\begin{definition}[Definition:Affine Transformation]
Let $\EE$ and $\FF$ be affine spaces with difference spaces $E$ and $F$ respectively.
Let $\LL: \EE \to \FF$ be a mapping.
Then $\LL$ is an '''affine transformation''' or '''affine mapping''' if there exists a linear transformation $L: E \to F$ such that for every pair of points $p, q \in \EE$:
:$\map \LL q = \map \LL p + \map L {\vec {p q} }$
\end{definition}