2
$\begingroup$

How to show that for a given square matrices $N(A) = R{(A^*)}^\perp$ and $N(A^*)=R{(A)}^\perp$ where $N(A) $ and $R(A) $ are the null and range spaces of matrix $A$, respectively?

I am not able to figure out how to start?I find difficulty when I have to deal with the orthogonal complement of subspaces.

Thanks for helping me.

  • 0
    @Marvis Thanks for pointing please edit that.2012-05-17

1 Answers 1

3

HINT Let $z \in N(A)$ and $x \in R(A^*)$. This gives us $Az = 0$ and $x = A^*b$, for some $b$. $x^*z = b^*Az = 0$. Hence, $x \perp z$.

  • 0
    Thanks a lot for helping me.2012-05-17