Linear span

In mathematics, the linear span of a set of elements of a vector space is the smallest linear subspace of that contains It is the set of all finite linear combinations of the elements of, and the intersection of all linear subspaces that contain It is often denoted or
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 is generally called the substructure generated by

Definition

Given a vector space over a field, the span of a set of vectors is defined to be the intersection of all subspaces of that contain. It is thus the smallest 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 of, and can be defined as such. That is,
When is empty, the only possibility is, and the previous expression for reduces to the empty sum. The standard convention for the empty sum implies thus a property that is immediate with the other definitions. However, many introductory textbooks simply include this fact as part of the definition.
When is finite, one has

Examples

The real vector space has as a spanning set. This particular spanning set is also a basis. If were replaced by, it would also form the canonical basis of.
Another spanning set for the same space is given by, but this set is not a basis, because it is linearly dependent.
The set is not a spanning set of, since its span is the space of all vectors in whose last component is zero. That space is also spanned by the set, as is a linear combination of and. Thus, the spanned space is not It can be identified with by removing the third components equal to zero.
The empty set is a spanning set of, since the empty set is a subset of all possible vector spaces in, and 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.

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.

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. 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, denoted by or, 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, 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.