Linear span

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In the mathematical subfield of linear algebra or more generally functional analysis, the linear span (also called the linear hull) of a set of vectors in a vector space is the intersection of all subspaces containing that set. The linear span of a set of vectors is therefore a vector space.

Definition[edit]

Given a vector space V over a field K, the span of a set S of vectors (not necessarily finite) 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 of S, which follows from the above definition.

\operatorname{span}(S) =  \left \{ {\sum_{i=1}^k \lambda_i v_i \Big| k \in \mathbb{N}, v_i  \in S, \lambda _i  \in \mathbf{K}} \right \}.

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. In the case of infinite S, infinite linear combinations (i.e. where a combination may involve an infinite sum, assuming such sums are defined somehow, e.g. if V is a Banach space) are excluded by the definition; a generalization that allows these is not equivalent.

Examples[edit]

The real vector space R3 has {(2,0,0), (0,1,0), (0,0,1)} as a spanning set. This particular spanning set is also a basis. If (2,0,0) 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; instead its span is the space of all vectors in R3 whose last component is zero.

Theorems[edit]

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.

Closed linear span[edit]

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\epsilon>0\,\exists x\in\operatorname{Sp}(E) : \|x-u\|<\epsilon\}.

Notes[edit]

The linear span of a set is dense in the closed linear span. Moreover, as stated in the below lemma, 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, consider Riesz's lemma).

A useful lemma[edit]

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

(a) \overline{\operatorname{Sp}}(E) is a closed linear subspace of X which contains E,

(b) \overline{\operatorname{Sp}}(E)=\overline{\operatorname{Sp}(E)}, viz. \overline{\operatorname{Sp}}(E) is the closure of \operatorname{Sp}(E),

(c) E^\perp=(\operatorname{Sp}(E))^\perp=(\overline{\operatorname{Sp}(E)})^\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.)

Matroids[edit]

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.

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