Since both linear transformations and translations are bijective transformations, also their compositions are bijective transformations.

The composition of a linear transformation with a translation is said affine transformation.

An affine transformation can be algebraically expressed in the following way:

The vector **v'***(x’;y’)* which corresponds to the vector
**v***(x;y)* is done by

- the product between the the matrix
and the vector
**v** -
the sum between this product and the vector
**τ***(c;c’)*

The same procedure can be more synthetically expressed by the matrical equation

The inverse transformation expresses **v** in terms of **v’**. One obtains

Affine transformations preserve all the properties of geometric linear transformations.

Example.

The affine transformation

is done by the composition of the linear transformation with the translation

The determinant of L is Δ = - 3; the inverse of L is

So we obtain

The same result is expressed by algebraic system

A fixed point of a transformation T is a point that is mapped to itself.

The fixed point **u** of an affinity done by the composition of L and τ can be
determined by solving the equation

that is

In the given example we have

The inverse matrix of (L-I) is

The opposite of the product between this inverse matrix and |τ> gives the fixed point: