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 u ₁ + u ₂ + ⋯+ u ₙ in V if there exist scalars a ₁, a ₂ , ⋯, a ₙ in K such that v = a 1 u 1 + a ₂ u ₂ + ⋯+ a ₙ u ₙ Alternatively, v is a linear combination of vectors u ₁ + u ₂ + ⋯+ u ₙ if there is a solution to the vector equation v = x 1 u 1 + x ₂ u ₂ + ⋯+ x ₙ u ₙ where x ₁ + x ₂ + ⋯+ x ₙ 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 u 1 = (1, 2, 3), u 2 = (2, 3, 7), u 3 = (3, 5, 6) Solution: We seek scalars x ,