General linear group

Let $V$ be a finite-dimensional real or complex vector space.

Definition (General linear group).
The general linear group of $V$ is

with group operation given by composition. After choosing a basis, $\mathrm{GL}(V)\cong \mathrm{GL}_n(\mathbb F)$ where $\mathbb F=\mathbb R$ or $\mathbb C$ and

Lie group structure.
Viewed as a subset of the affine space $M_n(\mathbb F)$, $\mathrm{GL}_n(\mathbb F)$ is open (since $\det$ is continuous and $\mathbb F^\times$ is open), hence it is a smooth manifold and a Lie group . Its Lie algebra is the general linear Lie algebra $\mathfrak{gl}_n(\mathbb F)$, identified with $T_I\mathrm{GL}_n(\mathbb F)$ (compare Lie algebra of a Lie group ).

Basic structure.
Over $\mathbb C$, $\mathrm{GL}_n(\mathbb C)$ is connected. Over $\mathbb R$, $\mathrm{GL}_n(\mathbb R)$ has two connected components distinguished by the sign of $\det$. The exponential map is the matrix exponential $\exp:\mathfrak{gl}_n(\mathbb F)\to \mathrm{GL}_n(\mathbb F)$.

Context.
Many linear representations of Lie groups are concretely homomorphisms into some $\mathrm{GL}(V)$; special subgroups such as SL_n , O(n) , and U(n) are defined by additional algebraic constraints.