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  "body": "\n## Definition\n\nLet $F$ be a field and let $V$ be a vector space over $F$ with a [[basis|0KCQ9Y3QKA]] $(e_1, \\dots, e_n)$.\nLet $B$ be a [[bilinear form|0SQZMXB6GP]] on $V$.\nThe *matrix* of $B$ in this basis is the $n \\times n$ matrix $M$ with entries\n$$M_{ij} = B(e_i, e_j) \\qquad (1 \\le i, j \\le n).$$\n\n## Remarks\n\nIf $x$ and $y$ are the coordinate vectors of $u$ and $v$, then $B(u, v) = x^T M y$.\nThe map $B \\mapsto M$ is a linear bijection from the bilinear forms on $V$ to the $n \\times n$ matrices over $F$.\nFor the dot product on $\\R^n$ and the standard basis, $M$ is the identity matrix.\n\n## Explanations\n\n### L1 human\n\nA bilinear form is fixed by its values on pairs of basis vectors.\nWe store these $n^2$ values in a matrix.\nThen the form becomes a product of a row vector, the matrix and a column vector.\n",
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