Condition for Linear Transformation

Theorem
Let $G$ be a unitary $R$-module.

Let $H$ be an $R$-module.

Let $\phi: G \to H$ be a mapping.

Then $\phi$ is a linear transformation :
 * $\forall x, y \in G: \forall \lambda, \mu \in R: \map \phi {\lambda x + \mu y} = \lambda \map \phi x + \mu \map \phi y$

Sufficient Condition
Any linear transformation clearly satisfies the condition.

Necessary Condition
Let $\phi$ be such that the condition is satisfied.

Let $\lambda = \mu = 1_R$.

Then:
 * $\map \phi {x + y} = \map \phi x + \map \phi y$

Now let $\mu = 0_R$.

Then:
 * $\map \phi {\lambda x} = \lambda \map \phi x$

Thus the conditions are fulfilled for $\phi$ to be a homomorphism, that is, a linear transformation.