Exercise 3.2.22

If $A$ times every column of $B$ is zero, the column space of $B$ is contained in the nullspace of $A$. An example is $A=\left[\begin{array}{cc}1\hfill & 1\hfill \\ 1\hfill & 1\hfill \end{array}\right]$ and $B=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill -1\hfill \end{array}\right]$. Here $C(B)$ equals $N(A)$. For $B=0,C(B)$ is smaller than $N(A)$