Exercise 4.1.8

Figure 4.3 has $x=x_r+x_n$, where $x_r$ is in the row space and $x_n$ is in the nullspace. Then $A{x}_n=0$ and $Ax=A{x}_r+A{x}_n=A{x}_r$. The example has $x=(1,0)$ and row space $=$ line through $(1,1)$ so the splitting is $x=x_r+x_n=\left(\frac{1}{2},\frac{1}{2}\right)+\left(\frac{1}{2},-\frac{1}{2}\right)$. All $Ax$ are in $C(A)$.