# rank proof

• Oct 6th 2009, 08:15 AM
hayter221
rank proof
If B and C are invertible then prove rank(BA)=rank(AC)=rank(A).
• Oct 6th 2009, 09:33 AM
tonio
Quote:

Originally Posted by hayter221
If B and C are invertible then prove rank(BA)=rank(AC)=rank(A).

Whether you mean matrices or linear maps is the same: as B,C are invertible they are 1-1 onto maps (matrices) so BA is an map ONTO the image of A ==> rk(BA) = rk(A), and the same with C.

Tonio
• Oct 6th 2009, 09:59 AM
aman_cc
Hi -Based on this post, I was trying to look if there is generic rule for finding rank(AB) based on rank(A) and rank(B). I was trying to work that out but not going anywhere.

All I could find was rank(AB)=< Min[rank(A),rank(B)]

Can we say more?
• Oct 6th 2009, 03:04 PM
hayter221
hey tonio, can you rephrase your response solely in terms of matrices, or would the statement be the same?
• Oct 6th 2009, 03:46 PM
tonio
Quote:

Originally Posted by hayter221
hey tonio, can you rephrase your response solely in terms of matrices, or would the statement be the same?

It'd be the same using matrix language: instead of dimension of image use rank, instead of kernel use nullity, etc.

Tonio