How many vectors are there in the basis of solution set for Ax=0?
Solution 1:
You are correct about using the Rank & Nullity theorem. Observe that $A$ has rank of $2$ because the rows of $A$ are linearly independent because the second row is not a multiple of the first row. And this shows that the dimension of the null space of $A$ is $3$. Alternatively, you can row reduce $A$ in RREF and the number of free variables equals to the nullity of $A$ which is $3$ in this example.