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
and the vector vThe 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:
