MATH 323 Lecture 13

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Brain dump on Vector Space

Review

1. span: $\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}(v_1,\dots ,v_n)\subseteq V$
2. linear independence: $c_1{v}_1+\dots +c_n{v}_n=0$
3. if 1 & 2, then $v$'s define a basis ($w=c_1{v}_1+\dots +c_n{v}_n\mathrm{\forall }w\in V$
4. cardinality of basis is dimension
   • if $\{v_i{\}}_{i=1}^n$ and $\{w_j{\}}_{j=1}^k$ are bases for $V$, then $n=k$.
   • Note: $\dim V=0⟺V=\{0\}$

Bases of Subspace

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

Infinite Dimension

Let $P=\bigcup _{n=1}^{\mathrm{\infty }}{P}_n$ represent the space of polynomials. Then $\dim P_n=|\{1,x,\dots ,x^{n-1}\}|=n$ and $\dim P=\mathrm{\infty }$

Column and Row Space

For a $m\times n$ matrix, $M=(v_1,\dots ,v_n)$, where $v$'s are columns.

$\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}(v_1,\dots ,v_n)\subseteq {\mathrm{R}}^m$ is called the column space of $M$.

$\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}(\overrightarrow{w}:\text{rows of }M)\subseteq {\mathrm{R}}^n$ is called row space of $M$.

Theorem 3.4.4

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

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

Bases in Euclidean Space

${\overrightarrow{v}}_1,\dots ,{\overrightarrow{v}}_n\in {\mathbb{ℝ}}^n$ is a basis iff $\left|M\right|\ne 0$, where $M=({\overrightarrow{v}}_1,\dots ,{\overrightarrow{v}}_n)$ is a $n\times n$ matrix.

Standard Bases

$\hat{ı},\hat{ȷ},\hat{k}$ is standard basis in ${\mathbb{ℝ}}^3$.

A standard basis for ${\mathbb{ℝ}}^n$ is a set $\{{\overrightarrow{e}}_1,\dots ,{\overrightarrow{e}}_n\}$ such that $i$^th entry of ${\overrightarrow{e}}_i$ is $1$, and all other entries are $0$

Let $[a,b]$ denote an ordered set representation, such that $[a,b]\ne [b,a]$. $[{\overrightarrow{e}}_1,{\overrightarrow{e}}_2]$ will denote an ordered basis for ${\mathbb{ℝ}}^2$, for example.

Conversion of Bases

To Standard Basis

For ${\overrightarrow{u}}_1,{\overrightarrow{u}}_2\in {\mathbb{ℝ}}^2$, $\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}({\overrightarrow{u}}_1,{\overrightarrow{u}}_2)={\mathbb{ℝ}}^2$, and let $[{\overrightarrow{u}}_1,{\overrightarrow{u}}_2]$ be a second basis (i.e. not standard).

We can use basis vectors to describe any point $\overrightarrow{x}=c_1{\overrightarrow{u}}_1+c_2{\overrightarrow{u}}_2$ in ${\mathbb{ℝ}}^2$.

1. Given $\overrightarrow{x}=⟨x_1,x_2⟩$, find its coordinates w.r.t. $[{\overrightarrow{u}}_1,{\overrightarrow{u}}_2]$
2. Given $c_1{\overrightarrow{u}}_1+c_2{\overrightarrow{u}}_2$, find its coordinates w.r.t. $[{\overrightarrow{e}}_1,{\overrightarrow{e}}_2]$.

Examples

${\overrightarrow{u}}_1=⟨3,2⟩=3{\overrightarrow{e}}_1+2{\overrightarrow{e}}_2$ and ${\overrightarrow{u}}_2=⟨1,1⟩={\overrightarrow{e}}_1+{\overrightarrow{e}}_2$.

Case 2: $c_1{\overrightarrow{u}}_1+c_2{\overrightarrow{u}}_2=c_1(3{\overrightarrow{e}}_1+2{\overrightarrow{e}}_2)+c_2({\overrightarrow{e}}_1+{\overrightarrow{e}}_2)=(3c_1+c_2){\overrightarrow{e}}_1+(2c_1+c_2){\overrightarrow{e}}_2$.

Case 1: $\overrightarrow{x}=⟨3c_1+c_2,2c_1+c_2⟩=\left[\begin{array}{cc}3& 1\\ 2& 1\end{array}\right]\left[\begin{array}{c}{c}_1\\ c_2\end{array}\right]$.

Note: The 2×2 matrix $U=({\overrightarrow{u}}_1,{\overrightarrow{u}}_2)$ is called the transition matrix.

Assuming $|U|\ne 0$, $\overrightarrow{c}=U^{-1}\overrightarrow{x}$

To Arbitrary Basis

$[{\overrightarrow{v}}_1,{\overrightarrow{v}}_2]\to [{\overrightarrow{u}}_1,{\overrightarrow{u}}_2]$

$\overrightarrow{x}=c_1{\overrightarrow{v}}_1+c_2{\overrightarrow{v}}_2=d_1{\overrightarrow{u}}_1+d_2{\overrightarrow{u}}_2$

Let $V=({\overrightarrow{v}}_1,{\overrightarrow{v}}_2)$ and $U=({\overrightarrow{u}}_1{\overrightarrow{u}}_2)$. Then (assuming $V$ and $U$ are nonsingular)

$\begin{array}{rl}V\overrightarrow{c}& =U\overrightarrow{d}\\ \overrightarrow{d}& =U^{-1}V\overrightarrow{c}\end{array}$

The product $S=U^{-1}V$ is the tranformation matrix from $[{\overrightarrow{v}}_1,{\overrightarrow{v}}_2]$ to $[{\overrightarrow{u}}_1,{\overrightarrow{u}}_2]$

Summary

Cases for transformation matrix $S$

1. $[{\overrightarrow{v}}_1,{\overrightarrow{v}}_2]\to [{\overrightarrow{e}}_1,{\overrightarrow{e}}_2]:S=V$
2. $[{\overrightarrow{v}}_1,{\overrightarrow{v}}_2]\to [{\overrightarrow{u}}_1,{\overrightarrow{u}}_2]:S=U^{-1}V$
3. $[{\overrightarrow{e}}_1,{\overrightarrow{e}}_2]\to [{\overrightarrow{u}}_1,{\overrightarrow{u}}_2]:S=U^{-1}$

Note that $S$ for standard basis $[{\overrightarrow{e}}_1,{\overrightarrow{e}}_2]$ is the identity matrix, so $S=U^{-1}I=U^{-1}$

Coordinate Vector

For vector space $V$ with dimension $\dim V=n$, let $E=[v_1,v_2,\dots ,v_n]$ be the ordered basis for $V$. Thus $v=c_1{v}_1+c_2{v}_2+\dots +c_n{v}_n\mathrm{\forall }v\in V$.

The vector $\overrightarrow{c}=⟨c_1,c_2,\dots ,c_n⟩\in {\mathbb{ℝ}}^n$ is called the coordinate vector of $V$ w.r.t. $E$ and is denoted $\overrightarrow{c}=[V{]}_E$.

Therefore, the previous section could be rewritten as

$[X{]}_F=S[X{]}_E$
where $E$ and $F$ are ordered bases for $U$ and $V$, respectively, and $S=U^{-1}V$

Review

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

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