Linear Algebra with Applications, 10th Edition

5.5

1. 1. (a) and (d)

2. 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. 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. 4. 

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

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

6. 6. 

   1. (a) 15;

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

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

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

8. 9. 

   1. (b) 

      1. 0,

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

      3. 0,

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

9. 14. Hint: H is symmetric since

   $$H^T={(I-2{\text{uu}}^T)}^T=I^T-2{\left({\text{u}}^T\right)}^T{\text{u}}^T=I-2{\text{uu}}^T=H$$

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

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

    $${\left[\text{det}\right(}^{}$$