Example BM A basis for the vector space of matrices

In the vector space $M_{mn}$ of matrices (Example VSM) define the matrices $B_{k\ell}$, $1\leq k\leq m$, $1\leq\ell\leq n$ by \begin{equation*} \matrixentry{B_{k\ell}}{ij}=\begin{cases} 1&\text{if }k=i,\,\ell=j\\ 0&\text{otherwise} \end{cases} \end{equation*}

So these matrices have entries that are all zeros, with the exception of a lone entry that is one. The set of all $mn$ of them, \begin{equation*} B=\setparts{B_{k\ell}}{1\leq k\leq m,\ 1\leq\ell\leq n} \end{equation*} forms a basis for $M_{mn}$. See Exercise B.M20.