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

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 $$\varphi(t), t \in T$$.