1. (a) and (d)

2. 

1. (a) Hint: You need to show that the vectors are mutually orthogonal, that is,

   $${\text{u}}_1^T{\text{u}}_2={\text{u}}_1^T{\text{u}}_3={\text{u}}_2^T{\text{u}}_3=0$$

   and also that the vectors are unit vectors, that is,

   $${\text{u}}_1^T{\text{u}}_1={\text{u}}_2^T{\text{u}}_2={\text{u}}_3^T{\text{u}}_3=1$$

2. (b)
   $\begin{array}{c}\mathbf{x}=-\frac{\sqrt{2}}{3}{\mathbf{u}}_1+\frac{5}{3}{\mathbf{u}}_2,\hfill \\ \parallel \mathbf{x}\parallel =[{(-\frac{\sqrt{2}}{3})}^2+{\left(\frac{5}{3}\right)}^2{]}^{1/2}=\sqrt{3}\end{array}$

3. $\mathbf{p}={(\frac{23}{18},\frac{41}{18},\frac{8}{9})}^T,\mathbf{p}-\mathbf{x}={(\frac{5}{18},\frac{5}{18},-\frac{10}{9})}^T$

4. 

1. (b) Hint: Make use of Theorem 5.5.2.

5. Hint: Make use of Theorem 5.5.2 and Parseval’s formula.

6. 

1. (a) 15;

2. (b) $\parallel \mathbf{u}\parallel =3,\parallel \mathbf{v}\parallel =5\sqrt{2}$;

3. (c) $\frac{\pi }{4}$

7. Hint: Make use of Theorem 5.5.2 and Parseval’s formula.

9. 

1. (b) 

   1. 0,

   2. $-\frac{\pi }{2}$,

   3. 0,

   4. $\frac{\pi }{8}$

14. Hint: H is symmetric since

Since H is symmetric, it follows that $H^{T}H=H^2$. So, to show that H is orthogonal, you need to show that $H^2=I$.

15. Hint: Since $\text{det}\left(Q^T\right)\text{= det}(Q)$, it follows that

18. Hint: If P is a symmetric permutation matrix, then P is orthogonal and $P^{-1}=P^T=P$.

21. 

1. (b)  

   1. ${(2,-2)}^T$,

   2. ${(5,2)}^T$,

   3. ${(3,1)}^T$

22. 

1. (a) $P=\left[\begin{array}{cccc}\frac{1}{2}& \frac{1}{2}& 0& 0\\ \frac{1}{2}& \frac{1}{2}& 0& 0\\ 0& 0& \frac{1}{2}& \frac{1}{2}\\ 0& 0& \frac{1}{2}& \frac{1}{2}\end{array}\right]$;

23. 

1. (b) $Q=\left[\begin{array}{cccc}\hfill \frac{1}{2}& \hfill -\frac{1}{2}& \hfill 0& \hfill 0\\ \hfill -\frac{1}{2}& \hfill \frac{1}{2}& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill \frac{1}{2}& \hfill -\frac{1}{2}\\ \hfill 0& \hfill 0& \hfill -\frac{1}{2}& \hfill \frac{1}{2}\end{array}\right]$

25.  

1. (a) Hint: If U is a matrix whose columns form an orthonormal basis for S, then the projection matrix P corresponding to S is given by $P=U{U}^T$.

26. Hint: Show that $A^{T}A$ is a diagonal matrix and that its diagonal entries are ${\mathbf{\text{a}}}_1^T{\mathbf{\text{a}}}_1,{\mathbf{\text{a}}}_2^T{\mathbf{\text{a}}}_2,\mathrm{...},{\mathbf{\text{a}}}_n^T{\mathbf{\text{a}}}_n$.

27. Hint: Represent v as a sum of orthogonal vectors and use the Pythagorean law.

28. Hint: $V=S\oplus S^{\perp }$.

29. 

1. (b) $\parallel 1\parallel =\sqrt{2},\parallel x\parallel =\frac{\sqrt{6}}{3}$;

2. (c) $l(x)=\frac{9}{7}x$

   Hint: Normalize 1 and x so as to make them unit vectors $u_1\left(x\right)$ and $u_2\left(x\right)$ and then determine the stet solution in terms of $u_1\left(x\right)$ and $u_2\left(x\right)$.