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, ..., vk is any vector of the form
 * $$c_1\textbf{v}_1 + \cdots + c_k\textbf{v}_k$$

where c1, ..., ck are scalars. The span of v1, ..., vk 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
 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, ..., en 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


 * $${ \rm span } \left(v_1,...,v_r\right) = \left\{ {\lambda _1 v_1 +  \cdots  + \lambda _r v_r |\lambda _1, \ldots ,\lambda _r  \in \mathbb K} \right\}.$$

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), (&minus;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.