  In this work we define the trajectory coset of an affine map and use it to
study the similarity classes of affine maps. We use the trajectory coset, a
tool which allows us to gain a deeper understanding of the interplay between
geometry (properties of affine maps) and algebra (properties of linear maps).
We first state the geometric problem of similarity of affine maps. We then
develop the algebraic tools. The main idea is the development of an invariant
which determines whether one coset can be taken to another coset, under
isomorphism of modules. After resolving this problem we go back to geometrical
questions, similarity and invariant flats.
