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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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 Spring '08
 KOSTROV
 Linear Algebra, Algebra, Linear Independence

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