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  "body": "\n## Definition\n\nLet $F$ be a field, let $n \\ge 1$, and let $V$ and $W$ be vector spaces over $F$.\nA [[multilinear map|0443XNARDV]] $f \\colon V^n \\to W$ is *alternating* when $f(v_1, \\dots, v_n) = 0$ for each family with $v_i = v_j$ for some $i \\ne j$.\n\n## Remarks\n\nAn alternating map changes sign when two arguments are exchanged.\nIf the characteristic of $F$ is not $2$, then this sign rule implies that the map is alternating.\nAn alternating map is zero on each linearly dependent family.\nThe [[determinant|00KDZRW4C0]] is an alternating multilinear map of the rows of a square matrix.\n\n## Explanations\n\n### L1 human\n\nAn alternating map gives zero as soon as two inputs are the same.\nIt follows that swapping two inputs changes the sign of the value.\nThe signed area of the parallelogram of two plane vectors is an alternating bilinear map.\n",
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