Examples
The real vector space R3 has {(2,0,0), (0,1,0), (0,0,1)} as a spanning set. This particular spanning set is also a basis. If (2,0,0) were replaced by (1,0,0), it would also form the canonical basis of R3.
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 a spanning set of R3; instead its span is the space of all vectors in R3 whose last component is zero.
Read more about this topic: Linear Span
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