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  "body": "\n## Summary\n\nA linear algebraic group is a group of invertible matrices defined by polynomial equations in the entries.\nEquivalently, it is an [[algebraic group|GH81SYAPPN]] whose underlying scheme is affine.\nExamples are the [[general linear group|0FCHE902D7]], the [[special linear group|0V9PMN0RBH]], the [[orthogonal groups|0TKYAEJ8N7]] and the [[symplectic groups|X12FS2JAP0]].\nTheir structure theory, through tori, Borel subgroups and root systems, is a large part of modern algebra and number theory.\n\n## Explanations\n\n### L1 human\n\nThink of a group of matrices cut out by equations, such as the matrices $A$ with $A^{T} A = I$, or those with determinant $1$.\nSuch a group is at the same time a group and a geometric shape in the space of matrices.\nLinear algebraic groups are the algebraic counterpart of matrix Lie groups, and they make sense over any field, including finite fields.\nOver a finite field they give many of the finite simple groups.\n",
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