----- > [!proposition] Proposition. ([[general linear group is a Lie group]]) > The [[general linear group|general linear groups]] $\text{GL}_{n}(\mathbb{R})$ and $\text{GL}_{n}(\mathbb{C})$ are [[Lie group|Lie groups]] of dimension $n^{2}$ and $2n^{2}$ respectively. ^proposition > [!proof]+ Proof. ([[general linear group is a Lie group]]) > First assume $\mathbb{R}$ is the base [[field|field]]. We know that $\text{GL}_{n}(\mathbb{R}) = \{ \boldsymbol A \in \mathbb{R}^{n^{2}} : \det A \neq 0 \}.$The [[determinant]] map $\mathbb{R}^{n^{2}} \to \mathbb{R}$ is [[continuous]] ([[determinant of a matrix|it is a polynomial]]). Since $\text{GL}_{n}(\mathbb{R})=\det ^{-1}(\{ \mathbb{R} \setminus 0 \})$, it is an open subset of $\mathbb{R}^{n^{2}}$ as the inverse image of the open set $\mathbb{R} \setminus \{ 0 \}$ under $\det$. We therefore can cover it with a single $n^{2}$-dimensional coordinate chart: the [[identity map]] on $\mathbb{R}^{n^{2}}$ restricted to this open subset. Multiplication and inversion are [[smooth maps between manifolds|smooth]] because both are [[polynomial 4|polynomials]] or [[matrix inverse is adjugate over determinant|ratios thereof]]. ----- #### ---- #### 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 > ```