# 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.