Vector space/K/Finite-dimensional/Endomorphism/Invariant linear subspaces/Orthogonal complement/Fact

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, endowed with an inner product ${}\left\langle -,-\right\rangle$. Let

$\varphi \colon V\longrightarrow V$

denote an endomorphism.

Then a

linear subspace ${}U\subseteq V$ is ${}\varphi$-invariant if and only if the orthogonal complement

${}U^{\perp }$ is invariant under ${}{\hat {\varphi }}$.

Proof, Write another proof