A set of n + 1 points a0,..., an ∈ Rn is said to be in general position if the differences ai - aj span Rn.
(a) Show that the points are in general position if and only if they do not all lie in a proper affine subspace A ⊊ Rn, cf. Exercise 2.2.30.
(b) Let a0,..., an and b0,..., bn be two sets in general position. Show that there is an isometry F: Rn → Rn such that F[ai] = bi for all i = 0,..., n, if and only if their interpoint distances agree: ||ai - aj|| = ||bi - bj|| for all 0 ≤ i < j ≤ n.