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

Then $$\phi = \psi$$.

Also see
Compare with Homomorphism of Generated Group.