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Difference between revisions of "Linear hull"

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(Comment: linear span, cite Grünbaum (2003))
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''of a set $A$ in a vector space $E$''
 
''of a set $A$ in a vector space $E$''
  
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====Comments====
 
====Comments====
This is also called the ''linear envelope''. The closure of the linear hull of a set $A$ is called the ''[[linear closure]]'' of $A$.
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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 $A$.
  
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 $A$.
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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 $M$.
  
 
====References====
 
====References====
* Grünbaum, Branko, ''Convex polytopes''.  Graduate Texts in Mathematics '''221'''.  Springer (2003) ISBN 0-387-40409-0 {{ZBL| 1033.52001}}
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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=35320
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article