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Homework Help: Vector Subspace Question

  1. Dec 13, 2012 #1
    1. The problem statement, all variables and given/known data

    When is it true that the only subspaces of a vector space V, are V and {0}?

    2. Relevant equations


    3. The attempt at a solution

    Because a subspace has to be closed under addition and scalar multiplication, it is my intuition that this is true only when there are no infinite subsets of V. However, I am not sure this is correct and I do not have a better attempt at an answer. Any help is greatly appreciated.
    Last edited: Dec 13, 2012
  2. jcsd
  3. Dec 13, 2012 #2


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    What about the real line?
  4. Dec 13, 2012 #3
    Infinity doesn't enter the matter, since there are finite vector spaces with subspaces "in between" {0} and themselves.

    Take a vector space V and some subspace S of V, where S is neither just {0} nor all of V.

    What does S≠{0} mean, what must S look like so it's not just {0}?

    And what must V be like, so that this subspace S automatically becomes equal to V if it's not just {0}?
  5. Dec 13, 2012 #4


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    By a "finite vector space" do you mean just {0}?
  6. Dec 13, 2012 #5
    S would have to have dimension greater than 0 and less than dimV. In which case it seems like it would be a subspace. Does this mean it's true that the only subspaces of V are V and {0} only when V is {0}? I'm probably way off track here...
  7. Dec 13, 2012 #6
    No, I'm thinking of vector spaces over finite fields.

    Have a look at some examples: Look at V = {0}, V = a line, V = a plane, V = ##\mathbb R^3##. In which of these cases are there no subspaces "in between" {0} and V? If you see any pattern there, how about ##\mathbb R^4## etc.?
  8. Dec 14, 2012 #7

    It seems like it should be true only when dimV<2. If dimV is 1, then a subspace of V could only have dim1 or 0, thus making it true that the only subspaces are V itself and {0}. If dimV≥2, then V can have at least have a subspace with dim1. Is this right?
  9. Dec 14, 2012 #8


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    Yes, it is. Dimension is the number of vectors in a basis. If dim V>=2 then pick any one of the vectors and its span is a subspace of dimension 1.
  10. Dec 15, 2012 #9
    Great, thanks!
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