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**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!

[tex] a_{1} x_{1} + a_{2} x_{2} + ... + a_{n} x_{n} = b_{1} y_{1} + ... + b_{n} y_{n} = V [/tex]

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

[tex] a_{1} x_{1} + a_{2} x_{2} = b_{1} y_{1} + b_{2} y_{2} = V [/tex]

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!