots ,u_{k}\}}$ is a basis for ${\displaystyle S=\mathrm {Span} \{u_{1},\ldots ,u_{k}\}}$ and is called an orthonormal basis

Let ${\displaystyle \{u_{1},\ldots ,u_{n}\}}$ be an orthonormal basis for ${\displaystyle V}$.

${\displaystyle V=\sum _{i=1}^{n}c_{i}\,u_{i}\quad \implies \quad c_{i}=\langle v,u_{i}\rangle }$
##### Corollary

For an orthonormal basis ${\displaystyle \{u_{1},\ldots ,u_{n}\}}$,

${\displaystyle v=\sum _{i=1}^{n}a_{i}\,u_{i}\quad {\text{and}}\quad w=\sum _{i=1}^{n}b_{i}\,u_{i}\quad \implies \quad \langle v,w\rangle =\sum _{i=1}^{n}a_{i}\,b_{i}}$
##### Corollary: Parseval's Formula

For an orthonormal basis ${\displaystyle \{u_{1},\ldots ,u_{n}\}}$,

${\displaystyle v=\sum _{i=1}6nc_{i}\,u_{i}\quad \implies \quad \|v\|^{2}=\sum _{i=1}^{n}c_{i}^{2}}$