User:Jim.belk/Draft:Linear span

In linear algebra, the linear span (or span) of a collection of vectors is the set of all linear combinations of those vectors. The span of vectors is a Euclidean subspace of Rⁿ, such a line or plane through the origin. More generally, the span of vectors from a vector space is a linear subspace.

Definition

A linear combination of vectors v₁, ..., v_k is any vector of the form

$c_{1}{\textbf {v}}_{1}+\cdots +c_{k}{\textbf {v}}_{k}$

where c₁, ..., c_k are scalars. The span of v₁, ..., v_k is the set of all possible linear combinations:

${\text{Span}}\{{\textbf {v}}_{1},\ldots ,{\textbf {v}}_{k}\}=\left\{c_{1}{\textbf {v}}_{1}+\cdots +c_{k}{\textbf {v}}_{k}:c_{1},\ldots ,c_{k}\in {\textbf {R}}\right\}$

This definition can be generalized to allow for infinite sets of vectors (see below).

Examples

This will be an illustration

This will be an illustration

• The span of the vectors (1, 0) and (0, 1) is all of R². Every vector in R² can be expressed as a linear combination of these two:

  $(x,y)=x(1,0)+y(0,1)\,$

  This is the smallest More generally, the standard basis vectors e₁, ..., e_n span Rⁿ
• The vectors (1, 0) and (1, 1) also span R²:

  $(x,y)=(x-y)(1,0)+y(1,1)\,$

  (See the picture at the top of the article.) However, the vectors (1, 1) and (–2, –2) span a one-dimensional subspace. In general, a set of n vectors in Rⁿ span all of Rⁿ if and only if they are linearly independent.
• The vectors (0, 1, 0) and (0, 0, 1) span the yz-plane in R³.

Span of infinitely many vectors

Given a vector space V over a field K, the span of a set S (not necessarily finite) is defined to be the intersection W of all subspaces of V which contain S. When S is a finite set, then W is referred to as the subspace spanned by the vectors in S.

Let $v_{1},...,v_{r}\in V$. The span of the set of these vectors is

${\rm {span}}\left(v_{1},...,v_{r}\right)=\left\{{\lambda _{1}v_{1}+\cdots +\lambda _{r}v_{r}|\lambda _{1},\ldots ,\lambda _{r}\in \mathbb {K} }\right\}.$

Notes

The span of S may also be defined as the collection of all (finite) linear combinations of the elements of S.

If the span of S is V, then S is said to be a spanning set of V. A spanning set of V is not necessarily a basis for V, as it need not be linearly independent. However, a minimal spanning set for a given vector space is necessarily a basis. In other words, a spanning set is a basis for V if and only if every vector in V can be written as a unique linear combination of elements in the spanning set.

Examples

The real vector space R³ has {(1,0,0), (0,1,0), (0,0,1)} as a spanning set. This spanning set is actually a basis.

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

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: Let V be a finite dimensional vector space. Any set of vectors that spans V can be reduced to a basis by discarding vectors if necessary.

This also indicates that a basis is a minimal spanning set when V is finite dimensional.

External links

• M.I. Voitsekhovskii (2001) [1994], "Linear hull", Encyclopedia of Mathematics, EMS Press