Prove that the following statements are equivalent:

(i) x1, . . . , xk Rⁿ are affinely independent.

(ii) x2 x1, . . . , xk x1 Rⁿ are linearly independent.

(iii) (x1, 1), . . . ,(xk, 1) Rⁿ⁺¹ are linearly independent.

Hint: Show that (i) implies (ii), (ii) implies (iii), and (iii) implies (i).