Let S be a collection of vectors {v₁ ,...,vₙ} in a vector space V. Show that if and only if every vector w in V can be expressed uniquely as a linear combination of vectors in S, then S is a basis of V . In other words: suppose that for every vector w in V , there is exactly one set of constants c₁,...,cₙ so that c₁v₁ +···+cₙvₙ = w. Show that this means that the set S is linearly independent and spans V , and vice versa.