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Homework Help: Dimension problem

  1. Jul 9, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove that if W is a subspace of an n-dimensional vector space V and dim(W) = n, then W=V

    2. Relevant equations

    3. The attempt at a solution

    I don't know where to start.
  2. jcsd
  3. Jul 9, 2009 #2


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    Homework Helper

    What does 'dim' mean, and what does subspace mean? Can you look it up? That should get you started.
    Last edited: Jul 9, 2009
  4. Jul 12, 2009 #3
    I understand why W would equal V. Every linear combination of vectors in W is in W, and since W and V have the same number of elements in a basis, and W is in V, then W=V. I just don't know how to illustrate that.
    Last edited: Jul 12, 2009
  5. Jul 12, 2009 #4
    Probably the text, just before this, has some results about "linear independence", "spanning sets", and "dimension". Probably there is a proof that shows any two bases for a given vector space are the same size. That material is what you need to understand in order to do this problem.
  6. Jul 12, 2009 #5


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    As g edgar suggested, and I will make even more specific:

    If any two of these are true, so is the third
    1) There exist a set of n independent vectors
    2) There exist a set of n vectors that span the space
    3) The dimension is n
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