Exercise 3.5.25

(a) True ( $A$ and $A^T$ have the same rank) $$ (b) False $A=\left[\begin{array}{cc}1\hfill & 0\hfill \end{array}\right]$ and $A^T$ have very different left nullspaces $$ (c) False ( $A$ can be invertible and unsymmetric even if $C(A)=C\left(A^T\right))$ (d) True (The subspaces for $A$ and $-A$ are always the same. If $A^T=A$ or $A^T=-A$ they are also the same for $A^T$ )