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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