since dim of zero vector is 0, does it mean that if W= {0, v} then the dim W = 1?
i thought, by definition of dim, dim W= 2 instead?
i dont get why dim of a zero vector is 0 when there is one element, namely the zero vector.
Do you mean W = span{0,v}? In this case, we do have dim W = 1.
The reason is that the vectors 0 and v are not linearly independent (for example, 7*0 + 0*v = 0).
Hence span{0,v} = span{v}, and so dim W = 1 (remember that the dimension of a vector space is the number of basis vectors in a given basis, or equivalently, the maximal number of linearly independent vectors in the vector space).
As for why the dimension of the vector space consisting of nothing but the zero vector is 0, you can think of it as a convenient convention. In R^3, the dimension of a plane is 2. Moving down in dimension, the line has dimension 1, and moving down in dimension again, you have the vector space {0}, which is assumed to have dimension 0.
Your statement that $\displaystyle \mathbb{R}^n$ always has dimension $\displaystyle n$ is correct.
Then you take a plane, say $\displaystyle P$ inside of $\displaystyle \mathbb{R}^3$. This plane is a subspace of $\displaystyle \mathbb{R}^3$, and as such, there is no reason why the plane should have dimension 3. Remember, it is all of $\displaystyle \mathbb{R}^3$ that has dimension 3 as a vector space. This does not imply that all subspaces of $\displaystyle \mathbb{R}^3$ have dimension 3.