Matrix Multiplication as Composition

A composition is a product of two matrices

• has the geometric meaning of applying one transform, then another

When you take the product of $M_2\ast M_1$ you’re applying the matrix on the right to $\hat{i}$ and $\hat{j}$ first, this stems from function notation

$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\cdot \left[\begin{array}{cc}w& x\\ y& z\end{array}\right]=\left[\begin{array}{cc}aw+by& ax+bz\\ cx+dy& cx+dz\end{array}\right]$

The first column of the resulting matrix will always equal the left matrix, times the first column of the right matrix

Likewise, the second column of the resulting matrix will always equal the left matrix, times the second column of the right matrix

Btw, the order the matrices are multiplied in matters.

(ch. 4)