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

Proof

Let

$r=det\phi ,$

and set

$s:=\sqrt[n]{|r|},$

where $n$ denotes the dimension of $V$. Let $\sigma$ be the homothety with factor $s$. We consider the mapping

$\psi :={\sigma }^{-1}\circ \phi .$

This mapping is still angle-preserving, and its determinant is $1$ or $-1$. Because of exercise, $\psi$ is an isometry.

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