Homomorphism from abelian group to torsion-free abelian group is completely determined by images of a maximal linearly independent subset

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Statement

Suppose A and B are abelian groups such that B is a torsion-free abelian group. Suppose T is a maximal linearly independent subset of A. Then, a homomorphism of groups \varphi:A \to B is completely determined by knowledge of the values \varphi(t), t \in T.