Difference between revisions of "Linear hull"
From Encyclopedia of Mathematics
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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 $A$. | 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 $A$. | + | 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}} | * Grünbaum, Branko, ''Convex polytopes''. Graduate Texts in Mathematics '''221'''. Springer (2003) ISBN 0-387-40409-0 {{ZBL|1033.52001}} | ||
Revision as of 19:40, 27 February 2021
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=45264
Linear hull. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_hull&oldid=45264
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article