\begin{definition}[Definition:Linear Transformation/Vector Space]
Let $V, W$ be vector spaces over a field (or, more generally, division ring) $K$.
A mapping $A: V \to W$ is a '''linear transformation''' {{iff}}:
:$\forall v_1, v_2 \in V, \lambda \in K: \map A {\lambda v_1 + v_2} = \lambda \map A {v_1} + \map A {v_2}$
That is, a homomorphism from one vector space to another.
\end{definition}