How to determine whether a given set of vectors

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Presentation transcript:

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

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

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

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

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

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

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

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 ?

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 ?

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 ?

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 ?

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 ?

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 ?

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

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

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

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

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

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

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

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 ?

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 ?