---- Let $R$ be a [[ring]] (probably at least an [[integral domain]]). > [!definition] Definition. ([[general linear group]]) > The $n^{th}$ **general linear group** over $R$, denoted $\text{GL}_{n}(R)$, is the [[group]] of [[unit|units]] in the [[matrix|matrix ring]] $\mathcal{M}_{n}(R)$, i.e., the [[group]] of [[inverse matrix|invertible]] $n \times n$ [[matrix|matrices]] with entries in $R$. > > $\text{GL}_{n}(\mathbb{R})$ is a [[Lie group]] of dimension $n^{2}$. $\text{GL}_{n}(\mathbb{C})$ is a [[Lie group]] of dimension $(2n)^{2}$. See [[general linear group is a Lie group]]. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```