Thread: Rank of a matrix

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.

Originally Posted by razack

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.

Which definition of rank are you most comfortable with?

Are you comfortable with the idea that as a linear transformation then ?

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?