Linear Transformation of Generated Module

Theorem
Let $R$ be a ring.

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

Let $\phi$ and $\psi$ be linear transformations from $G$ 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