Linear Dependence Example Problems 4

In this page linear dependence example problems4 we are going to see some example problems to understand how to test whether the given vectors are linear dependent.

Example 4:

Test whether the vectors (1,1,1), (1,0,1) and (0,2,0) are linearly dependent.If so write the relationship for the vectors

Solution:

Let the given vectors be X₁ (1,1,1),X₂ (1,0,1) and X₃ (0,2,0)

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

λ₁ (1,1,1) + λ₂ (1,0,1) + λ₃ (0,2,0) = 0

 1 λ₁ + 1 λ₂ + 0 λ₃ = 0 --------(1)

 1 λ₁ + 0 λ₂ + 2 λ₃ = 0 --------(2)

 1 λ₁ + 1 λ₂ + 0 λ₃ = 0 --------(3)

from the second equation 1λ₁+0λ₂+2λ₃ = 0 we come to know λ₁ = -2 λ₃.

Now we are going to plug this value in the first equation

1 λ₂ + 1 λ₂ + 0 λ₃ = 0  

1(-2 λ₃) + 1 λ₂ + 0 λ₃ = 0

-2 λ₃ + 1 λ₂ + 0 λ₃ = 0

-2 λ₃ = - 1 λ₂ 

 1 λ₂ = 2 λ₃

  λ₂ = 2 λ₃

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

1(-2 λ₃) + 1 (2 λ₃) + 0 (λ₃) = 0

- 2 λ₃ + 2 λ₃ + 0 λ₃ = 0

0 λ₃ = 0  ----- (4)     linear dependence example problems 4 linear dependence example problems 4 linear dependence example problems 4

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

Values of  λ₃ = 1

                  λ₂ = 2

                  λ₁ = -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    linear dependence example problems 4

applying the values in the equation we will get -2 X₁ + 2 X₂ + (1) X₃ = 0

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