Euclidean vector space/Isometry/Orthogonal/Fact

Let ${\displaystyle {}V}$ be a Euclidean vector space, and let ${\displaystyle {}u_{1},\ldots ,u_{n}}$ denote an orthonormal basis of ${\displaystyle {}V}$. Let

${\displaystyle \varphi \colon V\longrightarrow V}$

be a linear mapping, and let ${\displaystyle {}M}$ be the describing matrix of ${\displaystyle {}\varphi }$ with respect to the given basis.

Then ${\displaystyle {}\varphi }$ is an

isometry if and only if

${\displaystyle {}{M^{\text{tr}}}M=E_{n}\,}$

holds.