Corollary

Let E, F, G be three right (resp. left) modules over a ring A, $(e_i)_{i \in I}$, $(f_k)_{k \in K}$, $(g_l)_{l \in L}$ respective finite bases of E, F, G, $u : E \to F$, $v : F \to G$ two linear mappings, $M(u)$ the matrix of u relative to the bases $(e_i)$, $(f_k)$, $M(v)$ the matrix of v relative to the bases $(f_k)$, $(g_l)$ and $M(v \circ u)$ the matrix of $v \circ u$ relative to the bases $(e_i)$, $(g_l)$; then

(18 D)

(18 G)

We prove for example (18 G). For all $x \in E$, by (17 G):

by associativity; the corollary then follows from no. 3, Proposition 1 since the matrix ${}^t M(x)$ with one row is arbitrary.

Remark (1). Formula (17 D) can be considered as a special case of (18 D). For there corresponds canonically to every $x \in E$ the linear mapping $\theta_x : A_d \to E$ mapping every $\alpha \in A$ to $x \alpha$ (§ 2, no. 1). It is immediate that the matrix $M(\theta_x)$ with respect to the basis 1 of $A_d$ and the basis $(e_i)$ of E is just the matrix $M(x)$; similarly $M(\theta_{u(x)}) = M(u(x))$ and formula (17 D) can therefore be considered as a translation of the relation