Linear Dependence Rank Method 4





In this page linear dependence rank method 4 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 4:

Test whether the vectors (1,1,1), (1,0,1) and (0,2,0) are linearly dependent.

Solution:

linear dependence rank method 4

˜
 
1 1 1
1 0 1
0 2 0
 

R₂ => R₂ - R₁

         1        0        1

         1        1        1

        (-)      (-)     (-)

      _______________

       0       -1        0

      _______________

˜
 
1 1 1
0 -1 0
0 2 0
 


R₂ => R₂ - R₁

R₃ => R₃ + 2R

         0       2        0

         0      -2        0

      _______________

       0       0        0

      _______________

linear dependence rank method 4  linear dependence rank method 4

˜
 
1 1 1
0 -1 0
0 0 0
 


R₂ => R₃ + 2R

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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