Here is a solution. However from the solution I cannot see if the proof is complete. It seems that the proof only show that the list $z$, $p0$,$p1$,$...$,$pm$ are linearly dependent. But how can we see that the list is the linear dependent? In other words. It seems that the assumption that the list is linearly independent is not used to get the contradiction.
http://linearalgebras.com/2A.html problem 17.