Show that a Code has the length of 4 and the hamming distance 4

Question

Let $A$ be a $3\times 4$ matrix with the columns $v_1,v_2,v_3,v_4 \in \mathbb{F}_4^3$ and $C$ the set of solutions of $Ax=0$. Show that $C$ is a code with the alphabet $\mathbb{F}_4$ with the length 4 and hamming distance 4.

It is clear that the length is 4. How do I show that the hamming distance is 4?