let A be mxn matrix and B be the transpose of A. let P=BA. prove that A and P have the same null space and the same rank. please help.
Follow Math Help Forum on Facebook and Google+
Originally Posted by Kat-M let A be mxn matrix and B be the transpose of A. let P=BA. prove that A and P have the same null space and the same rank. please help. we only need to prove that the null space of A and P are equal, because then by the rank-nullity theorem they must also have the same rank. if then and hence conversely, if then i.e. hence
Originally Posted by NonCommAlg if then i.e. hence Alternatively, if then and so .
Originally Posted by Opalg Alternatively, if then and so . that's certainly an easier way!
Originally Posted by NonCommAlg we only need to prove that the null space of A and P are equal, because then by the rank-nullity theorem they must also have the same rank. if then and hence conversely, if then i.e. hence i am trying to see why . could you also help me with this?
Then look at Opalg's response.
View Tag Cloud