Linear Groups

General Linear Group

Claim that the set of all linear operators together with operation of ordinary matrix multiplication is a group. We write it as ${\text{GL}}_n(\mathbb{F})$, where $n$ denotes the dimension of the corresponding square matrix. That is:${\text{GL}}_n(\mathbb{F})=\{A\in {\mathbb{F}}^{n\times n}\mid \text{A is invertible}\}$

Special Linear Group

Claim that the set of all linear operators such that determinant is $1$ is a group. Write it as ${\text{SL}}_n(\mathbb{F})$where $n$ denotes the dimension of the corresponding square matrix.

Orthogonal Group

The orthogonal group, often denoted ${\mathrm{O}}_n$, is the group of all orthogonal transformations of $n$-dimensional real space ${\mathbb{R}}^n$ under the operation of composition.

Rotation Group

The rotation group, often denoted ${\mathrm{SO}}_n$, is the group of all rotations about the origin of $n$-dimensional Euclidean space ${\mathbb{R}}^n$ under the operation of composition.

e.g. In ${\mathrm{SO}}_3$, counterclockwise rotation about the positive axis by some angle $\varphi$ is given by$\begin{array}{r}{R}_x(\alpha )=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \alpha & -\sin \alpha \\ 0& \sin \alpha & \cos \alpha \end{array}\right]\\ R_y(\beta )=\left[\begin{array}{ccc}\cos \beta & 0& \sin \beta \\ 0& 1& 0\\ -\sin \beta & 0& \cos \beta \end{array}\right]\\ R_z(\varphi )=\left[\begin{array}{ccc}\cos \varphi & -\sin \varphi & 0\\ \sin \varphi & \cos \varphi & 0\\ 0& 0& 1\end{array}\right]\end{array}$

Proposition

${\mathrm{SO}}_3\le {\text{GL}}_3(\mathbb{R})$.