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  "body": "\n## Definition\n\nLet $F$ be a field and let $V$ be a [[vector space|000AYPA9AJ]] over $F$.\nA *bilinear form* on $V$ is a map $B \\colon V \\times V \\to F$ such that, for all $u, v, w \\in V$ and $a \\in F$:\n\n1. $B(u + v, w) = B(u, w) + B(v, w)$ and $B(au, w) = a B(u, w)$.\n2. $B(u, v + w) = B(u, v) + B(u, w)$ and $B(u, aw) = a B(u, w)$.\n\n## Remarks\n\nThe [[dot product|0S52TNNTH9]] on $\\R^n$ is a bilinear form.\nFor each $n \\times n$ matrix $M$ over $F$, the map $B(x, y) = x^T M y$ is a bilinear form on $F^n$.\nThe bilinear forms on $V$ form a vector space over $F$; if $\\dim V = n$, it has dimension $n^2$.\nMore generally, a bilinear map $V \\times W \\to F$ uses two different spaces.\n\n## Explanations\n\n### L1 human\n\nA bilinear form takes two vectors and gives a number.\nIt is linear in the first vector and also linear in the second vector.\nThe dot product is the main example, but bilinear forms need not be symmetric or positive.\n",
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