Lecture 6 Notes

V 1 v 2 v 3 is a linearly dependent set 2 10 0 x1 01

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Unformatted text preview: t set 2 10 0 x1  01 18 0 Í x2  00 (b) Reduced echelon form: 33 0 0 x3 Let x 3  _____ (any nonzero number). Then x 1  _____ and x 2  _____. 1 ____ 3 5 2  ____ 5 3  ____ 9 9 0  3 0 0 or ____v 1  ____v 2  ____v 3  0 (one possible linear dependence relation) 3 Linear Independence of Matrix Columns A linear dependence relation such as 1 33 3 2  18 5 5 3 1 9 9 0  0 3 0 can be written as the matrix equation: 12 3 33 35 9 18 59 3 1 0  0 . 0 Each linear dependence relation among the columns of A corresponds to a nontrivial solution to Ax  0. The columns of matrix A are linearly independent if and only if the equation Ax  0 has only the trivial solution. 4 Special Cases Sometimes we can determine linear independence of a set with minimal effort. 1. A Set of One Vector Consider the set containing one nonzero vector: v 1 The only solution to x 1 v 1  0 is x 1  _____. So v 1 is linearly independent when v 1 0. 5 2. A Set of Two Vectors EXAMPLE u1  Let 2 1 , u2  4 2 , v1  2 1 , v2...
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