Linear independence of vectors &Linear dependence of vectors

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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 ...