Cardinality and dimension

1. Jun 7, 2006

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

Last edited: Jun 7, 2006
2. Jun 17, 2006

jdstokes

Hi again,

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

Thanks

James.

3. Jun 17, 2006

StatusX

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