Exercise 4.1.17

If $S$ is the subspace of $R^3$ containing only the zero vector, then $S^{\text{⊥}}$ is all of $R^3$. If $S$ is spanned by $(1,1,1)$, then $S^{\text{⊥}}$ is the plane spanned by $(1,-1,0)$ and $(1,0,-1)$. If $S$ is spanned by $(1,1,1)$ and $(1,1,-1)$, then $S^{\text{⊥}}$ is the line spanned by $(1,-1,0)$.