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  "body": "\n## Statement\n\nLet $F$ be a field whose [[characteristic|0F6ES3BRYP]] is not $2$.\nLet $V$ be a [[finite-dimensional vector space|0ZHWKQMVX5]] over $F$ with a [[nondegenerate|01NHFZF2SK]] [[symmetric bilinear form|0AXZNE5XNH]] $B$.\nLet $U$ and $W$ be [[linear subspaces|00YC1D652K]] of $V$, and let $\\sigma \\colon U \\to W$ be a linear bijection with $B(\\sigma(x), \\sigma(y)) = B(x, y)$ for all $x, y \\in U$.\nThen there is a linear bijection $\\tau \\colon V \\to V$ with $B(\\tau(x), \\tau(y)) = B(x, y)$ for all $x, y \\in V$ and $\\tau(x) = \\sigma(x)$ for all $x \\in U$.\n\n## Remarks\n\nThe usual proof is by induction on $\\dim U$, and it builds $\\tau$ as a product of reflections in non-isotropic vectors; dividing by $2$ is needed to write these reflections.\nA consequence is Witt cancellation: if $V_1 \\perp V_2$ and $V_1 \\perp V_3$ are isometric and $V_1, V_2, V_3$ are nondegenerate, then $V_2$ and $V_3$ are isometric.\nAnother consequence: all maximal totally isotropic subspaces of $V$ have the same dimension, the Witt index.\nFor example, in $\\R^n$ with the dot product, any isometric map between two subspaces extends to an orthogonal map of $\\R^n$.\n\nThe hypothesis on the characteristic is necessary for symmetric forms.\nOver $\\F_2$, take $V = \\F_2^3$ with the dot product, $u = (1,1,1)$ and $e_1 = (1,0,0)$; both have $B(x,x) = 1$, so $e_1 \\mapsto u$ is an isometry between lines.\nAn extension to $V$ would map the orthogonal complement of $e_1$ onto that of $u$.\nBut the complement of $e_1$ contains $(0,1,0)$ with $B(x,x) = 1$, while every $x$ in the complement of $u$ has $B(x,x) = x_1 + x_2 + x_3 = 0$.\nThe theorem holds in every characteristic for nondegenerate [[alternating forms|0DGDS5XY84]] and for quadratic forms with nondegenerate polar form.\n\n## Explanations\n\n### L1 human\n\nA symmetric bilinear form measures something like lengths and angles, possibly with signs.\nWitt's theorem says that if two configurations of vectors look the same for the form, then a symmetry of the whole space carries one onto the other.\nSo the symmetries of the form can carry any configuration onto any other configuration that the form cannot tell apart.\n\n### L2 human\n\nWitt's theorem is the foundation of the algebraic theory of [[quadratic forms|0RBQS1JJCR]].\nCancellation reduces the classification of forms to anisotropic forms plus a number of hyperbolic planes, the Witt decomposition.\nThis decomposition leads to the [[Witt group|KFVY0RPGF2]] of a field.\nFor real forms, cancellation gives a short proof of Sylvester's law of inertia.\n",
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