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 $\forall x \in S: \phi \left({x}\right) = \psi \left({x}\right)$.

Then $\phi = \psi$.

Also see
Compare with Homomorphism of Generated Group.