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  "body": "\n## Definition\n\nLet $F$ be a field, let $n \\ge 1$, and let $V_1, \\dots, V_n$ and $W$ be vector spaces over $F$.\nA map $f \\colon V_1 \\times \\cdots \\times V_n \\to W$ is *multilinear* when it is linear in each variable separately.\nThat is, for each $i$ and each fixed choice of $v_j \\in V_j$ for $j \\ne i$, the map $v \\mapsto f(v_1, \\dots, v_{i-1}, v, v_{i+1}, \\dots, v_n)$ is a [[linear map|0NRYHJPQG2]] $V_i \\to W$.\n\n## Remarks\n\nFor $n = 1$, a multilinear map is a linear map, and for $n = 2$ it is a *bilinear map*.\nThe dot product on $\\R^n$ and the matrix product $M_n(F) \\times M_n(F) \\to M_n(F)$ are bilinear.\nThe [[determinant|00KDZRW4C0]] is multilinear in the $n$ columns of a matrix.\nA multilinear map is in general not linear on the product space: $f(2v_1, 2v_2) = 4 f(v_1, v_2)$ for a bilinear $f$.\n\n## Explanations\n\n### L1 human\n\nA multilinear map takes several vectors and is linear in each one, while the others stay fixed.\nThe product $xy$ of two real numbers is a simple example: it is linear in $x$ and linear in $y$.\nIt is not linear in the pair $(x, y)$.\n",
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