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 $\phi = \psi$.

Proof
Let $y \in G$ be arbitrary.

Then by definition of generator, $y$ is the linear combination of elements of $S$:


 * $\ds y = \sum_{k \mathop = 1}^n \lambda_k a_k$

for $a_1, a_2, \ldots, a_n \in S, \lambda_1, \lambda_2, \ldots, \lambda n \in R$

Then:

Also see

 * Homomorphism of Generated Group