In my algebra's workbook there is this exercise that I don't know how to approach ... so far I have only dealt with linear maps of the type $vector \mapsto vector $, I've never seen $matrix \mapsto vector$.
let $H:M_{2,2}(\mathbb{R}) \to \mathbb{R^{2}} $ a linear map defined as:
$$ \begin{pmatrix} r & s\\ t & u\\ \end{pmatrix} \mapsto (2r+t-u,s-2t).$$
find the matrix associated with $H$ with base
$B= \{ \begin{pmatrix} 0 & -1\\ 0 & 1\\ \end{pmatrix},\begin{pmatrix} 1 & 0\\ 0 & -2\\ \end{pmatrix},\begin{pmatrix} 0 & 0\\ 0 & 1\\ \end{pmatrix},\begin{pmatrix} 0 & 0\\ -1 & 0\\ \end{pmatrix} \} $ of $ M_{2,2}(\mathbb{R})$
and $B'=\{(0,-2),(-1,0) \}$ base of $\mathbb{R^{2}}$
For a vector space, it does not matter how the objects are ordered. Hence you can think of a matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ just as a vector $(a,b,c,d)$ in this context.