Exercise 4.1.10

(a) With $A^T=A$, the column and row spaces are the same. The nullspace is always $\begin{array}{cc}\text{ perpendicular to the row space. }\hfill & \text{ (b) }x\text{ is in the nullspace and }z\text{ is in the column }\hfill \end{array}$space $=$ row space: so these "eigenvectors" $x$ and $z$ have $x^{T}z=0$.