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  "body": "\n## Definition\n\nLet $V$ be an inner product space over $\\R$ or $\\C$, and let $W$ be a finite-dimensional subspace of $V$, with [[orthogonal projection|0B8KZRT28M]] $P_W$.\nThe *reflection* in $W$ is the linear map\n$$R_W \\colon V \\to V, \\qquad R_W(v) = 2 P_W(v) - v.$$\n\n## Remarks\n\n$R_W$ fixes each vector of $W$, negates each vector of $W^\\perp$, and satisfies $R_W^2 = \\id_V$.\nIt is a linear isometry.\nFor a hyperplane $W = a^\\perp$ with $a \\ne 0$, the formula is $R_W(v) = v - 2 \\frac{\\langle v, a \\rangle}{\\langle a, a \\rangle} a$.\nIn $\\R^n$, the matrix of a hyperplane reflection is orthogonal with determinant $-1$.\n\n## Explanations\n\n### L1 human\n\nA reflection is a mirror image in a subspace.\nVectors in the mirror stay where they are, and vectors perpendicular to it change direction.\nReflecting twice brings every vector back.\n",
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