Exercise 1.2.7

Example of $A$ and $B$ for which $\mathit{AB}$ has a smaller column space than $A$. Let $A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 4\hfill \\ \hfill 2\hfill & \hfill 5\hfill \\ \hfill 3\hfill & \hfill 6\hfill \end{array}\right]$, it hasa column space of dimension 2, and $B=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \end{array}\right]$, then $\mathit{AB}=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 2\hfill \\ \hfill 3\hfill \end{array}\right]$, which is clearly has a column space of dimension 1, and is smaller than $A$.