Uniqueness of linear combination coefficients (${\sum }_i{a}_i{v}_i={\sum }_i{b}_i{v}_i⟹a_i=b_i$)

A family is linearly independent if and only if every vector in its linear span has unique coefficients as a linear combination of that family. For a short proof, subtract two representations to get ${\sum }_i(a_i-b_i)v_i=0$; linear independence forces each difference to vanish. Conversely, uniqueness applied to the zero vector forces every linear relation to use only zero coefficients. This explains the practical role of linear independence: a coordinate vector relative to an ordered basis is unambiguous.