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

2. 

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

3. 

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

4. 

1. (a) 0;

2. (b) 5;

3. (c) 7;

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

7. 

1. (a) 1;

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

8. 

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

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

11. 

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

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

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

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

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

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

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 a norm are satisfied.

3. (c) Hint: This is how the infinity norm is defined for function spaces. It is the continuous version of the discrete infinity norm we use for ${ℝ}^n$. Show that the 3 conditions required in the definition of a norm are satisfied.

31. Hint: See Exercise 21 in Section 5.1.

32. (b) Hint: Apply the result from part (a) to the matrix $A+B$.