Exercise 2.2.1

Suppose $A^T\mathit{Ax}=0$, then $\mathit{Ax}$ is in the nullspace of $A^T$, but $\mathit{Ax}$ is always in the column space of $A$, and we know $C(A)$ is perpendicular to the $N(A^T)$, so if $A^T\mathit{Ax}=0$, then it has to be that $\mathit{Ax}=0$, i.e. $x$ is in the nullspace of $A$.

On the other side, if we have $\mathit{Ax}=0$, so $x$ is in the nullspace of $A$, it’s clear that $A^T\mathit{Ax}=0$, so $x$ is in the nullspace of $A^{T}A$ as well.

Combine both results, we see that $N(A^{T}A)=N(A)$