MATH 323 Lecture 17

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Linear Transformation

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

$A=({\vec {a}}_{i}=L({\vec {e}}_{i})$

Rotation Transformation

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

${\begin{aligned}L({\vec {e}}_{1})&={\begin{pmatrix}\cos {\theta }\\\sin {\theta }\end{pmatrix}}L({\vec {e}}_{2})&={\begin{pmatrix}-\sin {\theta }\\\cos {\theta }\end{pmatrix}}\\A&={\begin{pmatrix}\cos {\theta }&-\sin {\theta }\\\sin {\theta }&\cos {\theta }\end{pmatrix}}\end{aligned}}$

In General

For arbitrary $L:V\to W$, where $E=[v_{1},\ldots ,v_{n}]$ is a basis in $V$ and $F=[w_{1},\ldots ,w_{m}]$ is a basis in $W$.

$v\to [v]_{E}\in \mathbb {R} ^{n}$ is the coordinate vector of $v$ w.r.t $E$ so $v=\sum _{i=1}^{n}x_{i}\,v_{i}=\left\langle x_{1},\ldots ,x_{n}\right\rangle$

For some $\sum _{i=1}^{m}y_{i}\,w_{i}=L(v)\in W$, we can represent $[L(v)]_{F}$ as $\left\langle y_{1},\ldots ,y_{m}\right\rangle \in \mathbb {R} ^{m}$.

${\vec {y}}=A{\vec {x}}\iff [L(v)]_{F}=A[v]_{E}$ , so what is $A$?

${\vec {a}}_{j}=[L(v_{j})]_{F}$ for $j=1,\ldots ,n$

$A=({\vec {a}}_{i},\ldots ,{\vec {a}}_{n})$

Theorem 4.2.2

Matrix representation theorem

If $E=[v_{1},\ldots ,v_{n}]$ and $F=[w_{1},\ldots ,w_{m}]$ 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, $A=({\vec {a}}_{1},\ldots ,{\vec {a}}_{n})$, where ${\vec {a}}_{j}=[L(v_{j})]_{F}$ for $j=1,\ldots ,n$.

Examples

For ${\vec {x}}\in \mathbb {R} ^{3}$ and $[{\vec {b}}_{1}=\left\langle 1,1\right\rangle ,{\vec {b}}_{2}=\left\langle -1,1\right\rangle ]$ is a basis in $\mathbb {R} ^{2}$, find the matrix $A$ representing $L({\vec {x}})=x_{1}{\vec {b}}_{1}+(x_{2}+x_{3}){\vec {b}}_{2}$ w.r.t. ordered bases $[{\vec {e}}_{1},{\vec {e}}_{2},{\vec {e}}_{3}]$ (standard 3D basis) to $B=[{\vec {b}}_{1},{\vec {b}}_{2}]$

Just take $A=\left(a_{j}=L({\vec {e}}_{i})\right)={\begin{pmatrix}1&0&0\\0&1&1\end{pmatrix}}$

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$L(\alpha \,{\vec {b}}_{1}+\beta \,{\vec {b}}_{2})=(\alpha +\beta ){\vec {b}}_{1}+2\beta \,{\vec {b}}_{2}$, where $[{\vec {b}}_{1},{\vec {b}}_{2}]$ is a basis for $\mathbb {R} ^{2}$. Find $A$.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = (L(1 \cdot \vec{b}_1 + 0 \cdot \vec{b}_2), L(0 \cdot \vec{b}_1 + 1 \cdot \vec{b}_2)) = \begin{pmatrix}1&1\\0&2\end{pmatrix}}

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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D:P_3 \to P_2} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D(p(x)) = p'(x)} .

$[x^{2},x,1]$ forms basis for $P_{3}$, and $[x,1]$ forms basis for $P_{2}$

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

Thus $A={\begin{bmatrix}2&0&0\\0&1&0\end{bmatrix}}$

${\begin{aligned}{[p(x)]}_{[x^{2},x,1]}&=\left\langle a,b,c\right\rangle \\{[D(p(x))]}_{[x,1]}&=\left\langle 2a,b\right\rangle \end{aligned}}$

Reversing Linear Transformations

Theorem 4.2.3

Given $E=[{\vec {u}}_{1},\ldots {\vec {u}}_{n}]$ and $F=[{\vec {b}}_{1},\ldots ,{\vec {b}}_{m}]$ are bases for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^n} and $\mathbb {R} ^{m}$ respectively,

If $A$ is the matrix representing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L:\mathbb{R}^n \to \mathbb{R}^m} w.r.t. $E$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} , then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{a}_j = B^{-1} L(\vec{u}_j)} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j=1,\ldots,n} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B = (\vec{b}_1, \ldots,\vec{b}_m)}

Corollary 4.2.4

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is the matrix representing the linear transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L:\mathbb{R}^n \to \mathbb{R}^m} w.r.t. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} , then the rref of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\vec{b}_1, \ldots,\vec{b}_m \mid L(\vec{u}_1), \ldots, L(\vec{u}_n))} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (I \mid A)} .

Example

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L: \mathbb{R}^2 \to \mathbb{R}^3} ,

Basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\vec{u}_1,\vec{u}_2]} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u}_1 = \left\langle 1,2 \right\rangle} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_2 = \left\langle 3,1 \right\rangle}

Basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\vec{b}_1, \vec{b}_2,\vec{b}_3]} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{b}_1 = \left\langle 1,0,0 \right\rangle} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{b}_2 = \left\langle 1,1,0 \right\rangle} , ${\vec {b}}_{3}=\left\langle 1,1,1\right\rangle$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(\vec{x}) = \begin{pmatrix}x_2\\x_1+x_2\\x_1-x_2\end{pmatrix}}

What is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} w.r.t. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\vec{u}_1,\vec{u}_2]} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\vec{b}_1, \vec{b}_2,\vec{b}_3]} ?

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} L(\vec{u}_1) &= \left\langle 2,3,-1 \right\rangle \\ L(\vec{u}_2) &= \left\langle 1,4,2 \right\rangle \end{align}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ \begin{array}{ccc|cc} 1&1&1&2&1\\ 0&1&1&3&4\\ 0&0&1&-1&2 \end{array}\right] \quad\longrightarrow\quad \left[\begin{array}{ccc|cc} 1&0&0&-1&-3\\ 0&1&0&4&2\\ 0&0&1&-1&2 \end{array}\right]}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = \begin{pmatrix}-1&-3\\4&2\\-1&2\end{pmatrix}}