In this page linear dependence rank method 2 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 2:
Test whether the vectors (1,3,1), (1,1,1) and (3,1,1) are linearly dependent.If so write the relationship for the vectors
Solution:

R₂ => R₂ + R₁ 
1 1 1 1 3 1 ___________________ 0 4 2 __________________ 
linear dependence rank method 2 linear dependence rank method 2 
R₃ => R₃  3R₁ 
3 1 1 3 9 3 () () () _________________ 0 8 4 _________________ 

R₂ => R₂ + R₁ R₃ => R₃  3R₁ 
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. linear dependence2 rank method
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