Definition 3

In each of the above two cases, the matrix of u with respect to the bases $(e_i), (f_k)$ is the matrix $M(u) = (u_{ki})_{(k, i) \in K \times I}$ such that

which is written respectively as

(14 D)

(14 G)

The column of $M(u)$ of index $i$ is therefore equal to $M(u(e_i))$.

Clearly if $u, v$ are two linear mappings of E into F and $M(u), M(v)$ their matrices with respect to the same bases, then

and for every element $\gamma$ of the centre $\Gamma$ of A. In other words, once the bases $(e_i), (f_k)$ are fixed, the mapping $u \mapsto M(u)$ is a $\Gamma$-module isomorphism of $\mathrm{Hom}_A(E, F)$ onto a subset of the set $A^{K \times I}$, equal to $A^{K \times I}$ if K is finite.