Linear Dependence Rank Method 3

In this page linear dependence rank method 3 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 3:

Test whether the vectors (1, 3, 1), (-1, 1, 1) and (2, 6, 2) are linearly dependent.

Solution:

linear dependence rank method 3

˜

 1 3 1 -1 1 1 2 6 2

 R₂ => R₂ + R₁ -1         1       1         1        3        1      _________________       0         4       2      _________________ R₃ => R₃ - 2R₁ 2         6       2         2        6        2        (-)      (-)     (-)      ___________________       0       0       0      ___________________

linear dependence rank method 3  linear dependence rank method 3

˜

 1 3 1 0 4 2 0 0 0

R₂ => R₂ + R₁

R₃ => R₃ - 2R₁

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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