Linear Algebra with Applications, 10th Edition

3.1

1. 1. 

   1. (a) $\parallel {\mathbf{x}}_1\parallel =10,\parallel {\mathbf{x}}_2\parallel =\sqrt{17}$;

   2. (b) $\parallel {\mathbf{x}}_3\parallel =13<\parallel {\mathbf{x}}_1\parallel +\parallel {\mathbf{x}}_2\parallel$

2. 2. 

   1. (a) $\parallel {\mathbf{x}}_1\parallel =\sqrt{5},\parallel {\mathbf{x}}_2\parallel =3\sqrt{5}$;

   2. (b) $\parallel {\mathbf{x}}_3\parallel =4\sqrt{5}=\parallel {\mathbf{x}}_1\parallel +\parallel {\mathbf{x}}_2\parallel$

3. 4. Hint: As in the previous exercise you must check to see if the eight axioms hold. Theorem 1.3.1 guarantees that 5 of the axioms hold, so you need only show that the remaining 3 also hold.

4. 7. If $\mathbf{x}+\mathbf{y}=\mathbf{x}$ for all x in the vector space, then $\mathbf{0}=\mathbf{0}+\mathbf{y}=\mathbf{y}$.

5. 8. If $\mathbf{x}+\mathbf{y}=\mathbf{x}+\mathbf{z}$, then $-\mathbf{x}+(\mathbf{x}+\mathbf{y})=-\mathbf{x}+(\mathbf{x}+\mathbf{y})$ and the conclusion follows using axioms 1, 2, 3, and 4.

6. 9. 

   1. (a) Hint: Show first that if $\mathbf{\text{y =}}\beta \mathbf{0},$ then $\mathbf{y}+\mathbf{y}=\mathbf{y}\mathrm{.}$

   2. (b) Hint: Show that if $\alpha \mathbf{\text{x}}=\mathbf{0}\mathrm{and}\alpha \ne 0,$ then $\mathbf{\text{x}}=\mathbf{0.}$

7. 10. Hint: Since addition is defined in the standard way, check out the axioms that involve scalar multiplication.

8. 11. V is not a vector space. Axiom 6 does not hold.

9. 13. Hint: Four of the axioms fail to hold. ...