Editing Linear span (section)

== Examples ==
The [[real number|real]] vector space <math>\mathbb R^3</math> has {(−1, 0, 0), (0, 1, 0), (0, 0, 1)} as a spanning set. This particular spanning set is also a [[Basis (linear algebra)|basis]]. If (−1, 0, 0) were replaced by (1, 0, 0), it would also form the [[standard basis|canonical basis]] of <math>\mathbb R^3</math>.

Another spanning set for the same space is given by {(1, 2, 3), (0, 1, 2), (−1, {{frac|1|2}}, 3), (1, 1, 1)}, but this set is not a basis, because it is [[Linear dependency|linearly dependent]].

The set {{math|{(1, 0, 0), (0, 1, 0), (1, 1, 0)}}} is not a spanning set of <math>\mathbb R^3</math>, since its span is the space of all vectors in <math>\mathbb R^3</math> whose last component is zero. That space is also spanned by the set {(1, 0, 0), (0, 1, 0)}, as (1, 1, 0) is a linear combination of (1, 0, 0) and (0, 1, 0). Thus, the spanned space is not <math>\mathbb R^3.</math> It can be identified with <math>\mathbb R^2</math> by removing the third components equal to zero.

The empty set is a spanning set of {(0, 0, 0)}, since the empty set is a subset of all possible vector spaces in <math>\mathbb R^3</math>, and {(0, 0, 0)} is the intersection of all of these vector spaces.

The set of [[monomial]]s {{mvar|x<sup>n</sup>}}, where {{mvar|n}} is a non-negative integer, spans the space of [[polynomial]]s.