Definition:Affine Transformation

Definition
Let $\mathcal E$ and $\mathcal F$ be affine spaces with difference spaces $E$ and $F$ respectively.

Let $\mathcal L : \mathcal E \to \mathcal F$ be a mapping.

Then $\mathcal L$ is an affine transformation or affine mapping if there exists a linear transformation $L : E \to F$ such that for every pair of points $p,q \in \mathcal E$
 * $\displaystyle \mathcal L\left(q\right) = \mathcal L\left(p\right) + L\left(\vec{p q}\right)$