Linear Dependence Example Problems 3





In this page linear dependence example problems 3 we are going to see some example problems to understand how to test whether the given vectors are linear dependent.

Example 3:

Test whether the vectors (1, 3, 1), (-1, 1, 1) and (2, 6, 2) 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 (2, 6, 2)

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 (2, 6, 2) = 0

 1 λ₁ - 1 λ₂ + 2 λ₃ = 0 --------(1)

 3 λ₁ + 1 λ₂ + 6 λ₃ = 0 --------(2)

 1 λ₁ + 1 λ₂ + 2 λ₃ = 0 --------(3)

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

(1) + (2) =>   1 λ₁ - 1 λ₂ + 2 λ₃ = 0

                   3 λ₁ + 1 λ₂ + 6 λ₃ = 0

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

                  4 λ₁ + 8 λ₃ = 0

4 (λ₁ + 2 λ₃) = 0

λ₁ = - 2 λ₃

λ₁ = -2 λ₃

Substitute λ₁ = - 2 λ₃ in the third equation linear dependence example problems 3

(3) =>  - 2 λ₃ + 1 λ₂ - 1 λ₃ = 0

           -3 λ₃ + 1 λ₂  = 0

           -3 λ₃ = - 1 λ₂

            1 λ₂3 λ₃

              λ₂ =  3 λ₃

Substitute λ₁ = -2 λ₃ and λ₂ =  3 λ₃ in the second equation

(2) =>  3 (-2 λ₃) + 1 (3 λ₃) + 6 λ₃ = 0

          -6 λ₃ + 3 λ₃ + 6 λ₃ = 0  

           -3 λ₃ + 3 λ₂ = 0 

           -3 (λ₃ - λ₂) = 0

                       λ₃ =  λ₂  example3 of linear dependence

Now we are going to plug  λ₁ = -2 λ₃, λ₂ =  3 λ₃ and λ₃ =  λ₂ in the first equation

1 (-2 λ₃) - 1 (3 λ₃)  + 2 λ₃ = 0

-2 λ - 3 λ + 3 λ₃ = 0

-2 λ₃ = 0

0 λ₃  = 0 --------(4)

Equation (4) is true for any value of λ. So that let us assume λ = 1 and λ₁ = -2 λ₃ and λ₂ = λ₃

Values of  λ₃ = 1

                  λ₁ = -2

                 λ₂ = 1 

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 -2 X₁ + 1 X₂ + (1) X₃ = 0           linear dependence example problems 3 linear dependence example problems 3







Example3 of Linear Dependence to Matrix
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