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  "body": "\n## Definition\n\nLet $F$ be a field and let $V$ be a vector space over $F$.\nA [[bilinear form|0SQZMXB6GP]] $B$ on $V$ is *symmetric* when\n$$B(u, v) = B(v, u) \\qquad \\text{for all } u, v \\in V.$$\n\n## Remarks\n\nIn each basis, $B$ is symmetric if and only if its matrix is a [[symmetric matrix|0VR671PHE5]].\nThe dot product on $\\R^n$ is symmetric.\nIf the characteristic of $F$ is not $2$, symmetric bilinear forms correspond one to one with [[quadratic forms|0RBQS1JJCR]].\n\n## Explanations\n\n### L1 human\n\nA symmetric bilinear form gives the same value for $(u, v)$ and for $(v, u)$.\nInner products on real spaces are the main examples.\nTheir matrices are symmetric.\n",
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