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Homework Help: Subspace query

  1. Apr 24, 2009 #1
    how many 1 dimensional subspaces of Z_3^3 are there?

    Z_3^3 has 3^3 = 27 vectors

    26 of which are non zero

    then we can say v and 2v have the same span and so there are in fact 13 1 dimensional subspaces. is this true?
  2. jcsd
  3. Apr 24, 2009 #2


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    No that doesn't make sense.

    Hint: Draw the space as a lattice.
  4. Apr 24, 2009 #3
    for Z_3^2, my lecturer does the following:

    there are 3^2=9 elements
    8 are non zero
    v and 2v have the same span
    therefore there are 4 1 dimensional subspaces

    why can't this be extended to Z_3^3

    i dont know how to draw it as a lattice or what that means to be honest...
  5. Apr 25, 2009 #4


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    Ok that does make sense, sorry. Here's why he argument works: The non-zero elements of the group Z3 all have order three, so that means that if v is a non zero element of Z3xZ3, v and 2v will be distinct, and 3v will be (0, 0, 0), so {v, 2v, 3v} will be a one dimensional subgroup generated by v.
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