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
May 30, 23 11:19 AM
May 30, 23 10:38 AM
May 26, 23 12:27 PM