Proposition 4

Let E be a right A-module with finite basis $(e_i)_{1 \leq i \leq n}$ of n elements. For a family of n elements $e'_i = \sum_{j=1}^n e_j a_{ji} (1 \leq i \leq n)$ to be a basis of E, it is necessary and sufficient that the square matrix $P = (a_{ji})$ of order n be invertible.

$P$ is just the matrix, with respect to the basis $(e_i)$ of the endomorphism $u$ of E defined by $u(e_i) = e'_i (1 \leq i \leq n)$. Now, for $u$ to be an automorphism of E, it is necessary and sufficient that $(u(e_i))$ be a basis of E (§ 1, no. 11, Corollary 3 to Proposition 17); whence the proposition.

The invertible matrix $P$ is called the matrix of passage from the basis $(e_i)$ to the basis $(e'_i)$. It can also be interpreted as the matrix of the identity mapping $l_E$ with respect to the bases $(e'_i)$ and $(e_i)$ (in that order); then clearly the matrix of passage from the basis $(e'_i)$ to the basis $(e_i)$ is the inverse $P^{-1}$ of $P$.