Linear Algebra with Applications, 10th Edition

5.4

1. 1. $\parallel \mathbf{x}{\parallel }_2=2,\parallel \mathbf{y}{\parallel }_2=6,\parallel \mathbf{x}+\mathbf{y}{\parallel }_2=2\sqrt{10}$

2. 2. 

   1. (a) $\theta =\frac{\pi }{4};\mathbf{p}={(\frac{4}{3},\frac{1}{3},\frac{1}{3},0)}^T$

3. 3. 

   1. (b) $\parallel \mathbf{x}\parallel =1,\parallel \mathbf{y}\parallel =3$

4. 4. 

   1. (a) 0;

   2. (b) 5;

   3. (c) 7;

   4. (d) $\sqrt{74}$

5. 7. 

   1. (a) 1;

   2. (b) $\frac{1}{\pi }$

6. 8. 

   1. (a) $\frac{\pi }{6}$;

   2. (b) $\mathbf{p}=\frac{3}{2}x$

7. 11. 

   1. (a) $\frac{\sqrt{10}}{2}$;

   2. (b) $\frac{\sqrt{34}}{4}$

8. 15. 

   1. (a) ${\parallel \mathbf{x}\parallel }_1=7,{\parallel \mathbf{x}\parallel }_2=5,{\parallel \mathbf{x}\parallel }_{\infty }=4$;

   2. (b) ${\parallel \mathbf{x}\parallel }_1=4,{\parallel \mathbf{x}\parallel }_2=\sqrt{6},{\parallel \mathbf{x}\parallel }_{\infty }=2$;

   3. (c) ${\parallel \mathbf{x}\parallel }_1=3,{\parallel \mathbf{x}\parallel }_2=\sqrt{3},{\parallel \mathbf{x}\parallel }_{\infty }=1$

9. 16. $\begin{array}{c}{\parallel \mathbf{x}-\mathbf{y}\parallel }_1=5,{\parallel \mathbf{x}-\mathbf{y}\parallel }_2=3,{\parallel \mathbf{x}-\mathbf{y}\parallel }_{\infty }=2\end{array}$

10. 17. Hint: If x is orthogonal to y, then it is also orthogonal to –y. Note that $\parallel \mathbf{\text{x}}-\mathbf{\text{y}}\parallel =\parallel \mathbf{\text{x +}}(-\mathbf{\text{y)}}\parallel$.

11. 20. Hint: If A is nonsingular and $A\mathbf{\text{x = 0}}$, then x must be the zero vector.

12. 28. 

    1. (a) Hint: If

       $$\left|f\mathrm{(a)}\right|+\left|f\mathrm{(b)}\right|=0$$

       does this imply that f must be the zero function?

    2. (b) Hint: This is how the 1-norm is defined for function spaces. It is the continuous version of the discrete 1-norm we use for ${ℝ}^n$. Show that the 3 conditions required in the definition of ...