**Linear Independence of Vectors**

Suppose α_{1}, α_{2}, α_{3}, α_{4}…………α_{n } are vectors of V(F) where F is field. And there exist

scalers a_{1}, a_{2}, a_{3}, a_{4}………..a_{n} which

belong to field F. α_{1}, α_{2}, α_{3}, α_{4}…………α_{n } vectors are called lineary independent if

a_{1} α_{1} + a_{2} α_{2} + a_{3} α_{3}+ ……… + a_{n} α_{n} = 0

and all a_{1}=a_{2}=a_{3}=a_{4}………..a_{n}= 0

Example-

Suppose α_{1}=(1,0,0), α_{2}=(0,1,0) and α_{3}= (0,0,1)

Linear combination of these vectors with scalars a_{1}, a_{2 }, and a_{3 } is

a_{1} α_{1} + a_{2} α_{2} + a_{3} α_{3}=0

You will get a_{1}=a_{2}=a_{3}=0

This implies that vectors α_{1}, α_{2}, α_{3} are linearly independent

**Linear Dependence of Vectors**

Suppose α_{1}, α_{2}, α_{3}, α_{4}…………α_{n } are vectors of V(F) where F is field. And there exist

scalers a_{1}, a_{2}, a_{3}, a_{4}………..a_{n} which

belong to field F. α_{1}, α_{2}, α_{3}, α_{4}…………α_{n } vectors are called lineary independent if

a_{1} α_{1} + a_{2} α_{2} + a_{3} α_{3}+ ……… + a_{n} α_{n} = 0

and all a_{1},a_{2},a_{3},a_{4}………..,a_{n} are not zero (some of them may be zero).

Search vectors and scalars such that

a_{1} α_{1} + a_{2} α_{2} + a_{3} α_{3}=0

and in a_{1}, a_{2 or } a_{3 }atleast one of them is not zero.

Then vectors α_{1}, α_{2}, α_{3} will be linearly dependent.

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