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.