MATH 323 Lecture 17

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Linear Transformation

$\mathrm{\forall }L:{\mathbb{ℝ}}^n\to {\mathbb{ℝ}}^m\mathrm{\exists }A\in {\mathbb{ℝ}}^{m\times n}$

$A=({\overrightarrow{a}}_i=L({\overrightarrow{e}}_i)$

Rotation Transformation

Let $L$ be a rotation by angle $\theta$ about the origin.

$\begin{array}{rlr}L({\overrightarrow{e}}_1)& =\left(\begin{array}{c}\cos \theta \\ \sin \theta \end{array}\right)L({\overrightarrow{e}}_2)& =\left(\begin{array}{c}-\sin \theta \\ \cos \theta \end{array}\right)\\ A& =\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)\end{array}$

In General

For arbitrary $L:V\to W$, where $E=[v_1,\dots ,v_n]$ is a basis in $V$ and $F=[w_1,\dots ,w_m]$ is a basis in $W$.

$v\to [v{]}_E\in {\mathbb{ℝ}}^n$ is the coordinate vector of $v$ w.r.t $E$ so $v=\sum _{i=1}^n{x}_i{v}_i=⟨x_1,\dots ,x_n⟩$

For some $\sum _{i=1}^m{y}_i{w}_i=L(v)\in W$, we can represent $[L(v){]}_F$ as $⟨y_1,\dots ,y_m⟩\in {\mathbb{ℝ}}^m$.

$\overrightarrow{y}=A\overrightarrow{x}⟺[L(v){]}_F=A[v{]}_E$ , so what is $A$?

${\overrightarrow{a}}_j=[L(v_j){]}_F$ for $j=1,\dots ,n$

$A=({\overrightarrow{a}}_i,\dots ,{\overrightarrow{a}}_n)$

Theorem 4.2.2

Matrix representation theorem

If $E=[v_1,\dots ,v_n]$ and $F=[w_1,\dots ,w_m]$ are ordered bases for vector spaces $V$ and $W$ respectively, then corresponding to each linear transformation $L:V\to W$ there is an $m\times n$ matrix $A$ such that

$[L(v){]}_F=A[v{]}_E$ for each $v\in V$

$A$ is the matrix representing $L$ relative to the ordered bases $E$ and $F$.

In fact, $A=({\overrightarrow{a}}_1,\dots ,{\overrightarrow{a}}_n)$, where ${\overrightarrow{a}}_j=[L(v_j){]}_F$ for $j=1,\dots ,n$.

Examples

For $\overrightarrow{x}\in {\mathbb{ℝ}}^3$ and $[{\overrightarrow{b}}_1=⟨1,1⟩,{\overrightarrow{b}}_2=⟨-1,1⟩]$ is a basis in ${\mathbb{ℝ}}^2$, find the matrix $A$ representing $L(\overrightarrow{x})=x_1{\overrightarrow{b}}_1+(x_2+x_3){\overrightarrow{b}}_2$ w.r.t. ordered bases $[{\overrightarrow{e}}_1,{\overrightarrow{e}}_2,{\overrightarrow{e}}_3]$ (standard 3D basis) to $B=[{\overrightarrow{b}}_1,{\overrightarrow{b}}_2]$

Just take $A=(a_j=L({\overrightarrow{e}}_i))=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\end{array}\right)$

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$L(\alpha {\overrightarrow{b}}_1+\beta {\overrightarrow{b}}_2)=(\alpha +\beta ){\overrightarrow{b}}_1+2\beta {\overrightarrow{b}}_2$, where $[{\overrightarrow{b}}_1,{\overrightarrow{b}}_2]$ is a basis for ${\mathbb{ℝ}}^2$. Find $A$.

$A=(L(1\cdot {\overrightarrow{b}}_1+0\cdot {\overrightarrow{b}}_2),L(0\cdot {\overrightarrow{b}}_1+1\cdot {\overrightarrow{b}}_2))=\left(\begin{array}{cc}1& 1\\ 0& 2\end{array}\right)$

---

$D:P_3\to P_2$, where $D(p(x))=p^{\prime }(x)$.

$[x^2,x,1]$ forms basis for $P_3$, and $[x,1]$ forms basis for $P_2$

$\begin{array}{rl}D(x^2)& =2x=2\cdot x+0\cdot 1\\ D(x)& =1=0\cdot x+1\cdot 1\\ D(1)& =0=0\cdot x+0\cdot 1\\ D(a{x}^2+bx+c)& =2ax+b\end{array}$

Thus $A=\left[\begin{array}{ccc}2& 0& 0\\ 0& 1& 0\end{array}\right]$

$\begin{array}{rl}{[p(x)]}_{[x^2,x,1]}& =⟨a,b,c⟩\\ {[D(p(x))]}_{[x,1]}& =⟨2a,b⟩\end{array}$

Reversing Linear Transformations

Theorem 4.2.3

Given $E=[{\overrightarrow{u}}_1,\dots {\overrightarrow{u}}_n]$ and $F=[{\overrightarrow{b}}_1,\dots ,{\overrightarrow{b}}_m]$ are bases for ${\mathbb{ℝ}}^n$ and ${\mathbb{ℝ}}^m$ respectively,

If $A$ is the matrix representing $L:{\mathbb{ℝ}}^n\to {\mathbb{ℝ}}^m$ w.r.t. $E$ and $F$, then

${\overrightarrow{a}}_j=B^{-1}L({\overrightarrow{u}}_j)$ for $j=1,\dots ,n$, where $B=({\overrightarrow{b}}_1,\dots ,{\overrightarrow{b}}_m)$

Corollary 4.2.4

If $A$ is the matrix representing the linear transformation $L:{\mathbb{ℝ}}^n\to {\mathbb{ℝ}}^m$ w.r.t. $E$ and $F$, then the rref of $({\overrightarrow{b}}_1,\dots ,{\overrightarrow{b}}_m\mid L({\overrightarrow{u}}_1),\dots ,L({\overrightarrow{u}}_n))$ is $(I\mid A)$.

Example

$L:{\mathbb{ℝ}}^2\to {\mathbb{ℝ}}^3$,

Basis $[{\overrightarrow{u}}_1,{\overrightarrow{u}}_2]$ is ${\overrightarrow{u}}_1=⟨1,2⟩$, $u_2=⟨3,1⟩$

Basis $[{\overrightarrow{b}}_1,{\overrightarrow{b}}_2,{\overrightarrow{b}}_3]$ is ${\overrightarrow{b}}_1=⟨1,0,0⟩$, ${\overrightarrow{b}}_2=⟨1,1,0⟩$, ${\overrightarrow{b}}_3=⟨1,1,1⟩$

$L(\overrightarrow{x})=\left(\begin{array}{c}{x}_2\\ x_1+x_2\\ x_1-x_2\end{array}\right)$

What is $A$ w.r.t. $[{\overrightarrow{u}}_1,{\overrightarrow{u}}_2]$ and $[{\overrightarrow{b}}_1,{\overrightarrow{b}}_2,{\overrightarrow{b}}_3]$?

$\begin{array}{rl}L({\overrightarrow{u}}_1)& =⟨2,3,-1⟩\\ L({\overrightarrow{u}}_2)& =⟨1,4,2⟩\end{array}$

$\left[\begin{array}{ccccc}1& 1& 1& 2& 1\\ 0& 1& 1& 3& 4\\ 0& 0& 1& -1& 2\end{array}\right]⟶\left[\begin{array}{ccccc}1& 0& 0& -1& -3\\ 0& 1& 0& 4& 2\\ 0& 0& 1& -1& 2\end{array}\right]$

$A=\left(\begin{array}{cc}-1& -3\\ 4& 2\\ -1& 2\end{array}\right)$