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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