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Matthew S. Webster            msw9125                        Attendance #43

Week 5                        Problem 4.1.12

1st Draft

acs.tamu.edu/~msw9125/4_1_12.html

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Test these vectors for linear independence.

V₁ =(1,2,4,3) or in matrix form  |V₁| = | 1 2 4 3 |
V₂ =(2,1,3,4)                    |V₂|   | 2 1 3 4 |
V₃ =(1,2,3,4)                    |V₃|   | 1 2 3 4 |
V₄ =(4,3,2,1)                    |V₄|   | 4 3 2 1 |

Test by row reduction.

    ->      | 1  2  4  3 |       3*(1)-2*(2)->  | 3  0  2   5 |
2*(1)-(2)-> | 0  3  5  2 |            ->        | 0  3  5   2 |
(1)-(3)  -> | 0  0  1 -1 |            ->        | 0  0  1  -1 |
4*(1)-(4)-> | 0  5 14 11 |       5*(2)-3*(4)->  | 0  0 -17 23 |

(1)-2*(3) -> | 3  0  0  7 |       6*(1)-7*(4)-> | 18 0 0  0 |
(2)-5*(3) -> | 0  3  0  7 |       6*(1)-7*(4 -> | 0 18 0  0 |
    ->       | 0  0  1 -1 |       6*(3)+(4)  -> | 0 0  6  0 |
(4)+17*(1)-> | 0  0  0  6 |             ->      | 0 0  0  6 |
 

This shows all 4 of the 4 vectors were independent.
A basis for the span is:

V₁ =(1,2,4,3)
V₂ =(2,1,3,4)
V₃ =(1,2,3,4)
V₄ =(4,3,2,1)