Linear span

In mathematics, the linear span (also called the linear hull^[1] or just span) of a set $S$ of vectors (from a vector space), denoted $span(S)$ ,^[2]^[3] is the smallest linear subspace that contains the set.^[4] 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 $V$ .

Definition[]

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.^[5]^[6]^[7]^[8]

\operatorname {span} (S)=\left\{{\left.\sum _{i=1}^{k}\lambda _{i}v_{i}\;\right|\;k\in \mathbb {N} ,v_{i}\in S,\lambda _{i}\in K}\right\}.

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.

Examples[]

The cross-hatched plane is the linear span of u and v in R³.

The real vector space R³ 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 R³.

Another spanning set for the same space is given by {(1, 2, 3), (0, 1, 2), (−1, 1⁄2, 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 R³, since its span is the space of all vectors in R³ 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 R².(when interpreted as a subset of R³).

The empty set is a spanning set of {(0, 0, 0)}, since the empty set is a subset of all possible vector spaces in R³, and {(0, 0, 0)} is the intersection of all of these vector spaces.

The set of functions xⁿ where n is a non-negative integer spans the space of polynomials.

Theorems[]

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.

Generalizations[]

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^{[citation needed]}.

The vector space definition can also be generalized to modules.^[9]^[10] Given an R-module A and a collection of elements a₁, …, a_n of A, the submodule of A spanned by a₁, …, a_n is the sum of cyclic modules

Ra_{1}+\cdots +Ra_{n}=\left\{\sum _{k=1}^{n}r_{k}a_{k}{\Big |}r_{k}\in R\right\}

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.

Closed linear span (functional analysis)[]

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 ${\overline {\operatorname {Sp} }}(E)$ or ${\overline {\operatorname {Span} }}(E)$ , is the intersection of all the closed linear subspaces of X which contain E.

One mathematical formulation of this is

{\overline {\operatorname {Sp} }}(E)=\{u\in X|\forall \varepsilon >0\,\exists x\in \operatorname {Sp} (E):\|x-u\|<\varepsilon \}.

The closed linear span of the set of functions xⁿ on the interval [0, 1], where n is a non-negative integer, depends on the norm used. If the L² norm is used, then the closed linear span is the Hilbert space of square-integrable functions on the interval. But if the maximum norm is used, the closed linear span will be the space of continuous functions on the interval. In either case, the closed linear span contains functions that are not polynomials, and so are not in the linear span itself. However, the cardinality of the set of functions in the closed linear span is the cardinality of the continuum, which is the same cardinality as for the set of polynomials.

Notes[]

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).

A useful lemma[]

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

${\overline {\operatorname {Sp} }}(E)$ is a closed linear subspace of X which contains E,
${\overline {\operatorname {Sp} }}(E)={\overline {\operatorname {Sp} (E)}}$ , viz. ${\overline {\operatorname {Sp} }}(E)$ is the closure of $\operatorname {Sp} (E)$ ,
$E^{\perp }=(\operatorname {Sp} (E))^{\perp }=\left({\overline {\operatorname {Sp} (E)}}\right)^{\perp }.$

(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.)

Citations[]

^ Encyclopedia of Mathematics (2020). Linear Hull.
^ Axler (2015) pp. 29-30, §§ 2.5, 2.8
^ Math Vault (2021) Vector space related operators.
^ Axler (2015) p. 29, § 2.7
^ Hefferon (2020) p. 100, ch. 2, Definition 2.13
^ Axler (2015) pp. 29-30, §§ 2.5, 2.8
^ Roman (2005) pp. 41-42
^ MathWorld (2021) Vector Space Span.
^ Roman (2005) p. 96, ch. 4
^ Lane & Birkhoff (1999) p. 193, ch. 6

Sources[]

Textbook[]

Axler, Sheldon Jay (2015). Linear Algebra Done Right (3rd ed.). Springer. ISBN 978-3-319-11079-0.
Hefferon, Jim (2020). Linear Algebra (4th ed.). Orthogonal Publishing. ISBN 978-1-944325-11-4.
Lane, Saunders Mac; Birkhoff, Garrett (1999) [1988]. Algebra (3rd ed.). AMS Chelsea Publishing. ISBN 978-0821816462.
Roman, Steven (2005). Advanced Linear Algebra (2nd ed.). Springer. ISBN 0-387-24766-1.
Rynne, Brian P.; Youngson, Martin A. (2008). Linear Functional Analysis. Springer. ISBN 978-1848000049.

Web[]

Lankham, Isaiah; Nachtergaele, Bruno; Schilling, Anne (13 February 2010). "Linear Algebra - As an Introduction to Abstract Mathematics" (PDF). University of California, Davis. Retrieved 27 September 2011.
"Comprehensive List of Algebra Symbols". Math Vault. Retrieved 16 Feb 2021.
Weisstein, Eric Wolfgang. "Vector Space Span". MathWorld. Retrieved 16 Feb 2021.
"Linear hull". Encyclopedia of Mathematics. 5 April 2020. Retrieved 16 Feb 2021.

External links[]

Linear Combinations and Span: Understanding linear combinations and spans of vectors, khanacademy.org.
Sanderson, Grant (August 6, 2016). "Linear combinations, span, and basis vectors". Essence of Linear Algebra – via YouTube.

[1] Encyclopedia of Mathematics (2020). Linear Hull.

[:0-2] Axler (2015) pp. 29-30, §§ 2.5, 2.8

[3] Math Vault (2021) Vector space related operators.

[4] Axler (2015) p. 29, § 2.7

[5] Hefferon (2020) p. 100, ch. 2, Definition 2.13

[:02-6] Axler (2015) pp. 29-30, §§ 2.5, 2.8

[7] Roman (2005) pp. 41-42

[8] MathWorld (2021) Vector Space Span.

[9] Roman (2005) p. 96, ch. 4

[10] Lane & Birkhoff (1999) p. 193, ch. 6

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