How to use Linear Combinations

LINEAR COMBINATIONS

         Let V be a vector space over a field K. A vector v in V is a linear combination of vectors u₁+u₂+⋯+uₙ in V if there exist scalars a₁,a₂,⋯,aₙ in K such that 

                                                v = a1u1+a₂u₂+⋯+aₙuₙ

Alternatively, v is a linear combination of vectors u₁+u₂+⋯+uₙ  if there is a solution to the vector equation

                                                v = x1u1+x₂u₂+⋯+xₙuₙ

where x₁+x₂+⋯+xₙ are unknown scalars.

 

Linear Combinations in Rⁿ

         The question of expressing one vector in Rⁿ as linear combination of other vector in Rⁿ is equivalent to solving a nonhomogeneous system of linear equations, as illustrated below.

a)  Suppose we want to express v = (3, 7, -4) as a linear combination of the vectors

u1 = (1, 2, 3), u2 = (2, 3, 7), u3 = (3, 5, 6) 

 

Solution:

         We seek scalars x, y, z such that  v = x1u1+x₂u₂+⋯+xₙuₙ, that is,

                    

            Reducing the system to echelon form yields

            Back-substitution yields the solution

                                                

            Thus,