I'm stuck on this question;
Show that $Nul(A^tA) = Nul(A)$ for every matrix A.
I don't really know where to start on this. I know that rank is not changed by transposing, so nullity is also something I can figure out (rank nullity theorem).
But this is as far as I can get, and I have no idea how to tackle this in regards particularly to the matrix product, $A^tA$.
EDIT: I basically just added everything I know about the topic. Question doesn't change.
Best Answer
If $x \in \ker A$ then since $A^T(Ax) = A^T 0 = 0 $ we see that $x \in \ker A^T A$.
If $x \in \ker A^T A$, then $A^T A x = 0$ and so $x^T A^T A x = (Ax)^T (Ax) = \|Ax\|^2 = 0$ and so $x \in \ker A$.