  An affine iterated function system is a finite collection of affine
invertible contractions and the invariant set associated to the mappings is
called self-affine. In 1988, Falconer proved that, for given matrices, the
Hausdorff dimension of the self-affine set is the affinity dimension for
Lebesgue almost every translation vectors. Similar statement was proven by
Jordan, Pollicott, and Simon in 2007 for the dimension of self-affine measures.
In this article, we have an orthogonal approach. We introduce a class of
self-affine systems in which, given translation vectors, we get the same
results for Lebesgue almost all matrices. The proofs rely on Ledrappier-Young
theory that was recently verified for affine iterated function systems by
B\'ar\'any and K\"aenm\"aki, and a new transversality condition, and in
particular they do not depend on properties of the Furstenberg measure. This
allows our results to hold for self-affine sets and measures in any Euclidean
space.
