Exercise 5.5.8

Suppose $A$ is an $m\times n$-matrix and $B$ is an $n\times p$-matrix. Show that

Hint: Consider the subspace $\operatorname{col}(B)\cap\operatorname{null}(A)$ and suppose a basis for this subspace is $\left\{\mathbf{w}_{1},\ldots,\mathbf{w}_{k}\right\}$. Let $\left\{\mathbf{z}_{1},\ldots,\mathbf{z}_{k}\right\}$ be such that $B\mathbf{z}_{i}=\mathbf{w}_{i}$. Now suppose $\left\{\mathbf{u}_{1},\ldots,\mathbf{u}_{r}\right\}$ is a basis for $\operatorname{null}(B)$, and argue that $\operatorname{null}(AB)\subseteq\operatorname{span}\left\{\mathbf{u}_{1},\ldots,\mathbf{u}_{r},\mathbf{z}_{1},\ldots,\mathbf{z}_{k}\right\}$.