Corollary 2

— Two matrices X and $X'$ with n rows and m columns over a principal ideal domain A are equivalent if and only if they have the same invariant factors.

Note that when A is a field we can take the $\alpha_i$ to be equal to 1, and then we recover Prop. 13 of II, p. 360.

If X is the matrix of the linear mapping f with respect to an arbitrary basis of L and an arbitrary basis of L', then the columns of X are the coordinate vectors, with respect to the basis of L', of elements of L' which form a system of generators for $f(L)$. The following result is thus an immediate consequence of Prop. 4.