Differential Equations and Linear Algebra, 4th Edition

Section 4.3

1. 1. Linearly dependent

2. 2. Linearly dependent

3. 3. Linearly dependent

4. 4. Linearly dependent

5. 5. Linearly dependent

6. 6. Linearly dependent

7. 7. Linearly dependent

8. 8. Linearly dependent

9. 9. $\mathbf{w}=2{\mathbf{v}}_1-3{\mathbf{v}}_2$

10. 10. $\mathbf{w}=7{\mathbf{v}}_1+4{\mathbf{v}}_2$

11. 11. $\mathbf{w}={\mathbf{v}}_1-2{\mathbf{v}}_2$

12. 12. $\mathbf{w}=2{\mathbf{v}}_1+5{\mathbf{v}}_2$

13. 13. w cannot be expressed as a linear combination of ${\mathbf{v}}_1$ and ${\mathbf{v}}_2$

14. 14. w cannot be expressed as a linear combination of ${\mathbf{v}}_1,{\mathbf{v}}_2$ and ${\mathbf{v}}_3$

15. 15. $\mathbf{w}=3{\mathbf{v}}_1-2{\mathbf{v}}_2+4{\mathbf{v}}_3$

16. 16. $\mathbf{w}=6{\mathbf{v}}_1-2{\mathbf{v}}_2+3{\mathbf{v}}_3$

17. 17. The vectors ${\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3$ are linearly independent.

18. 18. $3{\mathbf{v}}_1+{\mathbf{v}}_2+5{\mathbf{v}}_3=0$

19. 19. The vectors ${\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3$ are linearly independent.

20. 20. The vectors ${\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3$ are linearly independent.

21. 21. ${\mathbf{v}}_1-2{\mathbf{v}}_2-{\mathbf{v}}_3=0$

22. 22. $7{\mathbf{v}}_1+5{\mathbf{v}}_2-9{\mathbf{v}}_3=0$