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''linear manifold, affine subspace''
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''linear manifold, affine subspace, flat''
  
A subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l0595301.png" /> of a (linear) [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l0595302.png" /> that is a translate of a [[Linear subspace|linear subspace]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l0595303.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l0595304.png" />, that is, a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l0595305.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l0595306.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l0595307.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l0595308.png" /> determines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l0595309.png" /> uniquely, whereas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l05953010.png" /> is defined only modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l05953011.png" />:
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A subset $M$ of a (linear) [[vector space]] $E$ that is a translate of a [[linear subspace]] $L$ of $E$, that is, a set $M$ of the form $x_0 + L$ for some $x_0$. The set $M$ determines $L$ uniquely, whereas $x_0$ is defined only modulo $L$:
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$$
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x_0 + L = x_1 + N
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$$
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if and only if $L = N$ and $x_1 - x_0 \in L$. The dimension of $M$ is the dimension of $L$. A linear variety corresponding to a subspace of [[codimension]] 1 is called a ''hyperplane''.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l05953012.png" /></td> </tr></table>
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A linear variety may be alternatively characterised as a non-empty subset $M$ of $E$ such that the set $L =\{ m_1 - m_2 : m_1,m_2 \in M \}$ is a linear subspace of $E$; or such that for a fixed $m_0 \in M$ the set $L =\{ m_1 - m_0 : m_1 \in M \}$ is a linear subspace; or as a set closed under linear combinations $\sum_i \lambda_i m_i$ where $m_i \in M$ and $\sum_i \lambda_i = 1$.
  
if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l05953013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l05953014.png" />. The dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l05953015.png" /> is the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l05953016.png" />. A linear variety corresponding to a subspace of codimension 1 is called a hyperplane.
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The intersection of any family of linear varieties is again a linear variety. 
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====References====
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* N. Bourbaki, "Algebra I: Chapters 1-3", Elements of mathematics, Springer (1998) ISBN 3-540-64243-9

Latest revision as of 20:38, 31 December 2014

linear manifold, affine subspace, flat

A subset $M$ of a (linear) vector space $E$ that is a translate of a linear subspace $L$ of $E$, that is, a set $M$ of the form $x_0 + L$ for some $x_0$. The set $M$ determines $L$ uniquely, whereas $x_0$ is defined only modulo $L$: $$ x_0 + L = x_1 + N $$ if and only if $L = N$ and $x_1 - x_0 \in L$. The dimension of $M$ is the dimension of $L$. A linear variety corresponding to a subspace of codimension 1 is called a hyperplane.

A linear variety may be alternatively characterised as a non-empty subset $M$ of $E$ such that the set $L =\{ m_1 - m_2 : m_1,m_2 \in M \}$ is a linear subspace of $E$; or such that for a fixed $m_0 \in M$ the set $L =\{ m_1 - m_0 : m_1 \in M \}$ is a linear subspace; or as a set closed under linear combinations $\sum_i \lambda_i m_i$ where $m_i \in M$ and $\sum_i \lambda_i = 1$.

The intersection of any family of linear varieties is again a linear variety.

References

  • N. Bourbaki, "Algebra I: Chapters 1-3", Elements of mathematics, Springer (1998) ISBN 3-540-64243-9
How to Cite This Entry:
Linear variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_variety&oldid=14877
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article