Linear span
In mathematics, the linear span (also called the linear hull[1] or just span) of a set
It is the set of all finite linear combinations of the elements of S,[2] and the intersection of all linear subspaces that contain
For example, in geometry, two linearly independent vectors span a plane.
To express that a vector space V is a linear span of a subset S, one commonly uses one of the following phrases: S spans V; S is a spanning set of V; V is spanned or generated by S; S is a generator set or a generating set of V. Spans can be generalized to many mathematical structures, in which case, the smallest substructure containing
is generally called the substructure generated by
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. It is thus the smallest (for set inclusion) subspace containing W. It 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. It follows from this definition that the span of S is the set of all finite linear combinations of elements (vectors) of S, and can be defined as such.
When S is empty, the only possibility is n = 0, and the previous expression for
[a] The standard convention for the empty sum implies thus
However, many introductory textbooks simply include this fact as part of the definition.
If (−1, 0, 0) were replaced by (1, 0, 0), it would also form the canonical basis of
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).
The set of monomials xn, where n is a non-negative integer, spans the space of polynomials.
The set of all linear combinations of a subset S of V, a vector space over K, is the smallest linear subspace of V containing S. 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. Let V be a finite-dimensional vector space.
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.
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[7] The vector space definition can also be generalized to modules.
[8][9] Given an R-module A and a collection of elements a1, ..., an of A, the submodule of A spanned by a1, ..., an is the sum of cyclic modules
consisting of all R-linear combinations of the elements ai.
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.
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
, is the intersection of all the closed linear subspaces of X which contain E. 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.
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).
Let X be a normed space and let E be any non-empty subset of X.
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.)
