# MATH 323 Lecture 13

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Brain dump on Vector Space

## Review

1. span: ${\displaystyle \mathrm {Span} (v_{1},\ldots ,v_{n})\subseteq V}$
2. linear independence: ${\displaystyle c_{1}\,v_{1}+\dots +c_{n}\,v_{n}=0}$
3. if 1 & 2, then ${\displaystyle v}$'s define a basis (${\displaystyle w=c_{1}\,v_{1}+\dots +c_{n}\,v_{n}\quad \forall w\in V}$
4. cardinality of basis is dimension
• if ${\displaystyle \{v_{i}\}_{i=1}^{n}}$ and ${\displaystyle \{w_{j}\}_{j=1}^{k}}$ are bases for ${\displaystyle V}$, then ${\displaystyle n=k}$.
• Note: ${\displaystyle \dim V=0\iff V=\{0\}}$

## Bases of Subspace

For a vector space ${\displaystyle V}$ with dimension less than ∞, if ${\displaystyle W\subset V}$, then ${\displaystyle \dim W<\dim V}$.

## Infinite Dimension

Let ${\displaystyle P=\bigcup _{n=1}^{\infty }P_{n}}$ represent the space of polynomials. Then ${\displaystyle \dim P_{n}=|\{1,x,\ldots ,x^{n-1}\}|=n}$ and ${\displaystyle \dim P=\infty }$

## Column and Row Space

For a ${\displaystyle m\times n}$ matrix, ${\displaystyle M=(v_{1},\ldots ,v_{n})}$, where ${\displaystyle v}$'s are columns.

${\displaystyle \mathrm {Span} (v_{1},\ldots ,v_{n})\subseteq \mathrm {R} ^{m}}$ is called the column space of ${\displaystyle M}$.

${\displaystyle \mathrm {Span} ({\vec {w}}:{\mbox{rows of }}M)\subseteq \mathrm {R} ^{n}}$ is called row space of ${\displaystyle M}$.

## Theorem 3.4.4

For vector space ${\displaystyle V}$ with dimension ${\displaystyle \dim V=n>0}$, then

1. no set of fewer than ${\displaystyle n}$ vectors can span ${\displaystyle V}$;
2. any subset of fewer than ${\displaystyle n}$ linearly independent vectors can be extended (i.e. vectors can be added) to form a basis for ${\displaystyle V}$; and
3. any spanning set containing more than ${\displaystyle n}$ vectors can be pared down (i.e. vectors can be removed) to form a basis for ${\displaystyle V}$.

## Bases in Euclidean Space

${\displaystyle {\vec {v}}_{1},\ldots ,{\vec {v}}_{n}\in \mathbb {R} ^{n}}$ is a basis iff ${\displaystyle \left|M\right|\neq 0}$, where ${\displaystyle M=\left({\vec {v}}_{1},\ldots ,{\vec {v}}_{n}\right)}$ is a ${\displaystyle n\times n}$ matrix.

### Standard Bases

${\displaystyle {\hat {\imath }},{\hat {\jmath }},{\hat {k}}}$ is standard basis in ${\displaystyle \mathbb {R} ^{3}}$.

A standard basis for ${\displaystyle \mathbb {R} ^{n}}$ is a set ${\displaystyle \{{\vec {e}}_{1},\ldots ,{\vec {e}}_{n}\}}$ such that ${\displaystyle i}$th entry of ${\displaystyle {\vec {e}}_{i}}$ is ${\displaystyle 1}$, and all other entries are ${\displaystyle 0}$

Let ${\displaystyle [a,b]}$ denote an ordered set representation, such that ${\displaystyle [a,b]\neq [b,a]}$. ${\displaystyle [{\vec {e}}_{1},{\vec {e}}_{2}]}$ will denote an ordered basis for ${\displaystyle \mathbb {R} ^{2}}$, for example.

## Conversion of Bases

### To Standard Basis

For ${\displaystyle {\vec {u}}_{1},{\vec {u}}_{2}\in \mathbb {R} ^{2}}$, ${\displaystyle \mathrm {Span} ({\vec {u}}_{1},{\vec {u}}_{2})=\mathbb {R} ^{2}}$, and let ${\displaystyle [{\vec {u}}_{1},{\vec {u}}_{2}]}$ be a second basis (i.e. not standard).

We can use basis vectors to describe any point ${\displaystyle {\vec {x}}=c_{1}\,{\vec {u}}_{1}+c_{2}\,{\vec {u}}_{2}}$ in ${\displaystyle \mathbb {R} ^{2}}$.

1. Given ${\displaystyle {\vec {x}}=\left\langle x_{1},x_{2}\right\rangle }$, find its coordinates w.r.t. ${\displaystyle [{\vec {u}}_{1},{\vec {u}}_{2}]}$
2. Given ${\displaystyle c_{1}\,{\vec {u}}_{1}+c_{2}\,{\vec {u}}_{2}}$, find its coordinates w.r.t. ${\displaystyle [{\vec {e}}_{1},{\vec {e}}_{2}]}$.

### Examples

${\displaystyle {\vec {u}}_{1}=\left\langle 3,2\right\rangle =3{\vec {e}}_{1}+2{\vec {e}}_{2}}$ and ${\displaystyle {\vec {u}}_{2}=\left\langle 1,1\right\rangle ={\vec {e}}_{1}+{\vec {e}}_{2}}$.

Case 2: ${\displaystyle c_{1}\,{\vec {u}}_{1}+c_{2}\,{\vec {u}}_{2}=c_{1}(3{\vec {e}}_{1}+2{\vec {e}}_{2})+c_{2}({\vec {e}}_{1}+{\vec {e}}_{2})=(3c_{1}+c_{2}){\vec {e}}_{1}+(2c_{1}+c_{2}){\vec {e}}_{2}}$.

Case 1: ${\displaystyle {\vec {x}}=\left\langle 3c_{1}+c_{2},2c_{1}+c_{2}\right\rangle ={\begin{bmatrix}3&1\\2&1\end{bmatrix}}\,{\begin{bmatrix}c_{1}\\c_{2}\end{bmatrix}}}$.

Note: The 2×2 matrix ${\displaystyle U=({\vec {u}}_{1},{\vec {u}}_{2})}$ is called the transition matrix.

Assuming ${\displaystyle |U|\neq 0}$, ${\displaystyle {\vec {c}}=U^{-1}\,{\vec {x}}}$

### To Arbitrary Basis

${\displaystyle [{\vec {v}}_{1},{\vec {v}}_{2}]\to [{\vec {u}}_{1},{\vec {u}}_{2}]}$

${\displaystyle {\vec {x}}=c_{1}\,{\vec {v}}_{1}+c_{2}\,{\vec {v}}_{2}=d_{1}\,{\vec {u}}_{1}+d_{2}\,{\vec {u}}_{2}}$

Let ${\displaystyle V=({\vec {v}}_{1},{\vec {v}}_{2})}$ and ${\displaystyle U=({\vec {u}}_{1}{\vec {u}}_{2})}$. Then (assuming ${\displaystyle V}$ and ${\displaystyle U}$ are nonsingular)

{\displaystyle {\begin{aligned}V\,{\vec {c}}&=U\,{\vec {d}}\\{\vec {d}}&=U^{-1}\,V\,{\vec {c}}\end{aligned}}}

The product ${\displaystyle S=U^{-1}\,V}$ is the tranformation matrix from ${\displaystyle [{\vec {v}}_{1},{\vec {v}}_{2}]}$ to ${\displaystyle [{\vec {u}}_{1},{\vec {u}}_{2}]}$

### Summary

Cases for transformation matrix ${\displaystyle S}$

1. ${\displaystyle [{\vec {v}}_{1},{\vec {v}}_{2}]\to [{\vec {e}}_{1},{\vec {e}}_{2}]:S=V}$
2. ${\displaystyle [{\vec {v}}_{1},{\vec {v}}_{2}]\to [{\vec {u}}_{1},{\vec {u}}_{2}]:S=U^{-1}\,V}$
3. ${\displaystyle [{\vec {e}}_{1},{\vec {e}}_{2}]\to [{\vec {u}}_{1},{\vec {u}}_{2}]:S=U^{-1}}$

Note that ${\displaystyle S}$ for standard basis ${\displaystyle [{\vec {e}}_{1},{\vec {e}}_{2}]}$ is the identity matrix, so ${\displaystyle S=U^{-1}I=U^{-1}}$

## Coordinate Vector

For vector space ${\displaystyle V}$ with dimension ${\displaystyle \dim V=n}$, let ${\displaystyle E=[v_{1},v_{2},\ldots ,v_{n}]}$ be the ordered basis for ${\displaystyle V}$. Thus ${\displaystyle v=c_{1}\,v_{1}+c_{2}\,v_{2}+\dots +c_{n}\,v_{n}\quad \forall v\in V}$.

The vector ${\displaystyle {\vec {c}}=\left\langle c_{1},c_{2},\ldots ,c_{n}\right\rangle \in \mathbb {R} ^{n}}$ is called the coordinate vector of ${\displaystyle V}$ w.r.t. ${\displaystyle E}$ and is denoted ${\displaystyle {\vec {c}}=[V]_{E}}$.

Therefore, the previous section could be rewritten as

${\displaystyle [X]_{F}=S[X]_{E}\,\!}$
where ${\displaystyle E}$ and ${\displaystyle F}$ are ordered bases for ${\displaystyle U}$ and ${\displaystyle V}$, respectively, and ${\displaystyle S=U^{-1}V}$

## Review

Maximum number of points = 50 + 7 pt. bonus question

Look over Determinants, null spaces, and solving linear systems of equations.