Linear Dependence and Independence of Vectors

Linear  Independence of Vectors

Suppose α1, α2, α3, α4…………αn are vectors  of V(F) where F is field.  And there exist
scalers a1, a2, a3, a4………..an which
belong to field F. α1, α2, α3, α4…………αn vectors are called lineary independent if
a1 α1 + a2 α2 + a3 α3+ ……… + an αn = 0
and all a1=a2=a3=a4………..an= 0

Example-

Suppose α1=(1,0,0), α2=(0,1,0) and α3= (0,0,1)

Linear combination of these vectors with scalars a1, a, and a3
is

a1 α1 + a2 α2 + a3 α3=0

You will get  a1=a2=a3=0

This implies that vectors α1, α2, α3 are linearly independent

Linear  Dependence of Vectors

Suppose α1, α2, α3, α4…………αn are vectors  of V(F) where F is field.  And there exist
scalers a1, a2, a3, a4………..an which
belong to field F. α1, α2, α3, α4…………αn vectors are called lineary independent if
a1 α1 + a2 α2 + a3 α3+ ……… + an αn = 0
and all a1,a2,a3,a4………..,an are not zero (some of them may be zero).

Search vectors and scalars such that

a1 α1 + a2 α2 + a3 α3=0

and in a1, a2 or  aatleast one of them is not zero.

Then vectors α1, α2, α3 will be linearly dependent.

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