As exam preparation we were trying to proof the following task:
Let $V=\mathbb{R}^2$ and let $\phi$ be an endomorphism of $V$ with $\phi \circ \phi = id$ and $\phi \neq id$ and $\phi \neq -id$. Proof that this implies the existence of a basis $B=(b_1,b_2)$ of $V$ with $\phi(b_1) = b_1$ and $\phi(b_2)=-b_2$
Unfortunately we aren't able to solve that task and would very much appreciate some proofs and and a "how to" of how to approach such problems.
Thanks for your help.