The set S = {x1, x2, . . . , xn} is affinely dependent if and only if the set {x2 - x1, x3 - x1, . . . , xn - x1} is linearly dependent.
Exercise 1.157 implies that the maximum number of affinely independent elements in an n dimensional space is n + 1. Moreover the maximum dimension of a proper affine subset is n. Analogous to exercise 1.133, we have the following alternative characterization of affine dependence.