Spanning Sets in Vector Spaces

[mlb2358]

Homework Statement

True or False: If S is a spanning set for a vector space V, then every vector v in V must be uniquely expressible as a linear combination of the vectors in S.

Homework Equations

The Attempt at a Solution

For some reason, the answer to this question is false, although to me it seems like this is almost the definition of a spanning set. I am unsure about what the word "uniquely" means here, so maybe that is causing my confusion. Any help would be appreciated!

 

[lanedance]

I think you're onto it, but i'd go back to your definition of a spanning set. If it only requires that the set spans V and uniqueness is not required, then there may exist more than one way to form a given vector v, e.g. consider if your set had both (1,0,0) and (2,0,0) in it..