Can Spanning Subsets X and Y in Vector Space V Share No Common Vector?

[stunner5000pt]

If X and Y are nonempty subsets of vector space V such taht span X = span Y = V , must there be a vector common to both X and Y? Justify.

my intuition thinks there need not be one, but i can be wrong!

to go about proving it is where the fun starts!
a_{1} x_{1} + a_{2} x_{2} + ... + a_{n} x_{n} = b_{1} y_{1} + ... + b_{n} y_{n} = V
suppose X contaiend only one vector then ax = by. That seems to say taht one is a scalar multiple of the other
suppose X contaiend two vectors then
a_{1} x_{1} + a_{2} x_{2} = b_{1} y_{1} + b_{2} y_{2} = V
here that means the resultant of the first addition yields the resultant of the second addition. However it doesn't mean that X = Y since ai is not bi for all i.
So there need not be any common vector

is this right? Can you help with formalizing it?
Thank you!

 

[Palindrom]

Maybe I got it wrong.

Let B be a basis for V. Then take X=B, Y=2B as a counter example.

Did I miss something?


*If you're not over R, replace 2 by any a!=0.

 

[AKG]

Is there a fixed field you're working over, or must this work for any field, or more generally, over any ring? When the field is F = {0, 1}, then there is a vector space over F such that if Span(X) = Span(Y) = V, then Y and X share a vector, and that is when V = F. In general, if F is any field, then there is a vector space such that if X and Y are nonempty, and span(X) = Span(Y) = V, then X and Y share a vector, and that is when V = {0}. However, there are fields and vectors spaces over them such that Span(X) = Span(Y) = V, but X and Y share no common vector. To formally prove this, you just need a counter-example: V = F = R, X = {1}, Y = {-1}.