linear manifold
Definition Suppose $V$ is a vector space^{} and suppose that $L$ is a non-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 we say that the dimension^{} of $L$ is the dimension of $L+v$ and write $dimL=dim(L+v)$. In the important case $dimL=dimV-1$, $L$ 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 ${\mathbb{R}}^{2}$ examples of linear manifolds are points, lines (which are hyperplanes), and ${\mathbb{R}}^{2}$ itself. In ${\mathbb{R}}^{n}$ hyperplanes naturally describe tangent planes to a smooth hyper surface.
References
- 1 R. Cristescu, Topological vector spaces^{}, Noordhoff International Publishing, 1977.
Title | linear manifold |
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Canonical name | LinearManifold |
Date of creation | 2013-03-22 14:04:32 |
Last modified on | 2013-03-22 14:04:32 |
Owner | matte (1858) |
Last modified by | matte (1858) |
Numerical id | 6 |
Author | matte (1858) |
Entry type | Definition |
Classification | msc 15A03 |
Classification | msc 15-00 |
Related topic | VectorSubspace |
Related topic | LineSegment |
Defines | hyperplane |