Endomorphism/Real/Two-dimensional invariant/Fact

Let ${}V\neq 0$ be a real finite-dimensional vector space, and let

$\varphi \colon V\longrightarrow V$

denote an endomorphism.

Then there exists a

${}\varphi$-invariant linear subspace ${}U\subseteq V$ of dimension

${}1$ or ${}2$.

Proof, Write another proof