Linear span

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.