  ## Encyclopedia > Linear span

Article Content

# Linear span

If V is a vector space and S is a subset of V, then S spans V if every vector in V can be written as a linear combination of (finitely many) elements from S. S is then called a spanning set or generating set of V.

Given any subset S of a vector space V, regardless of whether S is a spanning set of V, we can define the span of S to be the set of all linear combinations of elements of S. Then S spans V if and only if V is the span of S; in general, however, the span of S will only be a subspace of V.

A spanning set that is also linearly independent is a basis. In other words, S is a basis of V if and only if every vector in V can be written as a linear combination of elements of S in exactly one way.

### Examples

The real vector space R3 has {(1,0,0), (0,1,0), (0,0,1)} as 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 even a spanning set of R3; instead its span is the space of all vectors in R3 whose last component is zero.

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 242 ... Contents 242 Centuries: 2nd century - 3rd century - 4th century Decades: 190s 200s 210s 220s 230s - 240s - 250s 260s 270s 280s 290s Years: 237 238 239 240 ...  