Unique way of convex combination

Question

Assignment:

Prove that vectors x⁰, x¹,...,x^k are affine independent in Rn if and only if every vector y ∈ conv{x⁰, x¹,...,x^k} can be represented in a unique way as a convex combination of x⁰, x¹,...,x^k.

Provide complete and step by step solution for the question and show calculations and use formulas.