1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    NA

    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

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    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!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Vector Subspace Question
  1. Vector Subspace (Replies: 3)

  2. Vector Subspace Question (Replies: 12)

  3. Vector Subspaces (Replies: 5)

  4. Vector subspaces (Replies: 6)

Loading...