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 by $R$-algebraic structure homomorphism the conditions are fulfilled for $$\phi$$ to be a homomorphism, that is, a linear transformation.