linear manifold

DefinitionSuppose $V$ is a vector space and suppose that $L$ is anon-empty subset of $V$ . If there exists a $v\\in V$ such that $L+v=\\{v+l\\mid l\\in L\\}$ is a vector subspace of $V$ , then $L$ is a linear manifold of $V$ . Then wesay that the dimension of $L$ is the dimension of $L+v$ and write $\\dim L=\\dim(L+v)$ .In the important case $\\dim L=\\dim V-1$ , $L$ is called a hyperplane.

A linear manifold is, in other words, a linear subspace that has possibly beenshifted away from the origin.For instance, in $\\mathbb{R}^{2}$ examples of linearmanifolds are points, lines (which are hyperplanes), and $\\mathbb{R}^{2}$ itself.In $\\mathbb{R}^{n}$ hyperplanes naturally describe tangent planes to a smoothhyper surface.

References

1 R. Cristescu, Topological vector spaces,Noordhoff International Publishing, 1977.

单词	LinearManifold
释义	linear manifold DefinitionSuppose $V$ is a vector space and suppose that $L$ is anon-empty subset of $V$ . If there exists a $v\\in V$ such that $L+v=\\{v+l\\mid l\\in L\\}$ is a vector subspace of $V$ , then $L$ is a linear manifold of $V$ . Then wesay that the dimension of $L$ is the dimension of $L+v$ and write $\\dim L=\\dim(L+v)$ .In the important case $\\dim L=\\dim V-1$ , $L$ is called a hyperplane. A linear manifold is, in other words, a linear subspace that has possibly beenshifted away from the origin.For instance, in $\\mathbb{R}^{2}$ examples of linearmanifolds are points, lines (which are hyperplanes), and $\\mathbb{R}^{2}$ itself.In $\\mathbb{R}^{n}$ hyperplanes naturally describe tangent planes to a smoothhyper surface. References 1 R. Cristescu, Topological vector spaces,Noordhoff International Publishing, 1977.
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