special linear group

Given a vector space $V$ , the special linear group ${\operatorname{SL}}(V)$ 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 ${\operatorname{SL}}(V)$ is often denoted ${\operatorname{SL}}(n,\mathbb{F})$ or ${\operatorname{SL}}_{n}(\mathbb{F})$ , and if one identifies each linear transformation with its matrix with respect to the standard basis, then ${\operatorname{SL}}(n,\mathbb{F})$ consists of all $n\times n$ matrices with entries in $\mathbb{F}$ that have determinant 1.

Title	special linear group
Canonical name	SpecialLinearGroup
Date of creation	2013-03-22 12:25:38
Last modified on	2013-03-22 12:25:38
Owner	djao (24)
Last modified by	djao (24)
Numerical id	7
Author	djao (24)
Entry type	Definition
Classification	msc 20G15
Synonym	SL
Related topic	GeneralLinearGroup
Related topic	Group
Related topic	UnimodularMatrix