$\mbox{GL}(n,\mathbb{F})$ refers to a [group](Groups.md) of [invertible](Operator%20inverse.md) $n \times n$ matrices containing elements in a number field $\mathbb{F}$. $\mbox{GL}$ aptly stands for _general linear_ since the general linear group encompasses all invertible matrices on a given number field, $\mathbb{F}$. Thus it is referred to as the _general linear group of degree $n.$_ $\mathbb{F}$ is usually $\mathbb{R}$ or $\mathbb{C}.$ # Properties In addition to the definition, since $\mbox{GL}(n,\mathbb{F})$ is a [group](Groups.md), the following properties are met: 1) $\forall A, B, \in \mbox{GL}(n,\mathbb{F}),$ $AB \in \mbox{GL}(n,\mathbb{F})$ 2) $\forall A, B, C\in \mbox{GL}(n,\mathbb{F}),$ $(AB)C = A(BC)$. 3) There exists $\mathbb{1}\in\mbox{GL}(n,\mathbb{F})$ such that $A\mathbb{1}=A$, $\forall A \in \mbox{GL}(n,\mathbb{F})$. In addition properties specific to $\mbox{GL}(n;\mathbb{F})$ are as follows: 4) $\mbox{GL}(n;\mathbb{F})$ is only a [subgroup](Subgroups.md) of itself. 5) $\mbox{GL}(n;\mathbb{R})$, $\mbox{GL}(n;\mathbb{C})$, and $\mbox{GL}(n;\mathbb{H})$ are [topological groups](Topological%20group.md). Property 1. is proven by showing that the product of any two invertible matrices is also [invertible](Operator%20inverse.md#Proof%20of%20property%201%20of%20invertible%20matrices). 2. is true by the associativity of Matrix multiplication, and 3. is true by the existence of the identity matrix, $\mathbb{1}$, which is also clearly invertible ( $\mathbb{1}$ is its own inverse). # Lie subgroups of GL(n,F) All [matrix Lie groups](Matrix%20Lie%20group.md) are [subgroups](Subgroups.md) of $\mbox{GL}(n;\mathbb{C})$. The corresponding [Lie algebras](Lie%20algebras.md) are $\mathfrak{gl}(n;\mathbb{C})$ and $\mathfrak{gl}(n;\mathbb{R})$. The former contains all matrices $X\in M_n{(\mathbb{C})}$ ([complex vector spaces](Complex%20vector%20spaces.md)) and the latter contains all matrices $X\in M_n{(\mathbb{R})}$ ([[Real vector spaces]]) and in each case the [matrix exponential](Matrix%20exponentials.md), $e^{tX}$, the elements of the group, are invertible. # Non-Lie subgroups of GL(n,F) To illustrate that not all matrix subgroups of $\mbox{GL}(n;\mathbb{C})$ are Lie Groups, consider the following examples: * The subgroup of $\mbox{GL}(2;\mathbb{C}):$ $\mbox{G} = \left\{\begin{pmatrix} e^{i t } & 0 \\ 0 & e^{i t a} \\ \end{pmatrix} ｜ t \in \mathbb{R} \right\}$ where $a$ is an irrational real number. Its closure is, $\overline{\mbox{G}} = \left\{\begin{pmatrix} e^{i \theta} & 0 \\ 0 & e^{i \phi} \\ \end{pmatrix} ｜ \theta, \phi \in \mathbb{R} \right\}$ This is the so called _irrational line_ of a torus. #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/Operators/Matrices