Possible invariant subspaces of $\mathbb{Q}^2$

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.

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1 Answer

Any proper invariant subspace would be a $1$ - dimensional $\Bbb{Q}$ - subspace. In other words, it must be spanned by an eigenvector with rational eigenvalue. But the characteristic polynomial of your matrix above is irreducible over $\Bbb{Q}$, so what can you deduce?

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