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 $p, q \in \mathcal E$:

$\map {\mathcal L} q = \map {\mathcal L} p + \map L {\vec {p q} }$