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'linear manifold'
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| Title of object: |
linear manifold |
| Canonical Name: |
LinearManifold |
| Type: |
Definition |
| Created on: |
2003-11-26 14:00:03 |
| Modified on: |
2003-11-27 13:33:41 |
| Classification: |
msc:15-00, msc:15A03 |
| Defines: |
hyperplane |
Preamble:
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Content:
{\bf Definition} \cite{cristescu} 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 {\bf linear manifold} of $V$. Then we
say that the dimension of $L$ is the dimension of $L+v$ and write $\dim L = \dim (L+v)$.
If $\dim L = \dim V -1$, then $L$ is called a {\bf hyperplane}.
%\subsubsection{Examples}
A linear manifold is, in other words, a linear subspace that has possibly been
shifted away from the origin.
For instance, in $\sR^2$ examples of linear
manifolds are points, lines (which are hyperplanes), and $\sR^2$ itself.
\begin{thebibliography}{9}
\bibitem{cristescu} R. Cristescu, \emph{Topological vector spaces},
Noordhoff International Publishing, 1977.
\end{thebibliography} |
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