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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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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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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 ...

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