Linear Dependence Example Problems 2

In this page linear dependence example problems 2 we are going to see some example problems to understand how to test whether the given vectors are linear dependent.

Example 2:

Test whether the vectors (1,3,1), (-1,1,1) and (3,1,-1) are linearly dependent.If so write the relationship for the vectors

Solution:

Let the given vectors be X₁ (1,3,1),X₂ (-1,1,1) and X₃ (3,1,-1)

Now we have to write the given vectors in the form λ₁ X₁ + λ₂ X₂ + λ₃ X₃ = 0

λ₁ (1,3,1) + λ₂ (-1,1,1) + λ₃ (3,1,-1) = 0

 1 λ₁ -1 λ₂ + 3 λ₃ = 0 --------(1)

 3 λ₁ + 1 λ₂ + 1 λ₃ = 0 --------(2)

 1 λ₁ + 1 λ₂ - 1 λ₃ = 0 --------(3)

First let us take the equations (1) and (2)

(1) + (2) =>   1 λ₁ - 1 λ₂ + 3 λ₃ = 0

                   3 λ₁ + 1 λ₂ + 1 λ₃ = 0

                -----------------------

                 4 λ₁ + 4 λ₃ = 0

4 (λ₁ + λ₃) = 0

λ₁ + λ₃ = 0

λ₁ = -λ₃

Substitute λ₁ = -λ₃ in the third equation  linear dependence example problems 2

(3) =>  1 λ₁ + 1 λ₂ - 1 λ₃ = 0

           1 (-λ₃) + 1 λ₂ - 1 λ₃ = 0

           -1 λ₃ + 1 λ₂ - 1 λ₃ = 0

           - 2 λ₃ + 1 λ₂  = 0

           - 2 λ₃ = - 1 λ₂ 

             2 λ₃ =  1 λ₂

              λ₂ =  2 λ₃

Substitute λ₁ = -λ₃ and λ₂ =  2 λ₃ in the second equation

(2) =>  3 (-λ₃) + 1 (2λ₃) + 1 λ₃ = 0

          -3 λ₃ + 2 λ₂ + 1 λ₃ = 0  

           -3 λ₃ + 3 λ₂ = 0 

           -3 (λ₃ - λ₂) = 0

                       λ₃ =  λ₂ example2 of linear dependence

Equation (4) is true for any value of λ₂. So that let us assume λ₃ = 1 and

λ₁ = -λ₃ , λ₂ =  2 λ₃

Values of  λ₃ = 1 

                 λ₁ = -1

                 λ₂ = 2

Therefore we can say that the given vectors are linearly dependent. Now we have to find their relationship. For that let us take the equation

λ₁ X₁ + λ₂ X₂ + λ₃ X₃ = 0

applying the values in the equation we will get -1 X₁ + 2 X₂ + 1 X₃ = 0           linear dependence example problems 2

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