Editing Lie group (section)

=== First examples ===
* The 2×2 [[real number|real]] [[invertible matrix|invertible matrices]] form a group under multiplication, called [[general linear group|general linear group of degree 2]] and denoted by <math>\operatorname{GL}(2, \mathbb{R})</math> or by {{tmath|1= \operatorname{GL}_2(\mathbb{R}) }}: <math display="block">\operatorname{GL}(2, \mathbb{R}) = \left\{A = \begin{pmatrix}a & b\\c & d\end{pmatrix} : \det A = ad-bc \ne 0\right\}.</math> This is a four-dimensional [[compact space|noncompact]] real Lie group; it is an open subset of {{tmath|1= \mathbb R^4 }}. This group is [[connected space|disconnected]]; it has two connected components corresponding to the positive and negative values of the [[determinant]].
* The [[rotation (mathematics)|rotation]] matrices form a [[subgroup]] of {{tmath|1= \operatorname{GL}(2, \mathbb{R}) }}, denoted by {{tmath|1= \operatorname{SO}(2, \mathbb{R}) }}. It is a Lie group in its own right: specifically, a one-dimensional compact connected Lie group which is [[diffeomorphic]] to the [[circle]]. Using the rotation angle <math>\varphi</math> as a parameter, this group can be [[parametric equations|parametrized]] as follows: <math display="block">\operatorname{SO}(2, \mathbb{R}) = \left\{\begin{pmatrix} \cos\varphi & -\sin\varphi \\ \sin\varphi & \cos\varphi \end{pmatrix} : \varphi \in \mathbb{R}\ /\ 2\pi\mathbb{Z}\right\}.</math> Addition of the angles corresponds to multiplication of the elements of {{tmath|1= \operatorname{SO}(2, \mathbb{R}) }}, and taking the opposite angle corresponds to inversion. Thus both multiplication and inversion are differentiable maps.
* The [[Affine group#Matrix representation|affine group of one dimension]] is a two-dimensional matrix Lie group, consisting of <math>2 \times 2</math> real, upper-triangular matrices, with the first diagonal entry being positive and the second diagonal entry being 1. Thus, the group consists of matrices of the form <math display="block"> A= \left( \begin{array}{cc} a & b\\ 0 & 1 \end{array}\right),\quad a>0,\, b \in \mathbb{R}.</math>