I'm not sure where to start on part a, but part b I think I have a decent start.Let A be an M by N matrix, B be an N by P matrix
a) Show that AB is the sum of n matrices each of rank at most 1
b) If the rank of A is n, what is the rank of AB?
We know that A maps from Fn to Fm, and B maps from Fp to Fn. So we define the left multiplication of these guys as such,
So to find the range of AB we can let Z be some arbitrary vector in and see for what values is it good for
We can note that is a subset of so the following inequality results,
We can further note that if is a surjective map then its range is actually equal to turning the inequality into equality.
Was the above process correct for b, and if so, are there any hints for a?