1. How to determine whether a given set of vectors
SPANS R3
example 1:
This set does SPAN R 3
example 2:
This set does NOT SPAN R 3

4. Do solutions exist for all x, y, and z ?

6. Do solutions exist for all x, y, and z ?

7. Do solutions exist for all x, y, and z ?

8. Do solutions exist for all x, y, and z ?

9. 1 1 1
x-y -1
Do solutions exist for all x, y, and z ?

10. 1 1 1
x-y -1 -1 1
z-(x-y)
Do solutions exist for all x, y, and z ?

11. 1 1 1
x-y -1 -1 1
z-(x-y) -1 1 2
y-(x-y)
Do solutions exist for all x, y, and z ?

12. 1 1 1
x-y -1 -1 1
z-(x-y)
(x-y)-(2y-x) -1 -1 1 2
y-(x-y) -1
Do solutions exist for all x, y, and z ?

13. 1 1 1
x-y -1 -1 1
z-(x-y)
(x-y)-(2y-x) -1 -1 1 2
y-(x-y) -1
(2x-3y)+(z-(x-y)) +1
Do solutions exist for all x, y, and z ?

14. 1 1 1
x-y -1 -1 1
z-(x-y)
(x-y)-(2y-x) -1 -1 1 2
y-(x-y) -1
(2y-x)-2(z-(x-y)) -2
Do solutions exist for all x, y, and z ?

15. 1 1 1
x-y -1 -1 1
z-(x-y)
(x-y)-(2y-x) -1 -1 1 2
y-(x-y) -1
Do solutions exist for all x, y, and z ?

16. 1 1 1
x-y -1 -1 1
z-(x-y)
(x-y)-(2y-x) -1 -1 1 2
y-(x-y) -1 = c1 c2 = c3 =
YES
Do solutions exist for all x, y, and z ?

17. YES
return to outline = c1 c2 = c3 =
YES
Do solutions exist for all x, y, and z ?

20. Do solutions exist for all x, y, and z ?

22. Do solutions exist for all x, y, and z ?

23. Do solutions exist for all x, y, and z ?

24. Do solutions exist for all x, y, and z ?

25. 1 1 1
x-y -1
Do solutions exist for all x, y, and z ?

26. 1 1 1
x-y -1 -1
z-(x-y)
Do solutions exist for all x, y, and z ?

27. 1 1 1
x-y -1 -1
z-(x-y)
There are no solutions if z – x + y does not equal zero
Do solutions exist for all x, y, and z ?

28. 1 1 1
x-y -1 -1
z-(x-y)
There are no solutions if z – x + y does not equal zero
The set does
Not span R3
Do solutions exist for all x, y, and z ?