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
Procedure for Method III
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
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