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$\newcommand{\Z}{{\Bbb Z}} \newcommand{\Q}{{\Bbb Q}} \newcommand{\Hom}{{\rm Hom}} $ The following lemma is certainly known. Lemma (well-known). Let $B$ be a lattice (that is, a finitely generated ...

If $f: V \to W$ is a surjective homomorphism of vector spaces, and we have fixed a basis for $V$, it is always possible to find a basis for $W$ such that the matrix associated to $\phi$ in the two ...