\begin{definition}[Definition:Linear Transformation]
A '''linear transformation''' is a homomorphism from one module to another.
Hence, let $R$ be a ring.
Let $M = \struct {G, +_G, \circ}_R$ and $N = \struct {H, +_H, \otimes}_R$ be $R$-modules.
Let $\phi: G \to H$ be a mapping.
Then $\phi$ is a '''linear transformation''' {{iff}}:
:$(1): \quad \forall x, y \in G: \map \phi {x +_G y} = \map \phi x +_H \map \phi y$
:$(2): \quad \forall x \in G: \forall \lambda \in R: \map \phi {\lambda \circ x} = \lambda \otimes \map \phi x$
\end{definition}