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    Linear independence of vectors &Linear dependence of vectors

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    Linear dependence of vectors
    Linear independence of vectors
    Linear combinations of vectors

    Define :-
    (a) Linear dependence of vectors
    (b) Linear independence of vectors
    (c) Linear combination of vectors
    Illustrate each of them with examples.

    Determine whether the following vectors in R3(R) are linearly dependent or linearly independent.
    (a) (1,2,1), (2,1,4), (4,5,6), (1,8,-3)
    (b) (1,2,3), (4,1,5), (- 4,6,2)

    In the vector space R3, express the vector (1,-2,5) as a linear combination of the vectors (1,1,1),(1,2,3) and (2,-1,1).

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    Linear dependence of vectors,Linear independence of vectors,Linear combinations of vectors

    Written by :-Thokchom Sarojkumar Sinha

    Define :-
    (a) Linear dependence of vectors
    (b) Linear independence of vectors
    (c) Linear combination of vectors
    Illustrate each of them with examples.

    Solution :- (a) Linear dependence of vectors
    A set containing the vectors u1,u2,u3,...,ur defined over a field F is said to be linearly dependent
    if there exists scalars a1,a2,a3,...,ar є F (not all zero) such that
    a1u1 + a2u2 + a3u3 + ...+ arur = 0
    (b) Linear independence of vectors
    Any set containing the vectors u1,u2,u3,...,ur defined over a field F is said to be linearly
    independent (L.I.) if it is not linearly dependent I.e.,if every equation of the form
    a1u1 + a2u2 + a3u3 + ...+ arur = 0 implies ai = 0 for all i such that 1≤i≤r
    (d) Linear combination of vectors
    Any vector u expressible in the form
    u = a1u1 + a2u2 + a3u3 + ...+ arur is called a linear combination of vectors u1,u2,u3,...,ur where a1,a2,,a3,...,ar are any r scalars, not all zero.
    If the vectors u1,u2,u3,...,ur are linearly dependent, then one of the vectors is a linear combination of the remaining vectors.
    Let the vectors u1,u2,u3,...,ur be linearly dependent , then
    a1u1 + a2u2 + a3 u3 + ...+ ar ur = 0 ----------------------------------------- (1)
    Where a1,a2,a3,...,ar are scalars, not ...

    Solution Summary

    This solution is comprised of a detailed explanation for Linear dependence of vectors,Linear independence of vectors,
    Linear combinations of vectors.
    It contains step-by-step explanation for determining whether the following vectors in R3(R) are linearly dependent or
    linearly independent.

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