Remark

(1) We now start with a left $A$-module $E$ with two bases $(e_i), (e'_i)$, each with $n$ elements; if we write $e'_i = \sum_{j=1}^n a_{ji} e_i$, $P = (a_{ji})$ is also called the matrix of passing from $(e_i)$ to $(e'_i)$; it is also the matrix of the automorphism of $E$ such that $u(e_i) = e'_i$, with respect to the basis $(e_i)$ and also the matrix of $l_E$ with respect to the bases $(e'_i)$ and $(e_i)$ in that order. The above results then hold with only the following modifications: formulae (33 D) to (36 D) are respectively replaced by

LINEAR ALGEBRA

(36 G)

(2) Under the hypotheses of Proposition 4, consider an element $x^* \in \mathbf{E}^*$; as the matrix of passage from $(e_i^*)$ to $(e_{i'}^*)$ is $tP^{-1}$ (Proposition 5), for the matrices $M(x^*)$ and $M'(x^*)$ of $x^*$ with respect to these two bases respectively,

or also

(37 D)

which is equivalent to the system of equations

(38 D)

for the elements $(x_i^*)$ and $(x_{i'}^*)$ of the matrices $M(x^*)$ and $M'(x^*)$. The corresponding formulae for a left A-module E are

(37 G)

(38 G)

(3) Let A, B be two rings, $\sigma : A \to B$ a homomorphism of A into B, E a right (resp. left) A-module, $(e_i)$, $(e_i')$ two bases with $n$ elements of E, F a right (resp. left) B-module, $(f_j)$, $(f_j')$ two bases with $m$ elements of F and $P$ (resp. $Q$) the matrix of passage from $(e_i)$ to $(e_i')$ (resp. from $(f_j)$ to $(f_j')$).

For every semi-linear mapping $u : E \to F$, relative to $\sigma$, let $M(u)$ be the matrix of $u$ with respect to $(e_i)$ and $(f_j)$ and $M'(u)$ its matrix with respect to $(e_i')$ and $(f_j')$. Then

(39 D)

(resp. (39 G)

The proof is the same as that for (33 D) and (33 G), this time using formulae (27 D) and (27 G) (no. 6).