User:Jim.belk/Draft:Linear span

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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.

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[edit] 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\}[1]

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

[edit] Examples

This will be an illustration
This will be an illustration
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, ..., 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.

[edit] 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\}.

[edit] 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.

[edit] 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.

[edit] 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.

[edit] External links