## 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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