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 $p, q \in \mathcal E$:
 * $\mathcal L \left({q}\right) = \mathcal L \left({p}\right) + L \left({\vec{p q} }\right)$

Also see

 * Definition:Tangent Map of Affine Transformation