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Difference between revisions of "Linear hull"

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''of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059260/l0592601.png" /> in a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059260/l0592602.png" />''
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{{TEX|done}}{{MSC|15A03}}
  
The intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059260/l0592603.png" /> of all subspaces containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059260/l0592604.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059260/l0592605.png" /> is also called the subspace generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059260/l0592606.png" />.
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''of a set $A$ in a vector space $E$''
  
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The intersection $M$ of all subspaces containing $A$. The set $M$ is also called the subspace generated by $A$.
  
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====Comments====
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This is also called the ''linear envelope''. In a [[topological vector space]], the [[Closure of a set|closure]] of the linear hull of a set $A$ is called the ''[[linear closure]]'' of $A$; it is also the intersection of  all closed subspaces containing $A$.
  
====Comments====
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A further term is ''span'' or ''linear span''.  It is equal to the set of all finite [[linear combination]]s of elements $\{m_i : i=1,\ldots,n \}$ of $A$. If the linear span of $A$ is $M$, then $A$ is a ''[[spanning set]]'' for $M$.
This is also called the linear envelope. The closure of the linear hull of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059260/l0592607.png" /> is called the [[Linear closure|linear closure]] of this set.
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====References====
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* Grünbaum, Branko, ''Convex polytopes''. Graduate Texts in Mathematics '''221'''. Springer (2003) {{ISBN|0-387-40409-0}} {{ZBL|1033.52001}}

Latest revision as of 08:49, 26 November 2023

2020 Mathematics Subject Classification: Primary: 15A03 [MSN][ZBL]

of a set $A$ in a vector space $E$

The intersection $M$ of all subspaces containing $A$. The set $M$ is also called the subspace generated by $A$.

Comments

This is also called the linear envelope. In a topological vector space, the closure of the linear hull of a set $A$ is called the linear closure of $A$; it is also the intersection of all closed subspaces containing $A$.

A further term is span or linear span. It is equal to the set of all finite linear combinations of elements $\{m_i : i=1,\ldots,n \}$ of $A$. If the linear span of $A$ is $M$, then $A$ is a spanning set for $M$.

References

  • Grünbaum, Branko, Convex polytopes. Graduate Texts in Mathematics 221. Springer (2003) ISBN 0-387-40409-0 Zbl 1033.52001
How to Cite This Entry:
Linear hull. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_hull&oldid=12870
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article