Condition for Linear Transformation

Theorem
Let $G$ be a unitary $R$-module, and let $H$ be an $R$-module.

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

Then $\phi$ is a linear transformation iff:
 * $\forall x, y \in G: \forall \lambda, \mu \in R: \phi \left({\lambda x + \mu y}\right) = \lambda \phi \left({x}\right) + \mu \phi \left({y}\right)$

Proof
Any linear transformation clearly satisfies the condition.

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

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

Then $\phi \left({x + y}\right) = \phi \left({x}\right) + \phi \left({y}\right)$.

Now let $\mu = 0_R$.

Then $\phi \left({\lambda x}\right) = \lambda \phi \left({x}\right)$.

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