Remark

(2) Let E and F be two left A-modules with bases $(e_i)_{i \in I}$ and $(f_k)_{k \in K}$ respectively. For every A-linear mapping $u : E \to F$, by (14 G), $u(e_i) = \sum_k u_{ki} f_k$; these relations can also be interpreted by saying that the column matrix $(u(e_i))_{i \in I}$ with elements in F is equal to the product $^tM(u) \cdot (f_k)$, where $(f_k)_{k \in K}$ is considered as a column matrix with elements in F and the product is calculated for the mapping $A \times F \to F$ defining the law of action on the A-module F (no. 2).

(3) Let A, B be two commutative rings and $\sigma : A \to B$ a ring homomorphism. In the notation of Proposition 3, $(e_i \otimes 1)$ and $(f_k \otimes 1)$ are respective bases of $E_{(B)} = E \otimes_A B$ and $F_{(B)} = F \otimes_A B$ (§ 5, no. 1, Proposition 4); moreover, if $(e_i^*)$ and $(f_k^*)$ are respectively the dual bases of $(e_i)$ and $(f_k)$, then $(e_i^* \otimes 1)$ and $(f_k^* \otimes 1)$ are respectively the dual bases of $(e_i \otimes 1)$ and $(f_k \otimes 1)$ (§ 5, no. 4). For every A-linear mapping $u : E \to F$, let $M(u)$ and $M(u \otimes 1)$ be the matrix of $u$ with respect to $(e_i)$ and $(f_k)$ and the matrix of the B-linear mapping $u \otimes 1$ with respect to $(e_i \otimes 1)$ and $(f_k \otimes 1)$. It follows from § 5, no. 4, formula (20) that

Consider a system of a finite number of right scalar linear equations in a finite number of unknowns

with $a_{kt}, x_i, b_k$ in A.

Let $(e_i)_{i \in I}, (f_k)_{k \in K}$ be the canonical bases of $E = A_d^I$ and $F = A_d^K$; the system (19) is equivalent to the equation $u(x) = b$, where $x = \sum_i e_i x_i$, $b = \sum_k f_k b_k$ and $u : E \to F$ is the linear mapping such that the matrix $M(u)$ with respect to the bases $(e_i)$ and $(f_k)$ are equal to $A = (a_{kt})_{(k, i) \in K \times L}$. This matrix is called the matrix of the system of linear equations (19). Recall (§ 2, no. 8, Remarks 2 and 3), that, writing $c_i = \sum_k f_k a_{kt}$, the system (19) is equivalent to the unique vector equation

and as $c_i$ is the column of index $i$ in the matrix $A$, we see that to say that the system (19) admits a solution amounts to saying that the matrix $b = (b_{k0})$ with one column is a linear combination of the columns of the matrix $A$.

We leave to the reader the task of formulating the analogous definitions and remarks for systems of left linear equations.