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Linear Independence

Module by: Mark A. Davenport. E-mail the author

Definition 1

A set of vectors {vj}j=1N{vj}j=1N is said to be linearly dependent is there exists a set of scalars α1,...,αNα1,...,αN (not all 0) such that

j = 1 N α j v j = 0 . j = 1 N α j v j = 0 .
(1)

Likewise if j=1Nαjvj=0j=1Nαjvj=0 only when αj=0αj=0jj, then {vj}j=1N{vj}j=1N is said to be linearly independent.

Example 1: V = R 3 V = R 3

v 1 = 2 1 0 , v 2 = 1 1 0 , v 3 = 1 2 0 . v 1 = 2 1 0 , v 2 = 1 1 0 , v 3 = 1 2 0 .
(2)

Find α1,α2,α3α1,α2,α3 such that α1v1+α2v2+α3v3=0α1v1+α2v2+α3v3=0. [α1=1,α2=-3,α3=1α1=1,α2=-3,α3=1.] Note that any two vectors are linearly independent.

Note that if a set of vectors {vj}j=1N{vj}j=1N are linearly dependent then we can remove vectors from the set without changing the span of the set.

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