# 15.7 Types of bases

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This module discusses the different types of basis that leads up to the definition of an orthonormal basis. Examples are given and the useful of the orthonormal basis is discussed.

## Normalized basis

Normalized Basis
a basis $\{{b}_{i}\}$ where each ${b}_{i}$ has unit norm
$\forall i, i\in \mathbb{Z}\colon ({b}_{i})=1$
The concept of basis applies to all vector spaces . The concept of normalized basis applies only to normed spaces .
You can always normalize a basis: just multiply each basis vector by a constant, such as $\frac{1}{({b}_{i})}$

We are given the following basis: $\{{b}_{0}, {b}_{1}\}=\{\begin{pmatrix}1\\ 1\\ \end{pmatrix}, \begin{pmatrix}1\\ -1\\ \end{pmatrix}\}$ Normalized with ${\ell }^{2}$ norm: ${\stackrel{~}{b}}_{0}=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\ 1\\ \end{pmatrix}$ ${\stackrel{~}{b}}_{1}=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\ -1\\ \end{pmatrix}$ Normalized with ${\ell }^{1}$ norm: ${\stackrel{~}{b}}_{0}=\frac{1}{2}\begin{pmatrix}1\\ 1\\ \end{pmatrix}$ ${\stackrel{~}{b}}_{1}=\frac{1}{2}\begin{pmatrix}1\\ -1\\ \end{pmatrix}$

## Orthogonal basis

Orthogonal Basis
a basis $\{{b}_{i}\}$ in which the elements are mutually orthogonal $\forall i, i\neq j\colon {b}_{i}\dot {b}_{j}=0$
The concept of orthogonal basis applies only to Hilbert Spaces .

Standard basis for ${ℝ}^{2}$ , also referred to as ${\ell }^{2}(\left[0 , 1\right])$ : ${b}_{0}=\begin{pmatrix}1\\ 0\\ \end{pmatrix}$ ${b}_{1}=\begin{pmatrix}0\\ 1\\ \end{pmatrix}$ ${b}_{0}\dot {b}_{1}=\sum_{i=0}^{1} {b}_{0}(i){b}_{1}(i)=1\times 0+0\times 1=0$

Now we have the following basis and relationship: $\{\begin{pmatrix}1\\ 1\\ \end{pmatrix}, \begin{pmatrix}1\\ -1\\ \end{pmatrix}\}=\{{h}_{0}, {h}_{1}\}$ ${h}_{0}\dot {h}_{1}=1\times 1+1\times -1=0$

## Orthonormal basis

Pulling the previous two sections (definitions) together, we arrive at the most important and useful basis type:

Orthonormal Basis
a basis that is both normalized and orthogonal $\forall i, i\in \mathbb{Z}\colon ({b}_{i})=1$ $\forall i, i\neq j\colon {b}_{i}\dot {b}_{j}$
We can shorten these two statements into one: ${b}_{i}\dot {b}_{j}={\delta }_{ij}$ where ${\delta }_{ij}=\begin{cases}1 & \text{if i=j}\\ 0 & \text{if i\neq j}\end{cases}$ Where ${\delta }_{ij}$ is referred to as the Kronecker delta function and is also often written as $\delta (i-j)$ .

## Orthonormal basis example #1

$\{{b}_{0}, {b}_{2}\}=\{\begin{pmatrix}1\\ 0\\ \end{pmatrix}, \begin{pmatrix}0\\ 1\\ \end{pmatrix}\}$

## Orthonormal basis example #2

$\{{b}_{0}, {b}_{2}\}=\{\begin{pmatrix}1\\ 1\\ \end{pmatrix}, \begin{pmatrix}1\\ -1\\ \end{pmatrix}\}$

## Orthonormal basis example #3

$\{{b}_{0}, {b}_{2}\}=\{\frac{1}{\sqrt{2}}\begin{pmatrix}1\\ 1\\ \end{pmatrix}, \frac{1}{\sqrt{2}}\begin{pmatrix}1\\ -1\\ \end{pmatrix}\}$

## Beauty of orthonormal bases

Orthonormal bases are very easy to deal with! If $\{{b}_{i}\}$ is an orthonormal basis, we can write for any $x$

$x=\sum {\alpha }_{i}{b}_{i}$
It is easy to find the ${\alpha }_{i}$ :
$x\dot {b}_{i}=\sum {\alpha }_{k}{b}_{k}\dot {b}_{i}=\sum {\alpha }_{k}({b}_{k}\dot {b}_{i})$
where in the above equation we can use our knowledge of thedelta function to reduce this equation: ${b}_{k}\dot {b}_{i}={\delta }_{ik}=\begin{cases}1 & \text{if i=k}\\ 0 & \text{if i\neq k}\end{cases}$
$x\dot {b}_{i}={\alpha }_{i}$
Therefore, we can conclude the following important equation for $x$ :
$x=\sum (x\dot {b}_{i}){b}_{i}$
The ${\alpha }_{i}$ 's are easy to compute (no interaction between the ${b}_{i}$ 's)

Given the following basis: $\{{b}_{0}, {b}_{1}\}=\{\frac{1}{\sqrt{2}}\begin{pmatrix}1\\ 1\\ \end{pmatrix}, \frac{1}{\sqrt{2}}\begin{pmatrix}1\\ -1\\ \end{pmatrix}\}$ represent $x=\begin{pmatrix}3\\ 2\\ \end{pmatrix}$

## Slightly modified fourier series

We are given the basis $(n, )$ 1 T ω 0 n t on ${L}^{2}(\left[0 , T\right]())$ where $T=\frac{2\pi }{{\omega }_{0}}$ . $f(t)=\sum$ f ω 0 n t ω 0 n t 1 T Where we can calculate the above inner product in ${L}^{2}$ as $f\dot e^{i{\omega }_{0}nt}=\frac{1}{\sqrt{T}}\int_{0}^{T} f(t)\overline{e^{i{\omega }_{0}nt}}\,d t=\frac{1}{\sqrt{T}}\int_{0}^{T} f(t)e^{-(i{\omega }_{0}nt)}\,d t$

## Orthonormal basis expansions in a hilbert space

Let $\{{b}_{i}\}$ be an orthonormal basis for a Hilbert space $H$ . Then, for any $x\in H$ we can write

$x=\sum {\alpha }_{i}{b}_{i}$
where ${\alpha }_{i}=x\dot {b}_{i}$ .
• "Analysis": decomposing $x$ in term of the ${b}_{i}$
${\alpha }_{i}=x\dot {b}_{i}$
• "Synthesis": building $x$ up out of a weighted combination of the ${b}_{i}$
$x=\sum {\alpha }_{i}{b}_{i}$

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