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  "body": "\n## Definition\n\nLet $V$ and $W$ be [[normed vector spaces|03J5RAKB30]] over $\\R$ or $\\C$.\nA *linear isometry* from $V$ to $W$ is a [[linear map|0NRYHJPQG2]] $T \\colon V \\to W$ with\n$$\\norm{T(v)} = \\norm{v} \\qquad \\text{for all } v \\in V.$$\n\n## Remarks\n\nA linear isometry is injective, because $T(v) = 0$ implies $\\norm{v} = 0$.\nIt keeps distances: $\\norm{T(u) - T(v)} = \\norm{u - v}$.\nBetween inner product spaces, it keeps inner products, see [[the next statement|0R7VXTYHYZ]].\nBetween spaces of the same finite dimension, a linear isometry is bijective.\n\n## Explanations\n\n### L1 human\n\nA linear isometry is a linear map that keeps all lengths.\nRotations and reflections of the plane are examples.\nA map that keeps lengths cannot squeeze anything to zero.\n",
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