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Proving subspace homework

  1. Feb 21, 2006 #1
    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!
     
  2. jcsd
  3. Feb 21, 2006 #2
    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.
     
  4. Feb 21, 2006 #3

    AKG

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    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}.
     
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