# rank theory

• Mar 6th 2011, 11:22 AM
situation
rank theory
literally have no clue on how to do this one... :'(

Prove that if A and B are n x n matrices and if rank(A) = n and rank(B) = n then rank(AB) = n.

any help would be much appreciated here as i have a test tomorrow!
• Mar 6th 2011, 11:57 AM
TheEmptySet
Quote:

Originally Posted by situation
literally have no clue on how to do this one... :'(

Prove that if A and B are n x n matrices and if rank(A) = n and rank(B) = n then rank(AB) = n.

any help would be much appreciated here as i have a test tomorrow!

I don't know what tools you have exactly but here is a way.

A square matrix has full rank if and only if it has non zero determinant. (only the zero vector can be in the nullspace) We also know that

\$\displaystyle \det(AB)=\det(A)\det(B)\$
• Mar 6th 2011, 12:24 PM
HallsofIvy
Another way- the "rank" of a linear transformation, A, from vector space V to itself is the dimension of A(V). Here, A is given by an n by n matrix so it is from \$\displaystyle R^n\$ to \$\displaystyle R^n\$. Since the rank of A is n, \$\displaystyle A(R^n)= R^n\$. Similarly, the rank of B is n so that \$\displaystyle B(R^n)= R^n\$. Putting those together, AB maps \$\displaystyle R^n\$ into all of \$\displaystyle R^n\$: \$\displaystyle AB(R^n)= R^n\$ and so AB has rank n.
• Mar 6th 2011, 01:31 PM
situation
Quote:

Originally Posted by HallsofIvy
Another way- the "rank" of a linear transformation, A, from vector space V to itself is the dimension of A(V). Here, A is given by an n by n matrix so it is from \$\displaystyle R^n\$ to \$\displaystyle R^n\$. Since the rank of A is n, \$\displaystyle A(R^n)= R^n\$. Similarly, the rank of B is n so that \$\displaystyle B(R^n)= R^n\$. Putting those together, AB maps \$\displaystyle R^n\$ into all of \$\displaystyle R^n\$: \$\displaystyle AB(R^n)= R^n\$ and so AB has rank n.

please could you just briefly explain what is meant by the dimension, a linear transformation, and a vector space? i have so pretty awful lecture notes which just arent making sense to me in working out what your explanation means.

thank you both very much!