Question

Can two vector spaces share the same set of vectors?

Answer 1

seems a bit vague to me.

what do you mean by "share the same set"?

Answer 2

the set of 2x2 matrixes with non zero determinants represent the same R^2 vector space ....

Answer 3

but even the <0,1> may be a vector in one of them, does not mean that it represents the same "point" in another one due to a difference in basises