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).

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.
(a) (1,2,1), (2,1,4), (4,5,6), (1,8,-3)
(b) (1,2,3), (4,1,5), (- 4,6,2)

It also contains step-by-step explanation for expressing the vector (1,-2,5) as a linear combination of the vectors (1,1,1),(1,2,3) and (2,-1,1) in the vector space R3.

Solution contains detailed step-by-step explanation.