Definition:Linear Transformation

General Definition
A linear transformation is a homomorphism from one module to another.

Linear Operator
A linear operator is a linear transformation from a module into itself.

Definition in a Vector Space
Let $V, W$ be vector spaces over a field (or, more generally, division ring) $K$.

A mapping $A: V \to W$ is said to be a linear transformation or a linear mapping iff:


 * $\forall v_1, v_2 \in V, \lambda \in K: A \left({\lambda v_1 + v_2}\right) = \lambda A \left({v_1}\right) + A \left({v_2}\right)$

Linear Operator
When in fact $V = W$, a linear transformation is called a linear operator.

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