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Subspaces and dimension

  1. Nov 1, 2005 #1
    Hi, I'm wondering how I would decide how many "subspaces of each dimension [tex]Z_2^3 [/tex] has." The answer is: 1 subspace with dim = 0, 7 with dim = 1, 7 with dim = 2, 1 with dim = 3.

    I'm looking for subsets of [tex]Z_2^3 [/tex] which are closed under addition and scalar multiplication. An arbitrary vector in [tex]Z_2^3 [/tex] is (a,b,c) where [tex]a,b,c \in Z_2 [/tex]. I can't think of a general way to do this, trial and error is a possibility and quite time consuming. So I think that there is some concept being tested which I'm not seeing.

    Any set consisting of a single vector with the zero vector in Z_2 would be a subspace since it would be closed under addition. But what about sets with 3 or more elements. I'm not sure how to approach this question. Can someone please help me out?
  2. jcsd
  3. Nov 1, 2005 #2
    Obviously, there is only one subspace of dimension one and three:
    In general, one has a duality between subspaces of dimension k and n-k. So, you only need to count the number of one dimensional subspaces. There are 2^(3) - 1 = 7 nonzero vectors and they are all linearly independent.
  4. Nov 1, 2005 #3


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    Obviously "Careful" meant to say "there is only one subspace of dimension zero and three".

    Be careful, Careful!
  5. Nov 1, 2005 #4
    Hmm, ok thanks for your help.
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