>[!info] General Linear Group >For an $n$-dimensional [[Vector Space|vector space]] $V$ over a field $F$, the general linear group $\mathrm{GL}(V)\quad \text{or} \quad \mathrm{GL}(n,F)$is the [[Set|set]] of all [[Maps and Induced Structures - Functions, Pushforwards and Pullbacks|bijective linear transformations]] (invertible $n\times n$ [[Matrices|matrices]]) --- #### Important Subgroups >[!info] General Orthogonal Group >The orthogonal group of order $n$ $\mathrm{O}(n) \coloneqq \{ \mathbf{A} \in \mathbb{R}^{n \times n}\colon \mathbf{AA}^{T} =\mathbf{I} \},$describes all transformations that preserve ... . It is a [[Topology and Topological Space|compact]] [[Lie Group]]. - **Special Orthogonal Group** of order $n$ (rotations) $\mathrm{SO}(n) \coloneqq \{ \mathbf{A} \in \mathbb{R}^{n \times n}\colon \mathbf{AA}^{T} =\mathbf{I}, \, \det(\mathbf{A})=1 \}\subset \mathrm{O}(n)$ - **Special Euclidean Group** of order $n$ (roto-translations) $\mathrm{SE}(n)=\{ g=(\mathbf{A},\mathbf{b})|\mathbf{A} \in \mathrm{SO}(n), \mathbf{b} \in \mathbb{R}^{n} \}$ - E.g. [[Robot Kinematics - Frames and Twists|Homogeneous Coordinates]] with [[Matrices|matrix]] product and inverse