MATH 323 Lecture 16

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Linear Transformations

Let $L:V\to W$ be a linear transformation for all $v_i\in V$ and ${\alpha }_i\in \mathbb{ℝ}$:

1. $L({\overrightarrow{0}}_V)={\overrightarrow{0}}_W$
2. $L(\sum _{i=1}^n{\alpha }_i{v}_i)=\sum _{i=1}^n{\alpha }_{i}L(v_i)$
3. $L(-\overrightarrow{v})=-L(\overrightarrow{v})$

In general, linear transformations are of the form

$L_A(\overrightarrow{x})=A\overrightarrow{x}$.

Transdimensional Transformations

If $L_A$ is a transformation from ${\mathbb{ℝ}}^n$ to ${\mathbb{ℝ}}^m$, then $A$ will be a $m\times n$ matrix.

Let $L(\overrightarrow{x})=x_1+x_2$ be a linear transformation from 2D space to 1D space.

$\begin{array}{rl}L(\alpha \overrightarrow{x}+\beta \overrightarrow{y})& =L\left(\left[\begin{array}{c}\alpha x_1+\beta y_1\\ \alpha x_2+\beta y_2\end{array}\right]\right)\\ & =\alpha x_1+\beta y_1+\alpha x_2+\beta y_2\\ & =\alpha (x_1+x_2)+\beta (y_1+y_2)\\ & =\alpha L(\overrightarrow{x})+\beta L(\overrightarrow{y})\end{array}$

The following transformation is not linear because it fails on (1) above:

$M(\overrightarrow{x})=\sqrt{x_1^2+x_2^2}$

Identity Transformation

$I:V\to V$ such that $I(\overrightarrow{v})=\overrightarrow{v}$.

Image and Kernel

Giver $L:V\to W$,

kernel

Set of vectors in $V$ such that $L(v)=0$

Very analogous to null space of a matrix.

image

Written $L(V)$

Set of vectors in $W$ such that $w=L(v)$ for some vector in $V$

Subspace of $V$ will have image contained in $L(V)$

Theorem 4.1.1

Let $L:V\to W$ be a linear transformation, and $S\subseteq V$ be a subspace. Then

1. The kernel of $L$ is a subspace of $V$
2. $L(S)$ is a subspace of $W$. In particular, $L(V)$ is a subspace of $W$

Example

Let $L:{\mathbb{ℝ}}^3\to {\mathbb{ℝ}}^2$ be defined as follows:

$L(\overrightarrow{x})=\left(\begin{array}{c}{x}_1+x_2\\ x_2+x_3\end{array}\right)$

Kernel is $\{\overrightarrow{x}|L(\overrightarrow{x})=\overrightarrow{0}\}$.

$x_1+x_2=x_2+x_3=0$, therefore $⟨a,-a,a⟩$ is the kernel of $L$.

For a subspace $S=Span({\overrightarrow{e}}_1,{\overrightarrow{e}}_2)=⟨a,0b⟩$, The image $L(S)=⟨a+0,0+b⟩={\mathbb{ℝ}}^2$.

Matrix Representations of Linear Transformations

Let $A$ be a $m\times n$ matrix, and $L_A:{\mathbb{ℝ}}^n\to {\mathbb{ℝ}}^m$. Then $L_A(\overrightarrow{x})=A\overrightarrow{x}$

$A$ is called the standard matrix representation of $L$.

Theorem 4.2.1

If $L$ is a linear transformation mapping ${\mathbb{ℝ}}^n$ into ${\mathbb{ℝ}}^m$, there is a $m\times n$ matrix $A$ such that $L(\overrightarrow{x})=A\overrightarrow{x}$ for each $x\in {\mathbb{ℝ}}^n$. In fact, the $j$^th column vector of $A$ is given by

${\overrightarrow{a}}_j=L({\overrightarrow{e}}_j)$ for $j=1,\dots ,n$

Example

$L(\overrightarrow{x})=\left(\begin{array}{c}{x}_1+x_2\\ x_2+x_3\end{array}\right)$. Find the standard matrix representation.

• ${\overrightarrow{a}}_1=L({\overrightarrow{e}}_1)=\left(\begin{array}{c}1+0\\ 0+0\end{array}\right)=\left(\begin{array}{c}1\\ 0\end{array}\right)$
• ${\overrightarrow{a}}_2=L({\overrightarrow{e}}_2)=\left(\begin{array}{c}0+1\\ 1+0\end{array}\right)=\left(\begin{array}{c}1\\ 1\end{array}\right)$
• ${\overrightarrow{a}}_3=L({\overrightarrow{e}}_3)=\left(\begin{array}{c}0+0\\ 0+1\end{array}\right)=\left(\begin{array}{c}0\\ 1\end{array}\right)$

So $A=(a_1,a_2,a_3)=\left(\begin{array}{ccc}1& 1& 0\\ 0& 1& 1\end{array}\right)$