# 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 **R**^{3} 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 **R**^{3}; instead its span is the space of all vectors in **R**^{3} whose last component is zero.