The change of basis matrix $C$ is the matrix that allows us to express the new basis in an old basis:
$$
x_{old} = Cx_{new}
$$

Mnemonic is as follows: consider we have an initial basis $(x_{old}, y_{old})$. Now, we define the new basis in terms of the old basis, e.g.:
$$
\begin{aligned}
x_{new} &= 1x_{old} + 2y_{old} \\
y_{new} &= 3x_{old} + 4y_{old} \\
\end{aligned}
$$
which can be written in matrix form as:
$$
\begin{bmatrix}x_{new} \\ y_{new} \\\end{bmatrix} =
\begin{bmatrix}1 && 2 \\ 3 && 4 \\\end{bmatrix}
\begin{bmatrix}x_{old} \\ y_{old} \\\end{bmatrix}
$$
and so if we set:
$$
M = \begin{bmatrix}1 && 2 \\ 3 && 4 \\\end{bmatrix}
$$
we have:
$$
\vec{x_{new}} = M\vec{x_{old}}
$$

The usual question then is: given a vector in the new basis, how do we represent it in the old basis?

The answer is that we simply have to calculate the <matrix inverse> of $M$:
$$
\vec{x_{old}} =  M^{-1}\vec{x_{new}}
$$

That $M^{-1}$ is the matrix inverse.


Change of basis matrix