# Linear independence of vectors &Linear dependence of vectors

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 Ñ” 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 1â‰¤iâ‰¤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.