# Linear hull

From Encyclopedia of Mathematics

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=51655

This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article