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Linear span

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Definition
Examples
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Equivalence of definitio..
Size of spanning set is ..
Spanning set can be redu..

Generalizations
Closed linear span (func..

Notes
A useful lemma

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In mathematics, the linear span (also called the linear hull. Linear Hull. or just span) of a set $S$ of elements of a vector space $V$ is the smallest linear subspace of $V$ that contains $S.$ It is the set of all finite linear combinations of the elements of , p. 29, § 2.7 and the intersection of all linear subspaces that contain $S.$ It is often denoted pp. 29-30, §§ 2.5, 2.8 or $\langle S \rangle.$

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, in which case, the smallest substructure containing $S$ is generally called the substructure generated by $S.$

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 linear subspace 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. p. 100, ch. 2, Definition 2.13 pp. 29-30, §§ 2.5, 2.8 pp. 41-42 That is, $\operatorname{span}(S) = \biggl \{ \lambda_1 \mathbf v_1 + \lambda_2 \mathbf v_2 + \cdots + \lambda_n \mathbf v_n \mid n \in \N,\; \mathbf v_1,...\mathbf v_n \in S, \; \lambda_1,...\lambda_n \in K \biggr \}$

When is empty set, the only possibility is , and the previous expression for $\operatorname{span}(S)$ reduces to the empty sum. The standard convention for the empty sum implies thus $\text{span}(\empty) = \{\mathbf 0\},$ a property that is immediate with the other definitions. However, many introductory textbooks simply include this fact as part of the definition.

When $S=\{\mathbf v_1,\ldots, \mathbf v_n\}$ is finite set, one has $\operatorname{span}(S) = \{ \lambda_1 \mathbf v_1 + \lambda_2 \mathbf v_2 + \cdots + \lambda_n \mathbf v_n \mid \lambda_1,...\lambda_n \in K \}$

Examples

The real number vector space

\mathbb 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 standard basis of

\mathbb 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 } is not a spanning set of $\mathbb R^3$ , since its span is the space of all vectors in $\mathbb R^3$ 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). Thus, the spanned space is not $\mathbb R^3.$ It can be identified with $\mathbb R^2$ by removing the third components equal to zero.

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

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

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 .

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 is closed under scalar multiplication. Letting

S = \{ \mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n \}

, it is trivial that the zero vector of exists in , since

\mathbf 0 = 0 \mathbf v_1 + 0 \mathbf v_2 + \cdots + 0 \mathbf v_n

. Adding together two linear combinations of also produces a linear combination of :

(\lambda_1 \mathbf v_1 + \cdots + \lambda_n \mathbf v_n) + (\mu_1 \mathbf v_1 + \cdots + \mu_n \mathbf v_n) = (\lambda_1 + \mu_1) \mathbf v_1 + \cdots + (\lambda_n + \mu_n) \mathbf v_n

, where all

\lambda_i, \mu_i \in K

, and multiplying a linear combination of by a scalar

c \in K

will produce another linear combination of :

c(\lambda_1 \mathbf v_1 + \cdots + \lambda_n \mathbf v_n) = c\lambda_1 \mathbf v_1 + \cdots + c\lambda_n \mathbf v_n

. Thus is a subspace of .

It follows that

S \subseteq \operatorname{span} S

, since every is a linear combination of (trivially). Suppose that is a linear subspace of containing . Since is closed under addition and scalar multiplication, then every linear combination

\lambda_1 \mathbf v_1 + \cdots + \lambda_n \mathbf v_n

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 .

Proof. Let

S = \{ \mathbf v_1, \ldots, \mathbf v_m \}

be a spanning set and

W = \{ \mathbf w_1, \ldots, \mathbf w_n \}

be a linearly independent set of vectors from . We want to show that

m \geq n

Since spans , then

S \cup \{ \mathbf w_1 \}

must also span , and

\mathbf w_1

must be a linear combination of . Thus

S \cup \{ \mathbf w_1 \}

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 is linearly independent. The resulting set is

\{ \mathbf w_1, \mathbf v_1, \ldots, \mathbf v_{i-1}, \mathbf v_{i+1}, \ldots, \mathbf v_m \}

, which is a spanning set of . We repeat this step times, where the resulting set after the th step is the union of

\{ \mathbf w_1, \ldots, \mathbf w_p \}

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 's—i.e.

m \geq n

. To verify this, we assume by way of contradiction that

m < n

. Then, at the th step, we have the set

\{ \mathbf w_1, \ldots, \mathbf w_m \}

and we can adjoin another vector

\mathbf w_{m+1}

. But, since

\{ \mathbf w_1, \ldots, \mathbf w_m \}

is a spanning set of ,

\mathbf w_{m+1}

is a linear combination of

\{ \mathbf w_1, \ldots, \mathbf w_m \}

. 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. p. 96, ch. 4 p. 193, ch. 6 Given an -module and a collection of elements , ..., of , the submodule of spanned by , ..., is the sum of $Ra_1 + \cdots + Ra_n = \left\{ \sum_{k=1}^n r_k a_k \bigg| r_k \in R \right\}$ 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 vector space 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 , denoted by $\overline{\operatorname{Sp}}(E)$ or $\overline{\operatorname{Span}}(E)$ , is the intersection of all the closed linear subspaces of which contain .

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,, 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 (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  $\operatorname{Sp}(E)$ ,
|  $E^\perp = (\operatorname{Sp}(E))^\perp = \left(\overline{\operatorname{Sp}(E)}\right)^\perp.$ 
|  $(E^\perp)^\perp = ((\operatorname{Sp}(E))^\perp)^\perp = \overline{\operatorname{Sp}(E)}.$

(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.)

Linear span

( Abstract Algebra )

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Linear span ( Abstract Algebra )

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Linear span

( Abstract Algebra )