In Linear Algebra Done Right, Axler very briefly touches on affine subsets. My understanding is that they are subspaces that have been translated. He also gives a definition of parallel affine sets.
I'm trying to understand how we extend vector spaces to geometry. Does this mean that vector spaces can only describe $n$-dimensional Euclidean space, because lines are affine subsets and can be parallel to one another? What does this imply about the parallel postulate?
Edit: Axler gives a theorem (3.85) that two affine sets are either equal or disjoint. Isn't that just the parallel postulate? Does that mean we somehow implicitly assume the parallel postulate when defining vector spaces?
The fact that it appears that vector spaces can only describe Euclidean space is the result of the following two observations:
$\mathbb{R}^n$ gives a model for Euclidean space where the lines correspond to the one-dimensional (affine) subspaces
Any finite-dimensional vector space (over the real numbers) is isomorphic to $\mathbb{R}^m$ for some $m\in\mathbb{N}_0$
However this does not mean that vector spaces cannot model other types of geometries. As @LordSharkTheUnknown mentioned, a famous example is Minkowski space. As a vector space it is isomorphic to $\mathbb{R}^n$, but the Minkowski metric (which gives extra structure besides the vector space structure) turns it into a model for hyperbolic geometry. So the important thing (or at least one of them) which induces a certain geometry is the metric structure, not the vector space structure.