Linear Span

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## Definition

## Examples

## Theorems

## Generalizations

## Closed linear span (functional analysis)

### Notes

### A useful lemma

## See also

## Notes

## References

## External links

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Linear Span

In linear algebra, the **linear span** (also called the **linear hull** or just **span**) of a set S of vectors (from a vector space), denoted ,^{[1]} is the smallest linear subspace that contains the set. It can be characterized either as the intersection of all linear subspaces that contain S, or as the set of linear combinations of elements of S. The linear span of a set of vectors is therefore a vector space. Spans can be generalized to matroids and modules.

For expressing that a vector space V is a span of a set S, one commonly uses the following phrases: S spans V; S generates V; V is spanned by S; V is generated by S; S is a **spanning set** of V; S is a generating set of V.

Given a vector space *V* over a field *K*, the span of a set *S* of vectors (not necessarily infinite) is defined to be the intersection *W* of all subspaces of *V* that contain *S*. *W* is referred to as the subspace *spanned by* *S*, or by the vectors in *S*. Conversely, *S* is called a *spanning set* of *W*, and we say that *S* *spans* *W*.

Alternatively, the span of *S* may be defined as the set of all finite linear combinations of elements (vectors) of *S*, which follows from the above definition.

In particular, if *S* is a finite subset of *V*, then the span of *S* is the set of all linear combinations of the elements of *S*.^{[2]}^{[3]} In the case of infinite *S*, infinite linear combinations (i.e. where a combination may involve an infinite sum, assuming that such sums are defined somehow as in, say, a Banach space) are excluded by the definition; a generalization that allows these is not equivalent.

The real vector space **R**^{3} has {(-1, 0, 0), (0, 1, 0), (0, 0, 1)} as a spanning set. This particular spanning set is also a basis. If (-1, 0, 0) were replaced by (1, 0, 0), it would also form the canonical basis of **R**^{3}.

Another spanning set for the same space is given by {(1, 2, 3), (0, 1, 2), (-1, , 3), (1, 1, 1)}, but this set is not a basis, because it is linearly dependent.

The set {(1, 0, 0), (0, 1, 0), (1, 1, 0)} is not a spanning set of **R**^{3}, since its span is the space of all vectors in **R**^{3} whose last component is zero. That space is also spanned by the set {(1, 0, 0), (0, 1, 0)}, as (1, 1, 0) is a linear combination of (1, 0, 0) and (0, 1, 0). It does, however, span **R**^{2}.(when interpreted as a subset of **R**^{3}).

The empty set is a spanning set of {(0, 0, 0)}, since the empty set is a subset of all possible vector spaces in **R**^{3}, and {(0, 0, 0)} is the intersection of all of these vector spaces.

The set of functions *x ^{n}* where

**Theorem 1:** The subspace spanned by a non-empty subset *S* of a vector space *V* is the set of all linear combinations of vectors in *S*.

This theorem is so well known that at times, it is referred to as the definition of span of a set.

**Theorem 2:** Every spanning set *S* of a vector space *V* must contain at least as many elements as any linearly independent set of vectors from *V*.

**Theorem 3:** Let *V* be a finite-dimensional vector space. Any set of vectors that spans *V* can be reduced to a basis for *V*, by discarding vectors if necessary (i.e. if there are linearly dependent vectors in the set). If the axiom of choice holds, this is true without the assumption that *V* has finite dimension.

This also indicates that a basis is a minimal spanning set when *V* is finite-dimensional.

Generalizing the definition of the span of points in space, a subset *X* of the ground set of a matroid is called a *spanning set*, if the rank of *X* equals the rank of the entire ground set^{[]}.

The vector space definition can also be generalized to modules.^{[4]} Given an *R*-module *A* and a collection of elements a_{1}, ..., a_{n} of A, the submodule of *A* spanned by a_{1}, ..., a_{n} is the sum of cyclic modules

consisting of all *R*-linear combinations of the elements a_{i}. As with the case of vector spaces, the submodule of A spanned by any subset of A is the intersection of all submodules containing that subset.

In functional analysis, a closed linear span of a set of vectors is the minimal closed set which contains the linear span of that set.

Suppose that *X* is a normed vector space and let *E* be any non-empty subset of *X*. The **closed linear span** of *E*, denoted by or , is the intersection of all the closed linear subspaces of *X* which contain *E*.

One mathematical formulation of this is

The closed linear span of the set of functions *x ^{n}* on the interval [0, 1], where

The linear span of a set is dense in the closed linear span. Moreover, as stated in the lemma below, the closed linear span is indeed the closure of the linear span.

Closed linear spans are important when dealing with closed linear subspaces (which are themselves highly important, see Riesz's lemma).

Let *X* be a normed space and let *E* be any non-empty subset of *X*. Then

- is a closed linear subspace of
*X*which contains*E*, - , viz. is the closure of ,

(So the usual way to find the closed linear span is to find the linear span first, and then the closure of that linear span.)

**^**"Comprehensive List of Algebra Symbols".*Math Vault*. 2020-03-25. Retrieved .**^**"Linear Algebra basics".*homepages.rpi.edu*. Retrieved .**^**Weisstein, Eric W. "Vector Space Span".*mathworld.wolfram.com*. Retrieved .**^**Lane, Saunders Mac; Birkhoff, Garrett (1999-02-28).*Algebra: Third Edition*. EDS Publications Ltd. p. 168. ISBN 9780821816462.

- M.I. Voitsekhovskii (2001) [1994], "Linear hull",
*Encyclopedia of Mathematics*, EMS Press - Lankham, Isaiah; Nachtergaele, Bruno; Schilling, Anne (13 February 2010). "Linear Algebra - As an Introduction to Abstract Mathematics" (PDF). University of California, Davis. Retrieved 2011.
- Brian P. Rynne & Martin A. Youngson (2008).
*Linear Functional Analysis*, page 4, Springer ISBN 978-1848000049.

- Linear Combinations and Span: Understanding linear combinations and spans of vectors, khanacademy.org.
- "Linear combinations, span, and basis vectors".
*Essence of linear algebra*. August 6, 2016 – via YouTube.

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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