# Definition The general linear group of degree $n$ is the set of $n\times n$ [[Non-singular Matrix|invertible matrices]] together with ordinary [[Matrix]] multiplication. This forms a group denoted by $GL(n, \mathbb{F})$, where $\mathbb{F}$ is the [[field]] from which the entries of the matrices are taken. This group does not form a vector space. When $\mathbb{F} = \mathbb{R}$, we have $GL(n, \mathbb{R})$, the *real general linear group in $n$ dimensions*, and similarly when $\mathbb{F} = \mathbb{C}.$ # Vector Spaces Given a [[vector space]] $V$ over a field $\mathbb{F}$, the general linear group is written as $GL(V)$ or $\text{Aut}(V)$ and is the group of all [[Automorphism|automorphisms]] of $V$ together with functional composition as the group operation. In other words, $GL(V)$ is the subset of $\mathcal{L}(V)$ (see [[Linear Operator#Vector Space|linear operators as a vector space]]) consisting of all *invertible* linear operators on $V $. # The Isomorphism If $V$ is a [[vector space]] over a [[field]] $\mathbb{F}$ with finite [[dimension]] $n$, then $GL(n,\mathbb{F})$ and $GL(V)$ are [[Group Isomorphism|isomorphic]]. To see this, consider the following. If we pick a basis for $V$, then for each $T \in GL(V)$ we get an invertible matrix $[T] \in M_n(\mathbb{F})$. This matrix belongs to $GL(n, \mathbb{F})$ discussed above by virtue of being [[Non-singular Matrix|invertible]]. One can check that this map is both one-to-one and onto, and since one can show that $[TU]=[T][U]$ we have an isomorphism between the two groups and $ GL(n, \mathbb{F}) \cong GL(V) $ # Lie Algebra $GL(n,\mathbb{F})$ is a [[Matrix Lie Group]], thus, its [[Lie Algebra]] is defined as: $ \mathfrak{gl}(n, \mathbb{F}) \equiv \left\{X \in M_n(\mathbb{F}) \, | \, e^{tX} \in GL(n, \mathbb{F})\, \forall t \in \mathbb{R}\right\} $ However, note that, for a given $X \in M_n(\mathbb{F})$, $e^{tX}$is invertible for all $t \in \mathbb{R}$ since its inverse is just $e^{-tX}$. That is, $e^{tX} \in GL(n, \mathbb{F}) \forall t \in \mathbb{R}$. Therefore $ \begin{align} \mathfrak{gl}(n, \mathbb{C}) &= M_n(\mathbb{C})\\ \mathfrak{gl}(n, \mathbb{R}) &= M_n(\mathbb{R}) \end{align} $ Thus, the Lie algebra of the group of [[Non-singular Matrix|invertible matrices]] is simply the [[vector space]] of all [[matrix|matrices]]. Due to the [[group isomorphism]] discussed [[#The Isomorphism|above]], we also have that the Lie algebra of the group of invertible [[Linear Operator|linear operators]] is simply the vector space of all linear operators $\mathcal{L}(V)$ equipped with a Lie bracket, and it is denoted $\mathfrak{gl}(V)$. # Isomorphism of Lie Algebras There are two Lie algebras to talk about here, $\mathfrak{gl}(V)$, the Lie algebra of all linear operators on an arbitrary vector space $V$, and $\mathfrak{gl}(n, \mathbb{F})$, the Lie algebra of all linear operators (in this case, matrices) acting on an $n$-dimensional vector space over a field $\mathbb{F}$ (with $n$ finite). If $V$ above is finite dimensional over a field $\mathbb{F}$, and has dimension $n$, then we have: $ \mathfrak{gl}(V) \cong \mathfrak{gl}(n, \mathbb{F}) $ where the $\cong$ denotes a [[Lie Algebra Isomorphism]].