Euclidean vector space/Automorphism/Preserves orthogonality, no isometry/Exercise

Give an example of a bijective linear mapping

$\varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{3}$

that is not an isometry but fulfills for all ${}u,v\in V$ the relation

$\left\langle u,v\right\rangle =0{\text{ if and only if }}\left\langle \varphi (u),\varphi (v)\right\rangle =0.$

Create a solution