I'm looking at a vector space $V = F^3$ where $F = \{0,1,2\}$ the field with three elements. I have a subspace $W = \text{span}(1,2,1)$ and I'm trying to explicitly describe the quotient space $V/W$.
I know the equivalence class for the zero vector is $W$ which is to say $\{(0,0,0),(1,2,1),(2,1,2)\}$. I know I need two other vectors that form a basis with $(1,2,1)$ for V. Take $(1,1,0)$ and $(0,0,1)$. Then to find the equivalence class for these two vectors I look at $E(v) = v + w; \forall w \in W$. So we get $$E(1,1,0) = \{(1,1,0),(2,0,1),(0,1,2)\}$$ and $$E(0,0,1) = \{(0,0,1),(1,2,2),(2,1,0)\}$$
My question is I know dim($V/W$) = dim($V$) - dim($W$) so I'm looking for two vectors in the quotient space that are linearly independent to form a basis but I don't think the above is correct because if I look at $E(1,1,0) + E(0,0,1)$ I should get $E(1,1,1)$ which does not appear in any of my equivalence classes. Have I made a mistake or misunderstood what is supposed to happen?
If someone could point me in the right direction I would greatly appreciate it.
The two elements you have picked are a basis. I'm not sure how you are calculating $E(1, 1, 0) + E(0, 0, 1)$ but by definition it is $E((1, 1, 0) + (0, 0, 1)) = E(1, 1, 1)$.