Operations on Linear Transformations

Let and be linear transformations. The composition, if defined by:

for all .

What happens if we try a transformation? Lets say we rotate a vector by 90 degrees and then shear a vector. That means we first apply the rotation, then the shear afterwards:

But we can multiply the matrices first instead, so

which will yield the same result.

As we can see, matrix multiplication is the same as the composition of linear transformations.

Recall that matrix multiplication is NOT commutative, and recall the intuition that was presented. Remember to "read" the transformations from right to left.

Let , and be linear transformations. Then is a linear transformation, and:

(see standard matrix)

Let be a counterclockwise rotation about the origin by an angle of , and let be a projection onto the -axis. Find the standard matrices for and .

Since and are linear, we have:

$
\begin{align}
[L] & = \begin{bmatrix}
\cos(\pi/4) & -\sin(\pi/4) \
\sin(\pi/4) & \cos(\pi/4) \
\end{bmatrix} = \begin{bmatrix}
\sqrt{ 2 }/2 & -\sqrt{ 2 }/2 \
\sqrt{ 2 }/2 & \sqrt{ 2 }/2 \
\end{bmatrix} \

[M] & = \begin{bmatrix}
\proj_{\vec{e}{1}}\vec{e} & \proj_{\vec{e}{1}}\vec{e}
\end{bmatrix} = \begin{bmatrix}
1 & 0 \
0 & 0
\end{bmatrix}
\end{align}

\begin{align} \
[M \circ L] & = [M][L] = \begin{bmatrix}
\sqrt{ 2 }/2 & -\sqrt{ 2 }/2 \
0 & 0
\end{bmatrix} \

[L \circ M] & = [L][M] = \begin{bmatrix}
\sqrt{ 2 }/2 & 0 \
\sqrt{ 2 }/2 & 0
\end{bmatrix}
\end{align}
$