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Wikipedia / Linear span

In mathematics, the linear span (also called the linear hull or just span) of a set of elements of a vector space is the smallest linear subspace It is the set of all finite linear combinations of the elements of , It is often denoted

For example, in geometry, two linearly independent vectors span a plane.

To express that a vector space is a linear span of a subset , one commonly uses one of the following phrases: spans ; is a spanning set of ; is spanned or generated by ; is a generator set or a generating set of .

Spans can be generalized to many mathematical structures is generally called the substructure generated

Definition

Given a vector space over a field , the span of a set of vectors (not necessarily finite) is defined to be the intersection of all subspaces of that contain . It is thus the smallest (for set inclusion) subspace containing . It is referred to as the subspace spanned by , or by the vectors in . Conversely, is called a spanning set of , and we say that spans .

It follows from this definition that the span of is the set of all finite linear combinations of elements (vectors) of , and can be defined as such.

When is empty, the only possibility is reduces to the empty sum. a property that is immediate with the other definitions. However, many introductory textbooks simply include this fact as part of the definition.

is finite

Examples

The real 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.

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 by removing the third components equal to zero.

, and {(0, 0, 0)} is the intersection of all of these vector spaces.

The set of monomials , where is a non-negative integer, spans the space of polynomials.

Theorems

Equivalence of definitions

The set of all linear combinations of a subset of , a vector space over , is the smallest linear subspace of containing .

  1. Proof. We first prove that is a subspace of . Since is a subset of , we only need to prove the existence of a zero vector in , that is closed under addition, and that , it is trivial that the zero vector of exists in . Adding together two linear combinations of also produces a linear combination of , and multiplying a linear combination of will produce another linear combination of . Thus is a subspace of .

    , since every is a linear combination of (trivially). Suppose that is a linear subspace of containing . Since must be contained in . Thus, is contained in every subspace of containing , and the intersection of all such subspaces, or the smallest such subspace, is equal to the set of all linear combinations of .

Size of spanning set is at least size of linearly independent set

Every spanning set of a vector space must contain at least as many elements as any linearly independent set of vectors from .

  1. Proof. be a linearly independent set of vectors from .

    Since spans must also span must be a linear combination of is linearly dependent, and we can remove one vector from that is a linear combination of the other elements. This vector cannot be any of the , since , which is a spanning set of . We repeat this step times, where the resulting set after the and vectors of .

    It is ensured until the th step that there will always be some to remove out of for every adjoint of , and thus there are at least as many 's as there are . Then, at the is a spanning set of . This is a contradiction, since is linearly independent.

Spanning set can be reduced to a basis

Let be a finite-dimensional vector space. Any set of vectors that spans can be reduced to a basis for , 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 has finite dimension. This also indicates that a basis is a minimal spanning set when is finite-dimensional.

Generalizations

Generalizing the definition of the span of points in space, a subset of the ground set of a matroid is called a spanning set if the rank of equals the rank of the entire ground set

The vector space definition can also be generalized to modules. Given an -module and a collection of elements , ..., of , the submodule of spanned by , ..., is the sum of cyclic modules consisting of all R-linear combinations of the elements . 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 is a normed vector space and let be any non-empty subset of . The closed linear span of , is the intersection of all the closed linear subspaces of which contain .

One mathematical formulation of this is

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

(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.)

See also

Footnotes

Citations

Sources

Textbooks

  • Lay, David C. (2021) Linear Algebra and Its Applications (6th Edition). Pearson.

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