## FANDOM

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A transformation matrix is a matrix representing a linear transformation. If $T$ is a transformation from $\R^n$ to $\R^m$ , $A$ is the m×n transformation matrix of $T$ such that

$T(\vec x)=\mathbf{A}\vec x$

The matrix can found by taking the column vectors representing the transformations of unit vectors in each direction. For example,

$T\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}=\begin{bmatrix}3x_2\\0\\2x_1\end{bmatrix}$
$T\begin{bmatrix}1\\0\\0\end{bmatrix}=\begin{bmatrix}0\\0\\2\end{bmatrix}\, ,T\begin{bmatrix}0\\1\\0\end{bmatrix}=\begin{bmatrix}3\\0\\0\end{bmatrix}\, ,T\begin{bmatrix}0\\0\\1\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}$

Therefore, the transformation matrix for $T$ will be

$\begin{bmatrix}0&3&0\\0&0&0\\2&0&0\end{bmatrix}$

Which does indeed represent the transformation $T$ .

$\begin{bmatrix}0&3&0\\0&0&0\\2&0&0\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix} =\begin{bmatrix}3x_2\\0\\2x_1\end{bmatrix}$

## Common transformation matrices

### $\R^2\to\R^2$

Identity transformation:

$\begin{bmatrix}1&0\\0&1\end{bmatrix}$

Projection onto $x$-axis:

$\begin{bmatrix}1&0\\0&0\end{bmatrix}$

Projection onto $y$-axis:

$\begin{bmatrix}0&0\\0&1\end{bmatrix}$

Reflection over $y=x$

$\begin{bmatrix}0&1\\1&0\end{bmatrix}$

Reflection over $x$-axis:

$\begin{bmatrix}1&0\\0&-1\end{bmatrix}$

Reflection over $y$-axis:

$\begin{bmatrix}-1&0\\0&1\end{bmatrix}$

Stretch by factor $n$

$\begin{bmatrix}n&0\\0&n\end{bmatrix}$

Vector projection onto $y=mx$ :

$\frac1{1+m^2}\begin{bmatrix}1&m\\m&m^2\end{bmatrix}$

Reflection over y=mx:

$\frac1{1+m^2}\begin{bmatrix}1-m^2&2m\\2m&m^2-1\end{bmatrix}$

Rotation counterclockwise by $\theta$ :

$\begin{bmatrix}\cos(\theta)&-\sin(\theta)\\\sin(\theta)&\cos(\theta)\end{bmatrix}$

Rotation clockwise by $\theta$ :

$\begin{bmatrix}\cos(\theta)&\sin(\theta)\\-\sin(\theta)&\cos(\theta)\end{bmatrix}$