1. Solving homogeneous equations: Ax = 0 Putting answer in parametric vector form Isabel K. Darcy Mathematics Department Applied Math and Computational Sciences University of Iowa Fig from knotplot.com

2. Solving homogeneous equations: Ax = 0 Putting answer in parametric vector form or Determining the solution set for Ax = 0 Isabel K. Darcy Mathematics Department Applied Math and Computational Sciences University of Iowa Fig from knotplot.com

3. Solving homogeneous equations: Ax = 0 Putting answer in parametric vector form or Determining the solution set for Ax = 0 Nullspace of A = solution set for Ax = 0 Isabel K. Darcy Mathematics Department Applied Math and Computational Sciences University of Iowa Fig from knotplot.com

4. Solve: A x = 0 where A =

5. = Put A into echelon form and then into reduced echelon form: R 2 – R 1  R 2 R 3 + 2R 1  R 3 R 1 + 5R 2  R 1 R 2 /2  R 2 R 1 + 8R 3  R 1 R 1 - 2R 3  R 1 R 3 /3  R 3

6. Solve: A x = 0 where A Put A into echelon form and then into reduced echelon form: R 2 – R 1  R 2 R 3 + 2R 1  R 3 R 1 + 5R 2  R 1 R 2 /2  R 2 R 1 + 8R 3  R 1 R 1 - 2R 3  R 1 R 3 /3  R 3 ~

7. Solve: A x = 0 where A ~ 000000

8. ~ x 1 x 2 x 3 x 4 000000

9. Solve: A x = 0 where A ~ x 1 x 2 x 3 x 4 000000 x 1 -2 x 4 x 2 -2 x 4 x 3 - x 4 x 4 x 4 =

10. Solve: A x = 0 where A ~ x 1 x 2 x 3 x 4 000000 x 1 -2 x 4 -2 x 2 -2 x 4 -2 x 3 - x 4 -1 x 4 x 4 1 = = x4x4

11. Solve: B x = 0 where B ~

12. ~ x 1 x 2 x 3 x 4 x 5 000000 x1x2x3x4x5x1x2x3x4x5 =

13. Solve: B x = 0 where B ~ x 1 x 2 x 3 x 4 x 5 000000 x 1 x 2 x 3 x 4 x 5 =

14. Solve: B x = 0 where B ~ x 1 x 2 x 3 x 4 x 5 000000 x 1 x 1 x 1 0 1 0 x 2 -8 x 4 0 -8 x 4 0 -8 x 3 -6 x 4 0 -6 x 4 0 -6 x 4 x 4 0 x 4 0 1 x 5 0 0 0 0 0 = = + x1x1 x4x4

15. Solve: C x = 0 where C ~

16. ~ x 1 x 2 x 3 x 4 x 5 0000000000 x1x2x3x4x5x1x2x3x4x5 =

17. Solve: C x = 0 where C ~ x 1 x 2 x 3 x 4 x 5 0000000000 x 1 -5x 5 -5 x 2 3 x 5 3 x 3 0 0 x 4 x 5 1 x 5 x 5 1 = = x5x5

18. Solve: D x = 0 where D ~

19. ~ x 1 x 2 x 3 x 4 x 5 x 6 000000 = x1x2x3x4x5x6x1x2x3x4x5x6

20. Solve: D x = 0 where D ~ x 1 x 2 x 3 x 4 x 5 x 6 000000 = x 1 x 2 x 3 x 4 x 5 x 6

21. Solve: D x = 0 where D ~ x 1 x 2 x 3 x 4 x 5 x 6 000000 = x 1 7x 4 - 2x 5 x 2 -x 4 x 3 -5x 4 + 4x 5 x 4 x 5 x 6

22. Solve: D x = 0 where D ~ x 1 x 2 x 3 x 4 x 5 x 6 000000 == + + x 1 7x 4 - 2x 5 7 -2 0 x 2 -x 4 -1 0 0 x 3 -5x 4 + 4x 5 -5 4 0 x 4 x 4 1 0 0 x 5 x 5 0 1 0 x 6 x 6 0 0 1 x4x4 x6x6 x5x5

23. Solve: E x = 0 where E ~

24. ~ x 1 x 2 x 3 x 4 x 5 x 6 x 7 = x1x2x3x4x5x6x7x1x2x3x4x5x6x7

25. Solve: E x = 0 where E ~ x 1 x 2 x 3 x 4 x 5 x 6 x 7 == + + x 1 x 2 5x 4 - 5x 7 x 3 -7x 4 + 3x 7 x 4 x 5 0 x 6 x 7 x 7 x1x1 x7x7 x4x4

26. Solve: E x = 0 where E ~ x 1 x 2 x 3 x 4 x 5 x 6 x 7 == + + x 1 x 1 1 0 0 x 2 5x 4 - 5x 7 0 5 -5 x 3 -7x 4 + 3x 7 0 -7 3 x 4 x 4 0 1 0 x 5 0 0 0 0 x 6 x 7 0 0 1 x 7 x 7 0 0 1 x1x1 x7x7 x4x4