Let A be an m x n matrix. Show that if B is n x p, then rank AB <= rank b.
the matrix AB will be a m x p matrix.
I think Rank A is at most n while Rank AB is at most p.
I think I'm supposed to show that p<= n somehow, but I'm not really sure.
I have no idea how to approach this problem. Hints and Help would be much appreciated.
I think I have this.... I don't know how to use TeX. Sorry.
Let the columns of A be [A1 A2 ... An] and the columns of B be [B1 B2 ... Bp]
then the ith column of the matrix AB will be:
ABi = [A1 A2 ... An]Bi = A1bi1 + A2bi2 + .... +Anbin
From this, I can see that The span of the columns of AB are within the span of the columns of A, or is the span of A in the case where the b coeficents are 1.
So:
Col AB <= Col A
Dim Col AB <= Dim Col A
Rank AB <= Rank A
Comments?