Section_3_3 - 3.3 Linear Independence A homogeneous system...

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3.3 Linear Independence A homogeneous system such as 1 2 - 3 3 5 9 5 9 3 x 1 x 2 x 3 = 0 0 0 can be viewed as a vector equation x 1 1 3 5 + x 2 2 5 9 + x 3 - 3 9 3 = 0 0 0 . The vector equation has the trivial solution ( x 1 = 0, x 2 = 0, x 3 = 0), but is this the only solution? Definition A set of vectors { v 1 , v 2 , . . . , v n } in V is said to be linearly independent if the vector equation c 1 v 1 + c 2 v 2 + · · · + c n v n = 0 has only the trivial solution, c 1 = c 2 = · · · = c n = 0. Otherwise, { v 1 , v 2 , . . . , v n } is said to be linearly dependent . Geometric Interpre- tation 1
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Example Let v 1 = 1 3 5 , v 2 = 2 5 9 , v 3 = - 3 9 3 . a. Determine if { v 1 , v 2 , v 3 } is linearly independent. b. If possible, find a linear dependence relation among v 1 , v 2 , v 3 . 2
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Linear Independence of Matrix Columns A linear dependence relation such as - 33 1 3 5 + 18 2 5 9 + 1 - 3 9 3 = 0 0 0
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