Linear Transformations

$$\begin{array}{c}T:R^n\to R^m\text{ is linear if }\{\begin{array}{ll}T\left(u+v\right)& =T\left(u\right)+T\left(v\right)\\ T\left(cv\right)& =cT\left(v\right)\end{array}\end{array}$$

Create a $2\times 2$ matrix $A$ that applies a linear transformation that rotates by an angle $\theta$ counterclockwise

$$\begin{array}{rlr}A& =[\overrightarrow{a_1},\overrightarrow{a_2}]& \\ T\left(\overrightarrow{e_1}\right)& =\overrightarrow{a_1}& \\ T\left(\overrightarrow{e_1}\right)& =\left[\begin{array}{c}a\\ b\end{array}\right]& =\left[\begin{array}{c}\cos \theta \\ \sin \theta \end{array}\right]\\ \overrightarrow{a_1}& =\left[\begin{array}{c}\cos \theta \\ \sin \theta \end{array}\right]& \\ T\left(\overrightarrow{e_2}\right)& =\overrightarrow{a_2}& \\ T\left(\overrightarrow{e_2}\right)& =\left[\begin{array}{c}a\\ b\end{array}\right]& =\left[\begin{array}{c}-\sin \theta \\ \cos \theta \end{array}\right]\\ \overrightarrow{a_2}& =\left[\begin{array}{c}-\sin \theta \\ \cos \theta \end{array}\right]& \end{array}$$ $$\boxed{A=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]}$$

Geometric Transformations

Reflections

Through $x_1-\text{axis}$

$$T\left(\left[\begin{array}{c}1\\ 1\end{array}\right]\right)=\left[\begin{array}{c}1\\ -1\end{array}\right]$$

So, $A=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$

Through $x_2-\text{axis}$

$$T\left(\left[\begin{array}{c}1\\ 1\end{array}\right]\right)=\left[\begin{array}{c}-1\\ 1\end{array}\right]$$

So, $A=\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$

Through $x_2=x_1$

$$\begin{array}{rl}T\left(\left[\begin{array}{c}1\\ 0\end{array}\right]\right)& =\left[\begin{array}{c}0\\ 1\end{array}\right]\\ T\left(\left[\begin{array}{c}0\\ 1\end{array}\right]\right)& =\left[\begin{array}{c}1\\ 0\end{array}\right]\end{array}$$

So, $A=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

Through $x_2=-x_1$

$$\begin{array}{rl}T\left(\left[\begin{array}{c}1\\ 0\end{array}\right]\right)& =\left[\begin{array}{c}0\\ -1\end{array}\right]\\ T\left(\left[\begin{array}{c}0\\ 1\end{array}\right]\right)& =\left[\begin{array}{c}-1\\ 0\end{array}\right]\end{array}$$

So, $A=\left[\begin{array}{cc}0& -1\\ -1& 0\end{array}\right]$

Contractions and Expansions

Horizontal

Contractions ($|k|<1$)

$$\begin{array}{rl}T\left(\left[\begin{array}{c}1\\ 0\end{array}\right]\right)& =\left[\begin{array}{c}k\\ 0\end{array}\right]\\ T\left(\left[\begin{array}{c}0\\ 1\end{array}\right]\right)& =\left[\begin{array}{c}0\\ 1\end{array}\right]\end{array}$$

So, $A=\left[\begin{array}{cc}k& 0\\ 0& 1\end{array}\right]$

Expansions ($|k|>1$)

$$\begin{array}{rl}T\left(\left[\begin{array}{c}1\\ 0\end{array}\right]\right)& =\left[\begin{array}{c}k\\ 0\end{array}\right]\\ T\left(\left[\begin{array}{c}0\\ 1\end{array}\right]\right)& =\left[\begin{array}{c}0\\ 1\end{array}\right]\end{array}$$

So, $A=\left[\begin{array}{cc}k& 0\\ 0& 1\end{array}\right]$

Vertical

Contractions ($|k|<1$)

$$\begin{array}{rl}T\left(\left[\begin{array}{c}1\\ 0\end{array}\right]\right)& =\left[\begin{array}{c}1\\ 0\end{array}\right]\\ T\left(\left[\begin{array}{c}0\\ 1\end{array}\right]\right)& =\left[\begin{array}{c}0\\ k\end{array}\right]\end{array}$$

So, $A=\left[\begin{array}{cc}1& 0\\ 0& k\end{array}\right]$

Expansions ($|k|>1$)

$$\begin{array}{rl}T\left(\left[\begin{array}{c}1\\ 0\end{array}\right]\right)& =\left[\begin{array}{c}1\\ 0\end{array}\right]\\ T\left(\left[\begin{array}{c}0\\ 1\end{array}\right]\right)& =\left[\begin{array}{c}0\\ k\end{array}\right]\end{array}$$

So, $A=\left[\begin{array}{cc}1& 0\\ 0& k\end{array}\right]$

Shears

Horizontal Shear

Left($k<0$)

$$\begin{array}{rl}T\left(\left[\begin{array}{c}1\\ 0\end{array}\right]\right)& =\left[\begin{array}{c}1\\ 0\end{array}\right]\\ T\left(\left[\begin{array}{c}0\\ 1\end{array}\right]\right)& =\left[\begin{array}{c}k\\ 1\end{array}\right]\end{array}$$

So, $A=\left[\begin{array}{cc}1& k\\ 0& 1\end{array}\right]$

Right($k>0$)

$$\begin{array}{rl}T\left(\left[\begin{array}{c}1\\ 0\end{array}\right]\right)& =\left[\begin{array}{c}1\\ 0\end{array}\right]\\ T\left(\left[\begin{array}{c}0\\ 1\end{array}\right]\right)& =\left[\begin{array}{c}k\\ 1\end{array}\right]\end{array}$$

So, $A=\left[\begin{array}{cc}1& k\\ 0& 1\end{array}\right]$

Vertical Shear

Down($k<0$)

$$\begin{array}{rl}T\left(\left[\begin{array}{c}1\\ 0\end{array}\right]\right)& =\left[\begin{array}{c}1\\ k\end{array}\right]\\ T\left(\left[\begin{array}{c}0\\ 1\end{array}\right]\right)& =\left[\begin{array}{c}0\\ 1\end{array}\right]\end{array}$$

So, $A=\left[\begin{array}{cc}1& 0\\ k& 1\end{array}\right]$

Up($k>0$)

$$\begin{array}{rl}T\left(\left[\begin{array}{c}1\\ 0\end{array}\right]\right)& =\left[\begin{array}{c}1\\ k\end{array}\right]\\ T\left(\left[\begin{array}{c}0\\ 1\end{array}\right]\right)& =\left[\begin{array}{c}0\\ 1\end{array}\right]\end{array}$$

So, $A=\left[\begin{array}{cc}1& 0\\ k& 1\end{array}\right]$

Projections

On to the $x_1-\text{Axis}$

$$\begin{array}{rl}T\left(\left[\begin{array}{c}1\\ 0\end{array}\right]\right)& =\left[\begin{array}{c}1\\ 0\end{array}\right]\\ T\left(\left[\begin{array}{c}0\\ 1\end{array}\right]\right)& =\left[\begin{array}{c}0\\ 0\end{array}\right]\end{array}$$

So, $A=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$

On to the $x_2-\text{Axis}$

$$\begin{array}{rl}T\left(\left[\begin{array}{c}1\\ 0\end{array}\right]\right)& =\left[\begin{array}{c}0\\ 0\end{array}\right]\\ T\left(\left[\begin{array}{c}0\\ 1\end{array}\right]\right)& =\left[\begin{array}{c}0\\ 1\end{array}\right]\end{array}$$

So, $A=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$