general linear group

Given a vector space $V$ , the general linear group ${\\operatorname{GL}}(V)$ is defined to be the group of invertible linear transformations from $V$ to $V$ . The group operation is defined by composition: given $T:V\\longrightarrow V$ and $T^{\\prime}:V\\longrightarrow V$ in ${\\operatorname{GL}}(V)$ , the product $TT^{\\prime}$ is just the composition of the maps $T$ and $T^{\\prime}$ .

If $V=\\mathbb{F}^{n}$ for some field $\\mathbb{F}$ , then the group ${\\operatorname{GL}}(V)$ is often denoted ${\\operatorname{GL}}(n,\\mathbb{F})$ or ${\\operatorname{GL}}_{n}(\\mathbb{F})$ . In this case, if one identifies each linear transformation $T:V\\longrightarrow V$ with its matrix with respect to the standard basis, the group ${\\operatorname{GL}}(n,\\mathbb{F})$ becomes the group of invertible $n\\times n$ matrices with entries in $\\mathbb{F}$ , under the group operation of matrix multiplication.

One also discusses the general linear group on a module $M$ over some ring $R$ . There it is the set of automorphisms of $M$ as an $R$ -module. For example, one might take ${\\operatorname{GL}}(\\mathbb{Z}\\oplus\\mathbb{Z})$ ; this is isomorphic to the group of two-by-two matrices with integer entries having determinant $\\pm 1$ . If $M$ is a general $R$ -module, there need not be a natural interpretation of ${\\operatorname{GL}}(M)$ as a matrix group.

The general linear group is an example of a group scheme; viewing it in this way ties together the properties of ${\\operatorname{GL}}(V)$ for different vector spaces $V$ and different fields $F$ . The general linear group is an algebraic group, and it is a Lie group if $V$ is a real or complex vector space.

When $V$ is a finite-dimensional Banach space, ${\\operatorname{GL}}(V)$ has a natural topology coming from the operator norm; this is isomorphic to the topology coming from its embedding into the ring of matrices. When $V$ is an infinite-dimensional vector space, some elements of ${\\operatorname{GL}}(V)$ may not be continuous and one generally looks instead at the set of bounded operators.

单词	GeneralLinearGroup
释义	general linear group Given a vector space $V$ , the general linear group ${\\operatorname{GL}}(V)$ is defined to be the group of invertible linear transformations from $V$ to $V$ . The group operation is defined by composition: given $T:V\\longrightarrow V$ and $T^{\\prime}:V\\longrightarrow V$ in ${\\operatorname{GL}}(V)$ , the product $TT^{\\prime}$ is just the composition of the maps $T$ and $T^{\\prime}$ . If $V=\\mathbb{F}^{n}$ for some field $\\mathbb{F}$ , then the group ${\\operatorname{GL}}(V)$ is often denoted ${\\operatorname{GL}}(n,\\mathbb{F})$ or ${\\operatorname{GL}}_{n}(\\mathbb{F})$ . In this case, if one identifies each linear transformation $T:V\\longrightarrow V$ with its matrix with respect to the standard basis, the group ${\\operatorname{GL}}(n,\\mathbb{F})$ becomes the group of invertible $n\\times n$ matrices with entries in $\\mathbb{F}$ , under the group operation of matrix multiplication. One also discusses the general linear group on a module $M$ over some ring $R$ . There it is the set of automorphisms of $M$ as an $R$ -module. For example, one might take ${\\operatorname{GL}}(\\mathbb{Z}\\oplus\\mathbb{Z})$ ; this is isomorphic to the group of two-by-two matrices with integer entries having determinant $\\pm 1$ . If $M$ is a general $R$ -module, there need not be a natural interpretation of ${\\operatorname{GL}}(M)$ as a matrix group. The general linear group is an example of a group scheme; viewing it in this way ties together the properties of ${\\operatorname{GL}}(V)$ for different vector spaces $V$ and different fields $F$ . The general linear group is an algebraic group, and it is a Lie group if $V$ is a real or complex vector space. When $V$ is a finite-dimensional Banach space, ${\\operatorname{GL}}(V)$ has a natural topology coming from the operator norm; this is isomorphic to the topology coming from its embedding into the ring of matrices. When $V$ is an infinite-dimensional vector space, some elements of ${\\operatorname{GL}}(V)$ may not be continuous and one generally looks instead at the set of bounded operators.
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