matrix-vector multiplication as basis transformation

The result of multiplying a simple vector by a matrix

$$A\cdot \mathbf{v}=\mathbf{u}$$

is a geometric transformation. Here’s a 90° anti clockwise rotation

$$\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\cdot \left[\begin{array}{c}2\\ 1\end{array}\right]=\left[\begin{array}{c}-1\\ 2\end{array}\right]$$

That is to say

$$2\left[\begin{array}{c}0\\ 1\end{array}\right]+1\left[\begin{array}{c}-1\\ 0\end{array}\right]=\left[\begin{array}{c}-1\\ 2\end{array}\right]$$

the resulting vector $\mathbf{u}$ is a linear combination of columns of $A$ times rows of $\mathbf{v}$ (four-ways-to-look-at-matrix-multiplication).

The role of each matrix element in the transformation is better understood by interpreting matrix vector multiplication as basis transformation. For example, in a 2 dimensional plane with $x$ and $y$ axis, the basis is defined as

$$\left[\begin{array}{cc}{a}_x& c_x\\ b_y& d_y\end{array}\right]$$

where the coefficients identify the amount of movement—the standard unit—that happens on their respective axis, given $1$ unit of movement along the corresponding axis in the multiplied vector, and it can be read like

for every unit along the $x$ axis move $a_x$ along the $x$ axis and $b_y$ along $y$, and for every unit along the $y$ axis move $c_x$ along the $x$ axis and $d_y$ along $y$

In other words, each matrix column corresponds to an input dimension—what input dimension gets transformed—each matrix row to an output dimension—how the input dimension is transformed

$$\left[\begin{array}{ccc}& {\mathbf{v}}_{1,1}\downarrow & {\mathbf{v}}_{2,1}\downarrow \\ {\mathbf{u}}_{1,1}\leftarrow & a_x& c_x\\ {\mathbf{u}}_{1,2}\leftarrow & b_y& d_y\end{array}\right]$$

A look at the matrix representation of the familiar Cartesian plane makes it clearer

$$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$

The first column is the $x$ axis, the second the $y$.

The initial transformation can thus be interpreted as moving $1$ unit along the $y$ axis (and $0$ along the $x$ axis) for every unit of movement along the $x$ axis and $-1$ along the $x$ axis (and $0$ along the $y$ axis) for every unit on the $y$ axis.