Reference for computations with affine subspaces

Question

I'm writing a paper in logic that involves some affine geometry. Specifically, I need the fact that given (some finite representation of) affine subspaces $A,B$ of $\mathbb{Q}^n$ and an affine transformation $f\colon \mathbb{ℚ}^n \rightarrow \mathbb{Q}^m$, it is possible to compute (finite representations of) $A \vee B, A\cup B, f(A)$, and $f^{-1}(A)$. It's perfectly clear how this works e.g. if the spaces are represented by coordinate systems or linear equation systems, but writing out the calculations doesn't really achieve anything in the context of the paper. Therefore, I would rather cite a book on affine geometry. I've looked at Berger's Geometry I and Snapper and Troyer's Metric Affine Geometry, but neither contains the material I need. Can anyone recommend a reference?