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  "body": "\n## Definition\n\nLet $F$ be a field and let $V$ be a [[finite-dimensional vector space|0ZHWKQMVX5]] over $F$.\nA *symplectic form* on $V$ is a [[bilinear form|0SQZMXB6GP]] $\\omega \\colon V \\times V \\to F$ such that:\n\n1. $\\omega$ is [[alternating|0DGDS5XY84]]: $\\omega(v, v) = 0$ for all $v \\in V$.\n2. $\\omega$ is [[nondegenerate|01NHFZF2SK]]: if $\\omega(v, w) = 0$ for all $w \\in V$, then $v = 0$.\n\nA *symplectic vector space* is a pair $(V, \\omega)$ of such a space $V$ and a symplectic form $\\omega$ on $V$.\n\n## Remarks\n\nExample: on $F^{2n}$ with coordinates $(x_1, \\dots, x_n, y_1, \\dots, y_n)$, the standard form is $\\omega(u, u') = \\sum_{i=1}^n (x_i y'_i - x'_i y_i)$.\nIt is alternating, since each term vanishes when $u = u'$.\nIt is nondegenerate: if $u$ has $x_i \\ne 0$, then $\\omega(u, f_i) = x_i \\ne 0$, and if $u$ has $y_i \\ne 0$, then $\\omega(u, e_i) = -y_i \\ne 0$.\nHere $e_i$ and $f_i$ are the standard basis vectors for the coordinates $x_i$ and $y_i$.\nNon-example: the form $\\omega(u, u') = x_1 y'_1 - x'_1 y_1$ on $F^3$ with coordinates $(x_1, y_1, z)$ is alternating but degenerate, because $(0, 0, 1)$ pairs to zero with every vector.\nEvery symplectic vector space has even dimension $2n$ and a basis $e_1, \\dots, e_n, f_1, \\dots, f_n$ with $\\omega(e_i, f_j) = \\delta_{ij}$ and $\\omega(e_i, e_j) = \\omega(f_i, f_j) = 0$.\nIn such a basis, the matrix of $\\omega$ is $J = \\begin{pmatrix} 0 \u0026 I_n \\\\ -I_n \u0026 0 \\end{pmatrix}$.\nIn characteristic other than $2$, alternating is the same as skew-symmetric: $\\omega(v, w) = -\\omega(w, v)$.\n",
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