Exercise 4.4.4

(a) $Q=\left[\begin{array}{cc}1\hfill & 0\hfill \\ 0\hfill & 1\hfill \\ 0\hfill & 0\hfill \end{array}\right],Q{Q}^T=\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill \end{array}\right]\ne I$. Any $Q$ with $n<m$ has $Q{Q}^T\ne I$.

(b) $(1,0)$ and $(0,0)$ are orthogonal, not independent. Nonzero orthogonal vectors are independent.

(c) From $q_1=(1,1,1)∕\sqrt{3}$ my favorite is $q_2=(1,-1,0)∕\sqrt{2}$ and $q_3=(1,1,-2)∕\sqrt{6}$