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  "body": "\n## Statement\n\nLet $V$ and $W$ be inner product spaces over $\\R$ or $\\C$, with the induced norms, and let $T \\colon V \\to W$ be a [[linear isometry|07SJXTWVQW]].\nThen, for all $u, v \\in V$,\n$$\\langle T(u), T(v) \\rangle = \\langle u, v \\rangle.$$\n\n## Remarks\n\nThe proof applies the [[polarisation identity|05FKHRT0B4]], which writes inner products in terms of norms.\nConversely, a linear map that keeps inner products keeps norms.\nSo a linear isometry also keeps angles and orthogonality.\n\n## Explanations\n\n### L1 human\n\nIf a linear map keeps lengths, then it also keeps angles.\nThe reason is that lengths determine inner products, by polarisation.\nSo rigid linear motions keep the whole geometry.\n",
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