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  "body": "\n## Description\n\nLet $R$ be a commutative ring and let $n \\ge 1$ be an integer.\nThe *general linear group* $GL_n(R)$ is the set of [[invertible|0EB9N0S33C]] $n \\times n$ matrices over $R$, with [[matrix multiplication|032Q9FB8MT]] as operation.\nIt is a [[group|G5S1M9]]: the product of invertible matrices is invertible, the identity element is $I_n$, and the inverse of $A$ is $A^{-1}$.\nEquivalently, $GL_n(R)$ is the group of units of the [[matrix ring|0FHVPRAE60]] $M_n(R)$.\n\n## Remarks\n\nFor a field $F$, $GL_n(F)$ is isomorphic to the group of linear isomorphisms of $F^n$ to itself.\nFor $n \\ge 2$, the group $GL_n(F)$ is not abelian.\nFor a finite field with $q$ elements, $|GL_n(\\F_q)| = (q^n - 1)(q^n - q) \\cdots (q^n - q^{n-1})$.\nThe [[special linear group|0V9PMN0RBH]] is the subgroup of matrices with determinant $1$.\n\n## Explanations\n\n### L1 human\n\nThe general linear group collects all the invertible matrices of one size.\nWe can multiply two of them, and we can undo each of them, so they form a group.\nGeometrically, its elements are all the ways to transform space linearly without flattening it.\n",
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