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
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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 λ₂ = 3 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 λ₃ - 3 λ₃ + 3 λ₃ = 0
-2 λ₃ = 0
0 λ₃ = 0 --------(4)
Equation (4) is true for any value of λ₃. So that let us assume λ₃ = 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
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