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

-----------------------

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

**Equality of matrices****Operation on matrices****Algebraic properties of matrices****Addition properties****Multiplication properties****Minor of matrix****Determinant of matrix****Adjoint of matrix****Inverse of matrix****Solving linear equation by inversion method****Solving linear equations of two unknowns by cramer method****Solving linear equations of three unknowns by cramer method****Rank of matrix****Solve linear equation of three unknowns by rank method****Linear dependence of vector****Characteristic Equation of Matrix****Characteristic Vector of Matrix****Diagonalization of Matrix****Gauss Elimination Method****Worksheet of matrices**

HTML Comment Box is loading comments...