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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 *alternating* when\n$$B(v, v) = 0 \\qquad \\text{for all } v \\in V.$$\n\n## Remarks\n\nAn alternating form is *skew-symmetric*: $B(u, v) = -B(v, u)$ for all $u, v$.\nIf the characteristic of $F$ is not $2$, each skew-symmetric form is alternating.\nOn $F^2$, the form $B(x, y) = x_1 y_2 - x_2 y_1$ is alternating; it is the determinant of the matrix with the columns $x$ and $y$.\nA nondegenerate alternating form is a *symplectic form*.\n\n## Explanations\n\n### L1 human\n\nAn alternating form gives zero when both vectors are equal.\nThen swapping the two vectors changes the sign of the value.\nThe signed area of the parallelogram spanned by two plane vectors is an example.\n",
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