Suppose that A and B are n n matrices and that A is inverti

Suppose that A and B are n × n matrices and that A is invertible. Show that AB has the same row space as B.

Solution

The row space of B is the set of all x^TB, whereas the row space of AB is the set of all y^TAB. (Here, x and y are vectors of length n.) Every y^TA is a x^T . Because A is assumed to be invertible, every x^T is a y^TA.