linear manifold

Definition Suppose V is a vector spaceMathworldPlanetmath and suppose that L is a non-empty subset of V. If there exists a vV such that L+v={v+llL} is a vector subspace of V, then L is a linear manifold of V. Then we say that the dimensionPlanetmathPlanetmathPlanetmath 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 2 examples of linear manifolds are points, lines (which are hyperplanes), and 2 itself. In n hyperplanes naturally describe tangent planes to a smooth hyper surface.


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