Affine map ($x\mapsto Ax+b$)

A map between affine spaces whose difference map is linear. After choosing origins it is $x\mapsto Ax+b$, where $A$ is its linear part of an affine map and $b$ its translation. This definition allows noninvertible maps; invertibility requires $A$ invertible.

The map $A$ associated with an affine map $x\mapsto Ax+b$; it is independent of the translation $b$. An affine Euclidean isometry has an orthogonal linear part, and an isometry between flat tori has a lift whose linear part carries one Euclidean lattice onto the other.