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 a vector space over fields (or, more generally, division rings) $K, L$, respectively.

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


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

Note that it is understood that there is also a mapping $A: K \to L$.

Upon encountering linear mappings, it is very likely that in fact $K$ is equal to $L$.

It is then common to impose that the mapping $A: K \to L$ be the identity mapping.