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  "body": "\n## Definition\n\nLet $F$ be a field of characteristic not $2$, and let $V$ be a finite-dimensional vector space over $F$.\nLet $Q$ be a [[quadratic form|0RBQS1JJCR]] on $V$.\nThe *symmetric bilinear form associated with* $Q$ is\n$$B(u, v) = \\tfrac{1}{2} \\bigl( Q(u + v) - Q(u) - Q(v) \\bigr) \\qquad (u, v \\in V).$$\n\n## Remarks\n\n$B$ is a [[symmetric bilinear form|0AXZNE5XNH]] with $B(v, v) = Q(v)$, and it is the only one.\nConversely, each symmetric bilinear form $B$ gives the quadratic form $v \\mapsto B(v, v)$; the two constructions are inverse to each other.\nFor $Q(x) = x^T A x$ with $A$ symmetric, $B(x, y) = x^T A y$.\nIt is half of the [[polar form|0HYFWKKY3C]] of $Q$.\n\n## Explanations\n\n### L1 human\n\nOutside characteristic $2$, quadratic forms and symmetric bilinear forms are the same thing in two forms.\nFrom $B$, we get $Q$ by putting the same vector twice.\nFrom $Q$, we get $B$ by this formula.\n",
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