Affine map vs. affine transformation

I am reading the Wikipedia page about affine space here. To fix notation, we let $$A$$ be underlying set and $$\vec{A}$$ be the vector space that acts on the set (free and transitively).

The page mentions two types of functions between affine spaces. One is an "affine map," which is a function $$f\colon A\rightarrow B$$ such that the map $$\vec{f}\colon\vec{A}\rightarrow\vec{B}$$ via $$b-a\mapsto f(b)-f(a)$$ is a well-defined linear map. This seems like the most "natural" type of map for me, as it preserves all of the structure since we would then have $$f(a+v)=f(a)+\vec{f}(v)$$.

There is then another page about "affine transformations", which seems to have a very geometrical definition based on parallel lines. Right before the table of contents, the page says that an affine map is a generalization of an affine transformation.

Is there a relation between affine maps and affine transformations? Are the latter just invertible affine maps? I'm kind of confused by how all of this works, and if anyone had a good reference that would be even better.

1 Answer

What you call an affine transformation is an automorphism of an affine space, that is, a biyective affine map from an affine space $$A$$ into itself. Affine maps are a generalization of affine transformations because not every affine map is, for example, biyective, neither it has to go from an affine space into itself.

An example of an affine transformation could be $$f:\mathbb{R} \rightarrow \mathbb{R}$$ such that $$f(x)=-x$$ (it is indeed linear). An example of an affine map could be any constant function from an affine space to another one