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  "body": "\n## Definition\n\nLet $k$ be a field.\nA *linear algebraic group* over $k$ is an [[algebraic group|GH81SYAPPN]] $G$ over $k$ whose underlying scheme is [[affine|B3V3XJ2W8G]].\n\n## Remarks\n\nEvery linear algebraic group over $k$ is isomorphic to a closed subgroup scheme of $\\mathrm{GL}_n$ for some $n$.\nSo an equivalent definition is: a closed subgroup scheme of some [[general linear group|0FCHE902D7]] $\\mathrm{GL}_n$.\nIn the classical language, with $k$ algebraically closed, it is a subgroup of $\\mathrm{GL}_n(k)$ that is the zero set of a set of polynomials in the matrix entries.\nMany authors also require $G$ to be smooth.\n\nExample: $\\mathrm{SL}_n = \\operatorname{Spec} k[x_{ij}] / (\\det - 1)$.\nIt is a closed subgroup of $\\mathrm{GL}_n$, cut out by $\\det = 1$, so it is affine.\n\nNon-example: an elliptic curve $E$ over $k$ is an algebraic group, but it is projective of positive dimension and so not affine.\n",
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