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linear manifold
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(Definition)
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Definition Suppose  is a vector space and suppose that  is a non-empty subset of  . If there exists a  such that
 is a vector subspace of  , then  is a linear manifold of  . Then we say that the dimension of  is the dimension of  and write
 . In the important case
 ,  is called a hyperplane.
A linear manifold is, in other words, a linear subspace that has possibly been shifted away from the origin. For instance, in
 examples of linear manifolds are points, lines (which are hyperplanes), and
 itself. In
 hyperplanes naturally describe tangent planes to a smooth hyper surface.
- 1
- R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.
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"linear manifold" is owned by matte.
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Cross-references: surface, smooth, tangent planes, lines, points, origin, dimension, vector subspace, subset, vector space
There are 20 references to this entry.
This is version 3 of linear manifold, born on 2003-11-26, modified 2005-10-29.
Object id is 5435, canonical name is LinearManifold.
Accessed 8173 times total.
Classification:
| AMS MSC: | 15-00 (Linear and multilinear algebra; matrix theory :: General reference works ) | | | 15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank) |
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Pending Errata and Addenda
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