vector_spaces2

vector_spaces2 - LINEAR INDEPENDENCE DEFINITION If S = {v1,...

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1 LINEAR INDEPENDENCE DEFINITION If S = { v 1, v 2 ,…, v r ,} is a nonempty set of vectors, then the vector equation k 1 v 1 + k 2 v 2 + … + k r v r = 0 has at least one solution, namely k 1 = 0, k 2 = 0, k r = 0 If this is the only solution, then S is called a linearly independent set. If there are other solutions, then S is called a linearly dependent set. EXAMPLE 1 A Linearly Dependent Set If v 1 = (2, -1, 0, 3), v 2 = (1, 2, 5, -1), and v 3 = (7, -1, 5, 8), then the set of vectors S = { v 1 , v 2 , v 3 } is linearly dependent, since 3 v 1 + v 2 - v 3 = 3(2, -1, 0, 3) + (1, 2, 5, -1) + (7, -1, 5, 8) (3 ä 2 +1 - 7, 3 ä (-1) +2 +1, 3 ä 0 + 5 - 5, 3 ä 3 - 1 - 8) (0, 0, 0, 0) = 0 . So 3 v 1 + v 2 - v 3 = 0 EXAMPLE 2 A Linearly Dependent Set The polynomials p 1 = 1 - x, p 2 = 5 + 3x - 2x 2 , and p 3 = 1 + 3 x – x 2 form a linearly dependent set in P 2 since 3 p 1 - p 2 + 2 p 3 = 3(1 - x ) (5 + 3x - 2x 2 ) + 2(1 + 3 x – x 2 ) = (3 ä 1 - 5 + 2) + (-3x -3x +6x) + (2x 2 - 2x 2 ) = 0 + 0 + 0 = 0 So, 3 p 1 - p 2 + 2 p 3 = 0 EXAMPLE 4 Determining Linear Independence/Dependence
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2 Determine whether the vectors v 1 , = (1, -2, 3), v 2 = (5, 6, -l), v 3 = (3, 2, 1) form a linearly dependent set or a linearly independent set.
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This note was uploaded on 06/20/2008 for the course AM 025b taught by Professor Kiruscheva during the Winter '07 term at UWO.

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vector_spaces2 - LINEAR INDEPENDENCE DEFINITION If S = {v1,...

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