Proposition 6

Let E and F be two right A-modules, $(e_i)$ and $(e'_i)$ two bases of E with $n$ elements, $(f_j)$ and $(f'_j)$ two bases of $F$ with $m$ elements, $P$ the matrix of passage from $(e_i)$ to $(e'_i)$ and $Q$ the matrix of passage from $(f_j)$ to $(f'_j)$. For every linear mapping $u$ of $E$ into $F$, let $M(u)$ be the matrix of $u$ with respect to the bases $(e_i)$ and $(f_j)$ and $M'(u)$ the matrix of $u$ with respect to the bases $(e'_i)$ and $(f'_j)$; then

We may write $u = l_F \circ u \circ l_E$. Formula (33) follows immediately from no. 4, Corollary to Proposition 2 when the matrix of $l_E$ is taken with respect to $(e'_i)$ and $(e_i)$, that of $u$ with respect to $(e_i)$ and $(f_i)$ and that of $l_F$ with respect to $(f_i)$ and $(f'_j)$.