In this page linear dependence in rank method 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 1:
Test whether the vectors (1,1,1), (2,1,1) and (3,0,2) are linearly dependent using rank method. linear dependence in rank method
Solution:

R₂ => R₂  2R₁ 
2 1 1 2 2 2 () (+) () _________________ 0 3 1 ________________  
R₃ => R₃  3R₁ 
3 0 2 3 3 3 () (+) () _____________ 0 3 1 _____________ 

R₂ => R₂  3R₁ R₃ => R₃  2R₁ 
R₃ => R₃  R₂ 
0 3 1 0 3 1 () () (+) _______________ 0 0 0 _______________ 
linear dependence in rank method 

R₃ => R₃  R₂ 
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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