If $AB = 0$, prove that the columns of matrix $B$ are vectors in the kernel of $A$ 
Let $A,B$ be $n\times n$ matrices.
If $AB=0$, prove that the columns of matrix $B$ are vectors in the kernel of $Ax=0$.

I'm not sure how to approach this. I know that if $B = 0$ and $A$ isn't, then $Ax=0$ is when $x=0=B$. But what if $A=0$? Seems like in this case B doesn't have to be a part of the kernel.
Or perhaps I'm just missing something?

It results from the very definition of matrix multiplication: to obtain column $j$ in the product $AB$, you multiply each line of $A$ with column $j$ of $B$, i.e. you multiply $A$ by column $j$ of $B$.
