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# 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.

Defines:

hyperplane

Related:

VectorSubspace, LineSegment

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

15A03*no label found*15-00

*no label found*

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