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.:
which can be written in matrix form as:
and so if we set:
we have:

$x_{new}y_{new} =1x_{old}+2y_{old}=3x_{old}+4y_{old} $

$[x_{new}y_{new} ]=[13 24 ][x_{old}y_{old} ]$

$M=[13 24 ]$

$x_{new} =Mx_{old} $

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

That $M_{−1}$ is the matrix inverse.