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Union of two subspaces

  1. Aug 25, 2011 #1
    Prove that a vector space cannot be the union of two proper
    subspaces.

    Let V be a vector space over a field F where U and W are proper subspaces.

    I am not sure where to start with this proof.
     
  2. jcsd
  3. Aug 25, 2011 #2
    There seems to be something missing in this problem. For example, take the case where U,W are proper subspaces of V, and [itex]U \subset W[/itex], then [itex]U \cap W \equiv W[/itex] is also a proper subspace of V.

    Are you sure there's not some other restrictions regarding the subspaces?
     
  4. Aug 25, 2011 #3
    Would it matter that it is the union and you have written the intersection?
     
  5. Aug 26, 2011 #4

    Mark44

    Staff: Mentor

    What is the definition of "proper subspace"? You should have included that definition when you posted the problem.
     
  6. Aug 26, 2011 #5
    A proper subspace can't be equal to V.
     
  7. Aug 26, 2011 #6

    Mark44

    Staff: Mentor

    But how is this term defined? What you gave is not the definition.
     
  8. Aug 26, 2011 #7
    If U is a proper subspace, then the dim U < dim V and U isn't the subspace of just the 0 vector, i.e., not the trivial subspaces.
     
  9. Aug 26, 2011 #8

    HallsofIvy

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    So your definition of "U is a proper subspace of V" does not include requiring that U be a subset of V?
     
  10. Aug 26, 2011 #9

    vela

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    That was a typo. Coto meant [itex]U \cup W = W[/itex].
     
  11. Aug 26, 2011 #10

    jambaugh

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    I think the issue is that the union is not the same as the span. The union of two subspaces will not be a subspace unless one of the subspaces is contained within the other (is a subspace of the subspace) in which case the union is the larger subspace.

    So apply that to the question at hand...
     
  12. Aug 26, 2011 #11
    Thanks.
     
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