In this page linear dependence example problems4 we are going to see some example problems to understand how to test whether the given vectors are linear dependent.

**Example 4:**

Test whether the vectors (1,1,1), (1,0,1) and (0,2,0) are linearly dependent.If so write the relationship for the vectors

**Solution: **

Let the given vectors be X₁ (1,1,1),X₂ (1,0,1) and X₃ (0,2,0)

Now we have to write the given vectors in the form λ₁ X₁ + λ₂ X₂ + λ₃ X₃ = 0

λ₁ (1,1,1) + λ₂ (1,0,1) + λ₃ (0,2,0) = 0

**1** **λ₁** + **1** **λ₂** + **0** **λ₃** **= 0** --------(1)

** 1** **λ₁** + **0** **λ₂** + **2** **λ₃** **= 0** --------(2)

**1** **λ₁** + **1** **λ₂** + **0** **λ₃** **= 0** --------(3)

from the second equation 1λ₁+0λ₂+2λ₃ = 0 we come to know **λ₁ = -2 ****λ₃**.

Now we are going to plug this value in the first equation

1 λ₂ + 1 λ₂ + 0 λ₃ = 0

1(-2 λ₃) + 1 λ₂ + 0 λ₃ = 0

-2 λ₃ + 1 λ₂ + 0 λ₃ = 0

-2 λ₃ = - 1 λ₂

1 λ₂ = 2 λ₃

**λ₂ = 2 λ₃**

Substitute ** λ₁ = -2 λ₃ **and

1(-2 λ₃) + 1 (2 λ₃) + 0 (λ₃) = 0

- 2 λ₃ + 2 λ₃ + 0 λ₃ = 0

0 λ₃ = 0 ----- (4) linear dependence example problems 4 linear dependence example problems 4 linear dependence example problems 4

Equation (4) is true for any value of λ₃. So that let us assume λ₃ = 1 and ** λ₁ = -2 λ₃ **and

Values of **λ₃ = 1**

** λ₂ = 2**

** λ₁ = -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 linear dependence example problems 4

applying the values in the equation we will get **-2**** X₁** + **2** **X₂** +** (1)** **X₃** = 0

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