A linear map of the plane which preserves linear independence over Q but not over R

Question

This is directly from Stewart & Tall's Algebraic Number Theory & FLT, chapter 8, Exercise 6, page 150. I can construct maps which do weird things but haven't been able to do this. Matrix representations haven't paid off YET. This map obviously must be non-singular. I have tried to construct sequences whose limit loses linearity but to no avail.

Answer 1

I assume "linear" here means $\mathbb Q$-linear. Let $v_1=(1,0)$, $v_2=(\sqrt 2,0)$, $v_3=(0,1)$. On the $\mathbb Q$-span of these three vectors we have a linear automorphism that maps $v_1\to v_2$, $v_2\to v_3$, $v_3\to v_1$. Extend that to a $\mathbb Q$-linear automorphism $f$ of $\mathbb R^2$ (uses the axiom of choice). As $f(v_1)$ and $f(v_3)$ are $\mathbb R$-linear dependent but $v_1,v_3$ are not, this seems to be what you are looking for.