special linear group SpecialLinearGroup 2002-02-22 01:08:29 2005-05-04 23:37:16 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\SL}{{\operatorname{SL}}} Given a vector space $V$, the special linear group $\SL(V)$ is defined to be the subgroup of the general linear group $\operatorname{GL}(V)$ consisting of all invertible linear transformations $T: V \longrightarrow V$ in $\operatorname{GL}(V)$ that have determinant 1. If $V = \mathbb{F}^n$ for some field $\mathbb{F}$, then the group $\SL(V)$ is often denoted $\SL(n,\mathbb{F})$ or $\SL_n(\mathbb{F})$, and if one identifies each linear transformation with its matrix with respect to the standard basis, then $\SL(n,\mathbb{F})$ consists of all $n \times n$ matrices with entries in $\mathbb{F}$ that have determinant 1.