MAT223: 9&10_Linear Transformations

Linear Transformations §

• Transformation (map) - another word for a function, used to describe changing vectors

  $$F:R^n\to R^m\text{defined by}\left[\begin{array}{c}x\\ y\end{array}\right]\mapsto \left[\begin{array}{c}x(...)\\ y(...)\end{array}\right]$$

  For $n,m\in \mathbb{N}$, and $(...)$ are real number operations for the transformation

  • The transformation is a set of rules that assigns each vector $\vec{x}$ in ${\mathbb{R}}^n$ to a vector $T(\vec{x})$ in ${\mathbb{R}}^m$

  The transformation $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$ defined by $\left[\begin{array}{c}x\\ y\end{array}\right]\mapsto \left[\begin{array}{c}x+3\\ y\end{array}\right]$ §

  

• Image of a Set - Let $L:{\mathbb{R}}^n\to {\mathbb{R}}^m$ be a transformation and let $X\subseteq {\mathbb{R}}^n$ be a set. The image of the set $X$ under $L$, denoted $L(X)$, is the set:

  $$L(x)=\{\vec{y}\in {\mathbb{R}}^m:\vec{y}=L(X)\text{ for some }\vec{x}\in X\}$$

  • “The set of all outputs of $L$ when the inputs comes from $X$”

  • For $\vec{x}$ in ${\mathbb{R}}^n$, the vector $T(\vec{x})$ in ${\mathbb{R}}^m$ is the image of the set

  Let $R:{\mathbb{R}}^2\to {\mathbb{R}}^2$ be rotation counter clockwise by $30^{\circ }$ and $X=\left\{x{\vec{e}}_1+y{\vec{e}}_2:x,y\in [0,1]\right\}$ be the filled-in unit squre. Then $R(X)$ is the filled-in unit squre that meets the axis at an angle of $30^{\circ }$ Describe this using set builder notation §

  

  Use the vector $(0,1),(1,0)$ in $X$ to find a matrix that would transform $X$ into $R(X)$ Calculate the corresponding vector in $R(X)$ using $\sin (30)$ and $\cos (30)$

  $$\begin{gathered} A=\left[\begin{array}{cc}a& c\\ b& d\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right]=\left[\begin{array}{c}1/2\\ \sqrt{3}/2\end{array}\right] \\ A=\left[\begin{array}{cc}a& c\\ b& d\end{array}\right]\left[\begin{array}{c}1\\ 0\end{array}\right]=\left[\begin{array}{c}\sqrt{3}/2\\ 1/2\end{array}\right] \\ A=\left[\begin{array}{cc}\sqrt[]{3}/2& -1/2\\ 1/2& \sqrt[]{3}/2\end{array}\right] \end{gathered}$$

  It is mathematically incorrect to multiply a matrix to a set ($AX\ne R(X)$) Get $R(X)=\left\{A(x{\vec{e}}_1+y{\vec{e}}_2):x,y\in [0,1]\right\}$

Linear Transformation §

• Linear Transformation, Let $V,W$ be subspaces. Function $T:V\to W$ is linear transformation if:

  $$T(\vec{u},\vec{v})=T(\vec{u})+T(\vec{v})\text{and}T(\alpha \vec{v})=\alpha T(\vec{v})$$

  For all vectors in $\vec{u},\vec{v}\in V$ and all scalars $\alpha$

  • $T(\text{span}\{\vec{u}\})=\text{span}\{T(\vec{u})\}$

  • If $T:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is a linear trarnsformation, then $T$ takes subspaces to subspaces

  • If $T:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is a linear transformation, then $T(\vec{0})=\vec{0}$ (because $T(\vec{0})=T(0\vec{v})=\vec{0}$ )

  • If $T:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is a linear transformation, then $T$ takes parallel lines to parallel lines (or points)

  Let $S:{\mathbb{R}}^2\to {\mathbb{R}}^2$ and $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$ be defined by $\left[\begin{array}{c}x\\ y\end{array}\right]\stackrel{S}{\mapsto }\left[\begin{array}{c}2x\\ y\end{array}\right]\text{and}\left[\begin{array}{c}x\\ y\end{array}\right]\stackrel{T}{\mapsto }\left[\begin{array}{c}x\\ y+4\end{array}\right]$, for each $S,T$, determine whether the transformation is linear §

  Let $\vec{u}=\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right],\vec{v}=\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]$ be vectors, and let $\alpha$ be a scaler

  Consider $S$, need to verify that $S(\vec{u}+\vec{v})=S(\vec{u})+S(\vec{v})$ and $S(\alpha \vec{v})=\alpha S(\vec{v})$, computing:

  $$S(\vec{u}+\vec{v})=S\left(\left[\begin{array}{c}{u}_1+v_1\\ u_2+v_2\end{array}\right]\right)=\left[\begin{array}{c}2u_1+2v_1\\ u_2+v_2\end{array}\right]=\left[\begin{array}{c}2u_1\\ u_2\end{array}\right]+\left[\begin{array}{c}2v_1\\ v_2\end{array}\right]=S(\vec{u})+S(\vec{v})$$

  Then

  $$S(\alpha \vec{u})=\left[\begin{array}{c}2\alpha u_1\\ \alpha u_2\end{array}\right]=\alpha \left[\begin{array}{c}2u_1\\ u_2\end{array}\right]=\alpha S(\vec{u})$$

  So $S$ satisfies all the properties of a linear transformation. Considering $T$, notice that $T(\vec{u}+\vec{v})\left[\begin{array}{c}{u}_1+v_1\\ u_2+v_2+4\end{array}\right]$ dooes not look like $T(\vec{u})+T(\vec{v})=\left[\begin{array}{c}{u}_1+v_1\\ u_2+v_2+8\end{array}\right]$. Therefore, $T$ is not linear. Since $T$ violated a linear transformation property, then $T$ cannot be a linear transformation

Function Notation vs. Linear Transformation Notation §

• In linear transformations: $T:{\mathbb{R}}^n\to {\mathbb{R}}^m$, $T(\vec{x})=T\vec{x}$
• While, $f(x)\ne$ the number $f(x)$ in linearly algebra (it is only a number, does not rep. function anymore)

Linear Transformation with Proofs §

• Starting with a linearity proof with “let $\vec{x},\vec{y}\in {\mathbb{R}}^n$ and let $\alpha$ be a scalar” allows the argument about all vectors and scalars be developed simultaneously
• Then prove that 2 definitions are satisfied

Let $T:{\mathbb{R}}^n\to {\mathbb{R}}^n$ be defined by $T(\vec{x})=\vec{x}+{\vec{e}}_1$, Show that $T$ is not linear §

Proof Show that $T$ does not distribute with respect to scalar multiplication $T(2\vec{0})=T(\vec{0})={\vec{e}}_1\ne 2{\vec{e}}_2=2T(\vec{0})$ Therefore, $T$ cannot be a linear transformation. $\square$

Matrix Transformations §

• Let $M$ be a matrix, for a vector $\vec{v}\in {\mathbb{R}}^2$, $M\vec{v}$ is another vector in ${\mathbb{R}}^2$. Then multiplication of $M$ can be interpreted as a transformation: $T:{\mathbb{R}}^2\to {\mathbb{R}}^2\text{by}T(\vec{x})=M\vec{x}$
• All matrix transformations are linear transformations, BUT, only most linear transformations are matrix transformations (closely related, but not the same thing)
• Correct way to specify a linear transformations
  • The transformations $T$ defined by $T(\vec{x})=M\vec{x}$
  • The transformation given by multiplication by $M$
  • The transformation induced by $M$
  • The matrix transformation given by $M$
  • The linear transformation whose matrix is $M$
• Incorrect way to specify a linear transformation: $T:{\mathbb{R}}^2\to {\mathbb{R}}^2\text{by}T=M$

Finding a Matrix for a Linear Transformation §

• Know: for $T:{\mathbb{R}}^n\to {\mathbb{R}}^m$ as a linear transformation, matrix for $T$ must be $m\times n$
  $$T:{\mathbb{R}}^n\to {\mathbb{R}}^m\Rightarrow T(\vec{x})=A_T\vec{x},A_T=\left[\begin{array}{ccc}T(\vec{e}1)& \dots & T({\vec{e}}_n)\end{array}\right]$$

Let the linear transformation to be $T({\vec{e}}_1)={\vec{v}}_1,T({\vec{e}}_2)={\vec{v}}_2$. Find a matrix $A_T$ such that $T(\vec{x})=A_T\vec{x}$

1. Write $\vec{x}$ in the form of unit vectors ($x_1{\vec{e}}_1+x_2{\vec{e}}_2$)
2. Then $T(\vec{x})=T(x_1{\vec{e}}_1+x_2{\vec{e}}_2)=x_{1}T({\vec{e}}_1),x_{2}T({\vec{e}}_2)=x_1{\vec{v}}_1+x_2{\vec{v}}_2$
3. Then $T(\vec{x})=\left[\begin{array}{cc}{\vec{v}}_1& {\vec{v}}_2\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=\left[\begin{array}{cc}{\vec{v}}_1& {\vec{v}}_2\end{array}\right]\vec{x}$
4. Which means that $T(\vec{x})=A_T\vec{x}$, where $A_T=\left[\begin{array}{cc}{\vec{v}}_1& {\vec{v}}_2\end{array}\right]=\left[\begin{array}{cc}T({\vec{e}}_1)& T({\vec{e}}_2)\end{array}\right]$ _

Let $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$ be defined by $T\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}2x+y\\ x\end{array}\right]$. Find a matrix, $M$, for $T$ §

Because $T$ is a transformation for ${\mathbb{R}}^2\to {\mathbb{R}}^2$, $M$ will be a $2\times 2$ matrix. Let $M=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ Use input-output paris to “calibrate” $M$

$$T\left[\begin{array}{c}1\\ 1\end{array}\right]=\left[\begin{array}{c}3\\ 3\end{array}\right]\text{and}T\left[\begin{array}{c}0\\ 1\end{array}\right]=\left[\begin{array}{c}1\\ 0\end{array}\right]$$

Since $M$ is a matrix for $T$, then $T\vec{x}$ for all $\vec{x}$ and so

$$M\left[\begin{array}{c}1\\ 1\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{c}1\\ 1\end{array}\right]=\left[\begin{array}{c}a+b\\ c+d\end{array}\right]=\left[\begin{array}{c}3\\ 1\end{array}\right]$$

and

$$M\left[\begin{array}{c}0\\ 1\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right]=\left[\begin{array}{c}b\\ d\end{array}\right]=\left[\begin{array}{c}1\\ 0\end{array}\right]$$

This gives the system of equations

$$\{\begin{array}{ccc}a+b& & =3\\ & c+d& =1\\ b& & =1\\ & d& =0\end{array}$$

and solving this system get: $M=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}2& 1\\ 1& 0\end{array}\right]$

The Composition of Linear Transformations §

• Composition of Functions - let $f:A\to B$ and $g:B\to C$ The composition of $g$ and $f$, notated $g\circ f$ the function $h:A\to C$ defined by $h(x)=g\circ f=g\left(f(x)\right)$

Let $h:{\mathbb{R}}^2\to {\mathbb{R}}^2$ be the transformation given by $M=\left[\begin{array}{cc}\sqrt[]{2}/2& -\sqrt[]{2}.2\\ 0& 0\end{array}\right]$, let $f:{\mathbb{R}}^2\mapsto {\mathbb{R}}^2$ be rotation counter clockwise by $45^{\circ }$ and let $g:{\mathbb{R}}^2\mapsto {\mathbb{R}}^2$ be projection onto the x-aixs. Write $h$ as the composition (in some order) of $f,g$ §

Use ${\vec{e}}_1,{\vec{e}}_2$ to determine whether $h$ is $f\circ g$ or $g\circ f$

$$h({\vec{e}}_1)=\left[\begin{array}{c}\sqrt[]{2}/2\\ 0\end{array}\right]\text{and}h({\vec{e}}_2)=\left[\begin{array}{c}-\sqrt[]{2}/2\\ 0\end{array}\right]$$

For $f\circ g$: $f\circ g({\vec{e}}_1)=\left[\begin{array}{c}\sqrt[]{2}/2\\ \sqrt[]{2}/2\end{array}\right]$ and $f\circ g({\vec{e}}_2)=\left[\begin{array}{c}0\\ 0\end{array}\right]$

For $g\circ f$; $g\circ f({\vec{e}}_1)=\left[\begin{array}{c}\sqrt[]{2}/2\\ 0\end{array}\right]$ and $g\circ f({\vec{e}}_2)=\left[\begin{array}{c}-\sqrt[]{2}/2\\ 0\end{array}\right]$

Since the answer of $g\circ f$ agrees with $h$ , then $h=g\circ f$

Compositions and Matrix Products §

• If $\mathcal{P}:{\mathbb{R}}^a\to {\mathbb{R}}^b$ and $\mathcal{Q}:{\mathbb{R}}^c\to {\mathbb{R}}^a$ are matrix transformations with matrices $M_P$ and $M_Q$, then $\mathcal{P}\circ \mathcal{Q}$ is a matrix transformation whose matrix is given by the matrix product $M_P{M}_Q$

Find entries of the matrix $A$ below, assuming the equation must be true $\forall x_1,x_2,x_3$. For $A\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]=\left[\begin{array}{c}3x_1-x_3\\ -2x_1+4x_2\\ 6x_2-3x_3\end{array}\right]$. §

Let $v_1,v_2,v_3$ represent the columns of matrix $A$ Then:

x_1 v_1 + x_2 v_2 + x_3 v_3 = \begin{bmatrix}3x_1 & & - x_3 \ -2x_1 & + 4x_2 & \ & 6x_2 & - 3x_3\end{bmatrix}

Extracting the coefficients, then $v_1 = (3, -2, 0) v_2 = (0, 4, 0) v_3 = (-1, 0, -3)$, combining to form the matrix $A$:

A = \begin{bmatrix}3 & 0 & - 1 \ -2 & 4 & 0 \ 0 & 6 & - 3\end{bmatrix}

An application would be linear transformations with respect to basis. Let $T:{\mathbb{R}}^n\to {\mathbb{R}}^n$ §

• $T(\vec{x})=A\vec{x}$ where $A$ is the transformation matrix for T with respect to the standard basis
• Let $B=\left\{{\vec{v}}_1,\dots {\vec{v}}_n\right\}$ be a basis for ${\mathbb{R}}^n$. Then $[T(\vec{x}){]}_B=D[\vec{x}{]}_B$ where $D$ is the transformation matrix for $T$ with with respect to the basis $B$
• Let $C$ be the change of basis matrix for the basis $B$, then $C[\vec{x}{]}_B=\vec{x}$ and $[\vec{x}{]}_B=C^{-1}\vec{x}$

$$\begin{array}{rl}D[\vec{x}{]}_B& =[T(\vec{x}){]}_B\\ & =[A\vec{x}{]}_B\\ & =C^{-1}(A\vec{x})\\ & =C^{-1}A(C[\vec{x}{]}_B)\end{array}\text{\hspace{1em}(Using D[x]B=[T(x)]B)(Using T(x)=Ax)(Using [x]B=C−1x)(Using x=C[x]B)}$$

• $D$ is the transformation matrix for $T$ with respect to the basis $B$. And $C$ is the change of basis matrix for $B$. And $A$ is the transformation matrix for $T$ with respect to the standard basis. Then $D=C^{-1}AC$