# Linear span

In mathematics, 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.

">
 " size=20>

Browse articles alphabetically:
#0">0 | #1">1 | #2">2 | #3">3 | #4">4 | #5">5 | #6">6 | #7">7 | #8">8 | #9">9 | #_">_ | #A">A | #B">B | #C">C | #D">D | #E">E | #F">F | #G">G | #H">H | #I">I | #J">J | #K">K | #L">L | #M">M | #N">N | #O">O | #P">P | #Q">Q | #R">R | #S">S | #T">T | #U">U | #V">V | #W">W | #X">X | #Y">Y | #Z">Z