Definition:Linear Transformation

Definition
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 :
 * $(1): \quad \forall x, y \in G: \map \phi {x +_G y} = \map \phi x +_H \map \phi y$
 * $(2): \quad \forall x \in G: \forall \lambda \in R: \map \phi {\lambda \circ x} = \lambda \otimes \map \phi x$

Also denoted as
It is commonplace in the literature devoted to linear transformations for the argument not to be put in parenthesis:

That is, $A h$ would be used for $\map A h$, as long as the context makes this clear.

Also known as
The term linear mapping can sometimes be found, which means the same thing as linear transformation.

Some sources use the term module homomorphism.

Some authors, specifically in the field of functional analysis, use the term linear operator (or even just operator) for arbitrary linear transformations.

Some authors use the term linear functional, especially in the field of category theory.