What is the Cardinality and Dimension of \mathbb{Z}^{3}_{7}?

[jdstokes]

Find the cardinality and dimension of the vector space \mathbb{Z}^{3}_{7} over \mathbb{Z}_{7}.

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

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

Thanks

James

 

[jdstokes]

Hi again,

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

Thanks

James.

 

[StatusX]

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