Difference between dimension and rank of matrix

The null space is a subspace of the original vector space. Observe that the vector space in question is exactly $N(A)$, the null space of $A$.

As you observed, $rank(A) + null(A) = dim(V)$. So $2 + null(A) = 3$.

Hint:

The vector space of all $x\in\mathbb R^3$ such that $Ax = 0$ is called the kernel or nullspace of $A$.