Euclidean vector space/Orthonormal basis and isometry with R^n/Exercise

Let ${}V$ be a Euclidean vector space of dimension ${}n$. Show that a family of vectors ${}u_{1},\ldots ,u_{n}\in V$ is an orthonormal basis of ${}V$ if and only if the corresponding linear mapping

$\mathbb {R} ^{n}\longrightarrow V,e_{i}\longmapsto u_{i},$

is an isometry between ${}\mathbb {R} ^{n}$ and ${}V$.

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