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?

The columns of $B$ are exactly the vectors $B e_i$ where $e_i$ are the standard basis vectors.  Then since $AB = 0$, $AB e_i = 0$, so $A(B e_i) = 0$, so $B e_i \in \ker A$.
