Linear Transformation of Generated Module

Theorem
Let $G$ and $H$ be $R$-modules.

Let $\phi$ and $\psi$ be linear transformations $G$ into $H$.

Let $S$ be a generator for $G$.

Suppose that $\forall x \in S: \map \phi x = \map \psi x$.

Then $\phi = \psi$.

Also see

 * Homomorphism of Generated Group