Linear combinations of vectors:

The linear combination of finite set of vectors  is defined as a vector  such that 

 where k₁, k₂, ... k_n are any scalars.

LINEARLY DEPENDENT AND INDEPENDENT VECTORS

A system of vectors  is said to be linearly dependent if there exists a system of scalars k₁, k₂ ..., k_n such that

They are said to be linearly independent if every relation of type  implies that k₁ = k₂ =....=k_n = 0.

Notes:     

• The 2 collinear vectors are linearly dependent always.
• The 2 non-collinear non-zero vectors are always linearly independent
• The 3 coplanar vectors are always linearly dependent.
• The 3 non-coplanar non-zero vectors are always linearly independent.
• More than 3 vectors are linearly dependent always.
• The 3 points with position vectors   are collinear if   with 
• The 4 points with position vectors    are coplanar if 

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