Linear Dependence and Independence of Vectors

Linear  Independence of Vectors

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

Example-

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

Linear combination of these vectors with scalars a₁, a₂ , 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…………α_n are vectors  of V(F) where F is field.  And there exist
scalers a₁, a₂, a₃, a₄………..a_n which
belong to field F. α₁, α₂, α₃, α₄…………α_n vectors are called lineary independent if
a₁ α₁ + a₂ α₂ + a₃ α₃+ ……… + a_n α_n = 0
and all a₁,a₂,a₃,a₄………..,a_n 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, a_{2 or}  a₃  atleast one of them is not zero.

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