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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types of demand and the explanation
what is demand
other things remaining same if demend is increases supply is also decrease and if demend is decrease supply is also increases is called the demand
Mian
if the demand increase supply also increases
Thembi
you are wrong this is the law of demand and not the definition
Tarasum
Demand is the willingness of buy and ability to buy in a specific time period in specific place. Mian you are saying law of demand but not in proper way. you have to keep studying more. because its very basic things in Economics.
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founder , that is Adam Smith
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the wealth of nations, is it the first?
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Yes very sure it was released in 1759
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thank you Yusuf.
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then when did he died?
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17 July 1790 Born: 16 June 1723, Kirkcaldy, United Kingdom Place of death: Panmure House, Edinburgh, United Kingdom
Yusuf
1790
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that's my today questions, thank you Yusuf it's bed time see u after.
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mode is the most commonly occurring item in a set of data.
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is there monopsony word?
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I have no idea though
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monopsony is when there's only one buyer while monopoly is when there's only one producer.
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Monopoly is when there's excessively one seller and there is no entry in the market while monopsony is when there is one buyer
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(uncountable) Good humoured, playful, typically spontaneous conversation. verb (intransitive) To engage in banter or playful conversation. (intransitive) To play or do something amusing. (transitive) To tease mildly.
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wealth on nation, 1776
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