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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