Thread: Linear rank questions

IF M=AB is a product of the matrices A and B. If B has rank k, can M have rank greater than k? If A has rank k can M have rank greater than k? If the answer is yes, give an example. If the answer is no give an argument showing that it cannot be so.

Im not sure how the product can affect a new rank and the relation between a new rank and another matrix.

Originally Posted by Luck of the Irish

IF M=AB is a product of the matrices A and B. If B has rank k, can M have rank greater than k? If A has rank k can M have rank greater than k? If the answer is yes, give an example. If the answer is no give an argument showing that it cannot be so.

Im not sure how the product can affect a new rank and the relation between a new rank and another matrix.

for the first case, the nullity of M is at least the nullity of B suppose x is in the null space of B then ABx = A0 = 0, so by rank-nullity theorem the rank of m is at most k.

for the second case the answer are you sure the question isn't "can M have a rank less than k" ? because it is quiet obvious that the rank can't be more than k.

Bobak

Originally Posted by Luck of the Irish

IF M=AB is a product of the matrices A and B. If B has rank k, can M have rank greater than k? If A has rank k can M have rank greater than k? If the answer is yes, give an example. If the answer is no give an argument showing that it cannot be so.

Im not sure how the product can affect a new rank and the relation between a new rank and another matrix.

A liner transformation maps a vector space, of dimension n, onto a vector space of dimension less than or equal to n. If B has rank k, that means it maps the vector space it is applied to into into a vector space of dimension k. AB is A applied to the result of B: A applied to a vector space of dimension k and so the result must be a vector space of dimension less than or equal to k. The rank of M= AB cannot be larger than k.

Similarly, if A has dimension k, then no matter what the dimension of BV is, AB(V)= A(BV) has dimension no larger than k so, again, the rank of M= AB cannot be larger than k.

bobak its the right question, but Halls of Ivy thanks for the informative posts that cleared up my confusion. you both helped alot...

Originally Posted by Luck of the Irish

IF M=AB is a product of the matrices A and B. If B has rank k, can M have rank greater than k? If A has rank k can M have rank greater than k? If the answer is yes, give an example. If the answer is no give an argument showing that it cannot be so.

Im not sure how the product can affect a new rank and the relation between a new rank and another matrix.

the answer to both parts of your question is "no" because in general the rank of product of two matrices is at most the minimum of the rank of each matrix. this is very easy to see if you let T

and S be the linear transformations corresponding to A and B respectively and recall that the rank of A (respectively B) is the dimension of the range of T (respectively S).