Linear Dependence of Vectors





In this page linear dependence of vectors we are going to see how to check whether the given vectors are linearly dependent or independent.

Definition :

A system of vectors X₁,X₂,.........Xn are said to be linearly dependent,if at least one of the vectors is a linear combination of remaining vectors.Otherwise it is called linearly independent.

In other words we can say the system of vectors X₁,X₂,.........Xn are said to be linearly dependent,if there exists numbers λ₁, λ₂,............. λn in which at least one of them is non zero satisfying the equation λ₁ X₁ + λ₂ X₂ + ................ + λn Xn = 0.

There are three methods to test linear dependence or independence of vectors in matrix.

Procedure for Method I

Definition can be directly used to test linear dependence or independence of vectors in matrix.

Procedure for  Method II

  • First we have to write the given vectors as row vectors in the form of matrix.
  • Next we have to use elementary row operations on this matrix in which all the element in the nth column below the nth element are zero.
  • The row which is having every element zero should be below the non zero row.
  • Now we have to count the number of non zero vectors in the reduced form. If number of non zero vectors = number of given vectors,then we can decide that the vectors are linearly independent. Otherwise we can say it is linearly dependent.

Procedure for Method III

  • If the matrix formed by the given vectors as row vectors is the square matrix,then we have to find rank.
  • If the rank of the matrix = number of given vectors,then the vectors are said to be linearly independent otherwise we can say it is linearly dependent.

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

Method 1

Method 2

Method 3

2. Test whether the vectors (1,3,1), (-1,1,1) and (3,1,-1) are linearly dependent.If so write the relationship for the vectors

Method 1

Method 2

Method 3

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

Method 1

Method 2

Method 3

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

Method 1

Method 2

Method 3

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

Method 1

Method 2

Method 3








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