problem on rank

#1
Let A be a m*n matrix and B an n*k matrix. If $AB=0$ ,
prove than $rank A+rank B <=n$ .

#2
Let A be a m*n matrix with $rank A=m$ , B be an n*(n-m) matrix, with $rank B=(n-m)$ .
If $AB=0$ , and $X$ in $R^m$ is a solution to the linear system $AX=0$ ,prove there exists a unique $Y$ in $R^{n-m}$ such that $X=BY$ .