Exercise 1.3.3

If $x$ belongs to the nullspace of $C$, then we have $\mathit{Cx}=\left[\begin{array}{c}\hfill A\hfill \\ \hfill B\hfill \end{array}\right]x=\left[\begin{array}{c}\hfill \mathit{Ax}\hfill \\ \hfill \mathit{Bx}\hfill \end{array}\right]=0$ so it makes $\mathit{Ax}=0$ and $\mathit{Bx}=0$ at the same time. So the $x$ belongs to both the nullspace of $A$ and $B$.

The opposite is also true, i.e. if $x$ belongs to both the nullspace of $A$ and $B$, it also belongs to the nullspace of $C$. So we conclude that the nullspace of $C$ is the intersection of the nullspaces of $A$ and $B$.