Prove that a set B = {V₁, V2,..., Un} is a basis of a vector space V if and only if every vector in V can be represented uniquely as a linear combination of vectors in 3.

Recall: a set is a basis for V if
(1) it spans V, and
(2) it is linearly independent.
(1) holds for β by definition, so we have only to show (2).

Let {v1,...,vn}=β, and suppose a1.v1 + ... + an.vn = 0.
Clearly 0.v1 + ... + 0.vn = 0.
Since the representation of 0 belongs to V is unique we must have 0 = a1 = ... =an.
-β is linearly independent and therefore a basis for V.