15.7 Types of bases

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: $${\tilde{b}}_0=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\ 1\\ \end{pmatrix}$$ $${\tilde{b}}_1=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\ -1\\ \end{pmatrix}$$ Normalized with ${\ell }^1$ norm: $${\tilde{b}}_0=\frac{1}{2}\begin{pmatrix}1\\ 1\\ \end{pmatrix}$$ $${\tilde{b}}_1=\frac{1}{2}\begin{pmatrix}1\\ -1\\ \end{pmatrix}$$

Standard basis for ${ℝ}^2$ , also referred to as ${\ell }^2(\left[0 , 1\right])$ : $$b_0=\begin{pmatrix}1\\ 0\\ \end{pmatrix}$$ $$b_1=\begin{pmatrix}0\\ 1\\ \end{pmatrix}$$ $$b_0\cdot b_1=\sum_{i=0}^1 b_0(i)b_1(i)=1\times 0+0\times 1=0$$

Now we have the following basis and relationship: $$\{\begin{pmatrix}1\\ 1\\ \end{pmatrix}, \begin{pmatrix}1\\ -1\\ \end{pmatrix}\}=\{h_0, h_1\}$$ $$h_0\cdot h_1=1\times 1+1\times -1=0$$

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

$$\{b_0, b_2\}=\{\frac{1}{\sqrt{2}}\begin{pmatrix}1\\ 1\\ \end{pmatrix}, \frac{1}{\sqrt{2}}\begin{pmatrix}1\\ -1\\ \end{pmatrix}\}$$

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 $$f\cdot e^{i{\omega }_{0}nt}=\frac{1}{\sqrt{T}}\int_0^T f(t)\overline{e^{i{\omega }_{0}nt}}\,d t=\frac{1}{\sqrt{T}}\int_0^T f(t)e^{-(i{\omega }_{0}nt)}\,d t$$