Consider the linear map $T : \mathbb{Q}^2 \rightarrow \mathbb{Q}^2$, associated with the following matrix $T = \begin{pmatrix}2 & 1\\1&-2\end{pmatrix}$
I want to show that $\bf 0$ and $\mathbb{Q}^2$ are the only invariant subspaces under $T$. How does this change when $T$ is a real map instead of a rational?
I can't see how being a rational map has influence on the invariant subspaces.
This is an exercise for a course, but is not to be handed in.