Permutation matrix/Isometry/Exercise

Let ${}\pi \in S_{n}$ be a permutation, and let

$M_{\pi }\colon \mathbb {R} ^{n}\longrightarrow \mathbb {R} ^{n}$

denote the corresponding permutation matrix and the corresponding linear mapping. Show that ${}M_{\pi }$ is an isometry. When is it a proper isometry?

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