TITLE: Does anyone know an interesting introductory topic involving vector spaces over the rationals
QUESTION [6 upvotes]: Many introductory books on vector spaces mention that the scalars need not be reals, and might even have sections discussing complex vector spaces or vector spaces over the integers mod 2.  I have never seen any such book mention that all of the theory goes through as well if one restricts the scalars to be just rational numbers.  Perhaps this is because there is a dearth of interesting problems about such vector spaces accessible at this level that couldn't simply be discussed in the context of real scalars.
I wonder if there is an interesting introductory-level problem or topic about vector spaces that would be most naturally conducted by allowing rational number scalars.  Does anyone know of such, perhaps one with a number-theoretic aspect?
(By introductory:  I envision a first course on linear algebra, including non-math majors.  They would be seeing vector spaces (and that level of abstraction) for the first time.  Perhaps they would be seeing matrix multiplication for the first time.  Usually, in my experience, such courses primarily use the real numbers as scalars.)

REPLY [5 votes]: Continuing Akhil's answer, let's prove a theorem of Dehn: if a rectangle is tiled by squares, the the ratio of the lengths of its sides is rational.
Suppose to the contrary that the sides of the rectangle $x,y$ are not rationally dependent. Then we can find some linear homomorphism $f\colon \mathbb{R} \rightarrow \mathbb{Q}$ such that $f(x) = 1$ and $f(y) = -1$.
We define the $f$-area $A(R)$ of a rectangle $R$ with edge lengths $h,v$ to be $f(h)f(v)$. If a rectangle $R$ is tiled by rectangles $R_i$ forming a grid, then from linearity it immediately follows that $$A(R) = \sum_i A(R_i).$$
Denote the big rectangle $R$ and the squares $S_i$. Take your tiling and extend all the lines to form a grid inside the rectangle. Denote the grid rectangles by $G_j$. Then $$A(R) = \sum_j A(G_j) = \sum_i A(S_i).$$ Since a square has both sides equal, $A(S_i) \geq 0$. On the other hand, by construction $A(R) < 0$. This contradiction shows that the two sides of the big rectangle are, in fact, rationally dependent.
Instead of taking a linear mapping from $\mathbb{R}$ to $\mathbb{Q}$ we could take a linear mapping from a smaller, finite dimensional domain by only considering the lengths in the grid $G_j$ - then everything becomes beginner's linear algebra.