Linear Dependence Example Problems 1

In this page linear dependence example problems 1 we are going to see some example problems to understand how to test whether the given vectors are linear dependent.

Example 1:

Test whether the vectors (1,-1,1), (2,1,1) and (3,0,2) are linearly dependent.If so write the relationship for the vectors

Solution:

Let the given vectors be X₁ (1,-1,1),X₂ (2,1,1) and X₃ (3,0,2)

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

λ₁ (1,-1,1) + λ₂ (2,1,1) + λ₃ (3,0,2) = 0

 1 λ₁ + 2 λ₂ + 3 λ₃ = 0 --------(1)

-1 λ₁ + 1 λ₂ + 0 λ₃ = 0 --------(2)

 1 λ₁ + 1 λ₂ + 2 λ₃ = 0 --------(3)

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

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

λ₂ + 2 λ₂ + 3 λ₃ = 0

3 λ₂ + 3 λ₃ = 0

3 (λ₂ + λ₃)= 0

λ₂ + λ₃ = 0

λ₂ = - λ₃ 

λ₃  = - λ₂ 

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

1 λ₂ + 1 λ₂ + 2 (-λ₂) = 0

2 λ₂ - 2 λ₂ = 0

0 λ₂ = 0  ----- (4)

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

Values of  λ₂ = 1

                  λ₁ = 1

                 λ₃ = -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 1 X₁ + 1 X₂ + (-1) X₃ = 0            linear dependence example problems 1 linear dependence example problems 1

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