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

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