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.

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- Jan 9th 2009, 07:49 PMKat-Mnull space and rank
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. - Jan 10th 2009, 11:30 AMNonCommAlg
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 $\displaystyle Ax=0,$ then $\displaystyle Px=0$ and hence $\displaystyle \text{null}(A) \subseteq \text{null}(P).$ conversely, if $\displaystyle x \in \text{null}(P),$ then $\displaystyle Px=A^TAx=0,$ i.e. $\displaystyle Ax \in \text{null}(A^T)=(\text{col}(A))^{\perp}.$ hence $\displaystyle Ax \in \text{col(A)} \cap (\text{col}(A))^{\perp}=(0). \ \Box$ - Jan 10th 2009, 01:18 PMOpalg
- Jan 10th 2009, 01:33 PMNonCommAlg
- Jan 10th 2009, 01:43 PMKat-M
- Jan 10th 2009, 03:05 PMHallsofIvy
Then look at Opalg's response.