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

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

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

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

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

• Types of matrices
• Equality of matrices
• Operation on matrices
• Algebraic properties of matrices
• Multiplication properties
• Minor of a matrix
• Practice questions of minor of matrix
• Adjoint of matrix
• Adjoint of matrix worksheets
  
• Determinant of matrix
• Inverse of a matrix
• Practice questions of inverse of matrix
• Inversion method
• Inversion method worksheets
• Cramer's Rule for 3 equations
• Cramer's Rule for 2 equations
• Solving 2 equations using Cramer's method
• Rank Method in Matrix
• Rank of a matrix
• Solving using rank method
• Linear dependence of vectors
• Linear dependence of vectors in rank method
• Characteristic Equation of matrix
• Characteristic vector of matrix
• Diagonalization of matrix
• Gauss elimination method
  
• Matrix calculator
• Matrix worksheets