Show that the vector space $M_2$ has a basis $$e_{11} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \\ \end{pmatrix},\quad e_{12} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \\ \end{pmatrix} ,\quad e_{21} = \begin{pmatrix} 0 & 0 \\ 1 & 0 \\ \end{pmatrix},\quad e_{22} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \\ \end{pmatrix} $$ by doing the following steps:
$a.$ Show that the matrices $e_{11}, e_{12}, e_{21}, e_{22}$ span $M_2$.
$b.$ Show that the matrices $e_{11}, e_{12}, e_{21}, e_{22}$ are linearly independent.
I'm not sure how you would find the span and linear independence of matrices.