Let A be an n x n matrix with the property that the intersection of its column space and its null space is the 0 vector. That is: nullA(intersection)colA={0}. Show that A and A^2 have the same rank........
Hi
First, you can show that , which might be useful. (rank-nullity theorem)
Then, showing that and share the same rank is equivalent to proving that (rank-nullity theorem again) Here, we are in a special case : as , (find a proof ) you can try to show that which will give you and the equality of the dimensions.
For , you might take which can be decomposed as with and (because ) and develop