Linear Dependence Example Problems 5

In this page  linear dependence example problems 5 we are going to see some example problems to understand how to how to test whether the given vectors are linear dependent.

Example 5:

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

Solution:

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

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

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

 1 λ₁ + 1 λ₂ + 1 λ₃ = 0 --------(1)

 1 λ₁ + 2 λ₂ + 3 λ₃ = 0 --------(2)

 2 λ₁ - 1 λ₂ + 1 λ₃ = 0 --------(3)

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

2 x (1)   =>   2 λ₁ + 2 λ₂ + 2 λ₃ = 0

(1) - (2) =>  1 λ₁ + 2 λ₂ + 3 λ₃ = 0

                  (-)    (-)       (-)  

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

                  1 λ₁ - 1 λ₃ = 0 ---------(4)

1 λ₁ =  1 λ₃

 λ₁ = λ₃

Now we have to take the equations (1) and (3)

(1)   =>     1 λ₁ + 1 λ₂ + 1 λ₃ = 0

(3)   =>     2 λ₁ - 1 λ₂ + 1 λ₃ = 0

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

               3 λ₁ + 2 λ₃ = 0 ---------(5)

2 x (4) =>  2 λ₁ - 2 λ₃ = 0

               3 λ₁ + 2 λ₃ = 0

               --------------

                5 λ₁ = 0

                  λ₁ = 0

λ₃ = 0

Substitute λ₁ = 0 and λ₃ = 0 in the first equation

 1 (0) + 1 λ₂ + 1 (0) = 0

               0 + 1 λ₂ + 0 = 0

                       1 λ₂ = 0

                          λ₂ = 0

Values of  λ₃ = 0

                  λ₂ = 0

                  λ₁ = 0

linear dependence example problems 5 linear dependence example problems 5

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  example5 of linear dependence

applying the values in the equation we will get 0 X₁ + 0 X₂ + 0 X₃ = 0

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