# MATH 323 Lecture 22

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## Inner Product and Norm

### Cauchy-Schwarz Inequality

$|\langle u,v\rangle |\leq \|u\|\|v\|$ Can be rewritten as

$-1\leq \underbrace {\frac {\langle u,v\rangle }{\|u\|\|v\|}} _{\cos {\theta }}\leq 1$ $V$ is normed linear space if with each $v\in V$ , the number \|v\| is associated:

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

### Theorem 5.4.3

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

### Euclidean Norm

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

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

$\|x\|_{p}=\left(\sum _{i=1}^{n}|x_{i}|^{p}\right)^{\frac {1}{p}}$ Supreme norm $\|x\|_{\infty }=\max _{1\leq i\leq n}|x_{i}|$ #### Example

For ${\vec {x}}=\langle 4,-5,3\rangle \in \mathbb {R} ^{3}$ • $\|x\|_{1}=4+5+3=12$ • $\|x\|_{2}={\sqrt {16+25+9}}=5{\sqrt {2}}$ • $\|x\|_{\infty }=5$ ### Applications in Orthogonality

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

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

$\langle u_{i},u_{j}\rangle =\delta _{ij}={\begin{cases}0&i\neq j\\1i=j\end{cases}}$ This is called an orthonormal set

#### Theorem 5.5.1

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

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

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

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

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

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

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

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