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 $\dim L=\dim(L+v)$ . 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 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
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