Euclidean vector space/Linear mapping/Angle-preserving/Structure/Fact

Let

$\varphi \colon V\longrightarrow V$

denote an angle-preserving linear mapping on the Euclidean vector space ${}V$.

Then there exists an

isometry

$\psi \colon V\longrightarrow V$

and a homothety

$\sigma \colon V\longrightarrow V$

such that

${}\varphi =\sigma \circ \psi \,.$

Proof, Write another proof