Let $\mathbb{F}_3$ be the field with 3 elements. I have defined the vector space which contains 9 vectors as; $$V=\mathbb{F}_3^2=\left\{(x_1,x_2)\mid x_1,x_2 \in \mathbb{F}_3\right\}=\left\{(0,0),(0,1),(0,2)...(2,2)\right\}.$$
Similarly, for $V=\mathbb{F}_3^3$, we have; $$V=\mathbb{F}_3^3=\left\{(x_1,x_2,x_3)\mid x_1,x_2,x_3 \in \mathbb{F}_3\right\}= \left\{(0,0,0),(0,0,1),(0,0,2)...(2,2,2)\right\}.$$ This vector space contains 27 vectors. Have I defined these sets spaces correctly? And if so, how would I go about working out the number of ordered bases in these?
Thanks