#1

Let A be a m*n matrix and B an n*k matrix. If $\displaystyle AB=0$,

prove than $\displaystyle rank A+rank B <=n$.

#2

Let A be a m*n matrix with $\displaystyle rank A=m$, B be an n*(n-m) matrix, with $\displaystyle rank B=(n-m)$.

If $\displaystyle AB=0$, and $\displaystyle X$ in $\displaystyle R^m$ is a solution to the linear system $\displaystyle AX=0$,prove there exists a unique $\displaystyle Y$ in $\displaystyle R^{n-m}$ such that $\displaystyle X=BY$.