Linear Dependence in Rank Method





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

  • First we have to write the given vectors as row vectors in the form of matrix.
  • Next we have to use elementary row operations on this matrix in which all the element in the nth column below the nth element are zero.
  • The row which is having every element zero should be below the non zero row.
  • Now we have to count the number of non zero vectors in the reduced form. 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.

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:

˜
 
1 -1 1
2 1 1
3 0 2
 

R₂ => R₂ - 2R₁

         2         1         1

         2        -2        2

        (-)       (+)     (-)

      _________________

       0         3       -1

      ________________

R₃ => R₃ - 3R₁

         3         0        2

         3        -3        3

        (-)       (+)     (-)

       _____________

       0         3       -1

      _____________


˜
 
1 -1 1
0 3 -1
0 3 -1
 


R₂ => R₂ - 3R₁

R₃ => R₃ - 2R₁

R => R - R

       0        3         -1

       0        3         -1

      (-)      (-)      (+)

     _______________

       0         0        0   

      _______________

linear dependence in rank method


˜
 
1 1 -1
0 3 -1
0 0 0
 


R₃ => R₃ - R

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 Dependence1 Rank Method to Method 1
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