Linear Dependence Rank Method 5





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

  • 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 5:

Test whether the vectors (1,1,1), (1,2,3) and (2,-1,1) are linearly dependent.

Solution:

˜
 
1 1 1
1 2 3
2 -1 1
 

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

      _______________


˜
 
1 3 1
0 4 2
0 -8 -4
 


R₂ => R₂ + R₁

R₃ => R₃ - 2R₁

R₃ => R₃ + 2R

       0       -8        -4

       0        8         4

     _______________

       0         0        0   

      _______________


˜
 
1 3 1
0 -3 1
0 0 0
 


R₃ => 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.







Linear Dependence5 Rank Method to Method 1
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