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Homework Help: Cardinality and dimension

  1. Jun 7, 2006 #1
    Find the cardinality and dimension of the vector space [itex]\mathbb{Z}^{3}_{7}[/itex] over [itex]\mathbb{Z}_{7}[/itex].

    [itex]\mathbb{Z}^{3}_{7} = \{ (a,b,c) \; | \; a,b,c \in \mathbb{Z}_{7} \}[/itex].

    Then since [itex]\mathbb{Z}_{7}[/itex] is a field [itex]1 \cdot a = a \; \forall \; a[/itex], so [itex]B = \{ (1,0,0), (0,1,0) , (0,0,1) \}[/itex] is a basis of [itex]\mathbb{Z}^{3}_{7}[/itex], so [itex]\dim \mathbb{Z}^{3}_{7} = 3[/itex]. ans = 9, what the?

    Thanks

    James
     
    Last edited: Jun 7, 2006
  2. jcsd
  3. Jun 17, 2006 #2
    Hi again,

    Does anyone have any clues on this one? I'm really stuck.

    Thanks

    James.
     
  4. Jun 17, 2006 #3

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    Are you sure you copied the problem correctly? Because 3 certainly seems to be the right answer.
     
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