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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}

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