## Affine Transformations

An affine transformation between two vector spaces consists of a linear transformation followed by a translation:

whereis a matrix andis a vector.
In two dimensions this becomes

Affine transformations send lines to lines and preserve ratios of distances.

In general, an affine transformation is composed of linear transformations (rotation, scaling or shear)and a translation. Several linear transformations can be combined into a single one, so that the general formula given above is still applicable. This implies that the set of all affine transformations with A invertible forms a group.

Proof (for affine transformations in 2 dimensions):

Closure:

Identity: The identity is and this is affine.

Inverses:Ifthen

$T^{-1}$
is

Similarly for

Other properties follow from the properties of matrix multiplication and addition of vectors.