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.

单词	SpecialLinearGroup
释义	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.
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