# Proving subspace homework

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 doesnt 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!

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