In this page linear dependence rank method 5 we are going to see some example problem to understand how to test whether the given vectors are linear dependent.
Procedure for Method II
Example 5:
Test whether the vectors (1,1,1), (1,2,3) and (2,-1,1) are linearly dependent.
Solution:
|
R₂ => R₂ + R₁ |
1 2 3 1 1 1 (-) (-) (-) _____________ 0 1 2 _____________ |
linear dependence rank method 5 linear dependence rank method 5 |
R₃ => R₃ - 2R₁ |
2 -1 1 2 2 2 (-) (-) (-) _______________ 0 -3 -1 _______________ |
|
R₂ => R₂ + R₁ R₃ => R₃ - 2R₁ | |||||||||||||||
R₃ => R₃ + 2R₂ |
0 -8 -4 0 8 4 _______________ 0 0 0 _______________ | |||||||||||||||
|
R₃ => R₃ + 2R₂ |
Number of non zero rows is 2. So rank of the given matrix = 2.
If number of non zero vectors = number of given vectors,then we can decide that the vectors are linearly independent. Otherwise we can say it is linearly dependent.
Here rank of the given matrix is 2
which is less than the number of given vectors.So that we can decide the
given vectors are linearly dependent.
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