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  "body": "\n## Summary\n\nA symplectic vector space is a vector space with a nondegenerate [[alternating bilinear form|0DGDS5XY84]], called the symplectic form.\nA finite-dimensional symplectic vector space has even dimension and a standard basis in which the form has one fixed shape.\nThe linear maps that keep the form make up the [[symplectic group|X12FS2JAP0]].\nSymplectic vector spaces are the linear model of phase space in classical mechanics and of symplectic manifolds.\n\n## Explanations\n\n### L1 human\n\nAn inner product measures lengths and angles; a symplectic form measures signed areas instead.\nOn $\\R^2$, the form $\\omega((x_1, y_1), (x_2, y_2)) = x_1 y_2 - x_2 y_1$ gives the signed area of the parallelogram spanned by two vectors.\nEvery vector has area zero with itself, but each nonzero vector has nonzero area with some other vector.\nIn higher dimensions, the space splits into planes like this one, so the dimension is always even.\n",
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