Exercise 4.1.22

$(1,1,1,1)$ is a basis for the line $P^{\text{⊥}}$ orthogonal to $P$. $A=\left[\begin{array}{cccc}1\hfill & 1\hfill & 1\hfill & 1\hfill \end{array}\right]$has $P$ as its nullspace and $P^{\text{⊥}}$ as its row space.