k-means with nonnegative, proportions, or Boolean vectors. Suppose that the vectors x1, . . . , xN are clustered using k-means, with group representatives z1, . . . , zk. (a) Suppose the original vectors xi are nonnegative, i. e., their entries are nonnegative. Explain why the representatives zj are also nonnegative. (b) Suppose the original vectors xi represent proportions, i. e., their entries are nonneg- ative and sum to one. (This is the case when xi are word count histograms, for example.) Explain why the representatives zj also represent proportions, i. e., their entries are nonnegative and sum to one. (c) Suppose the original vectors xi are Boolean, i. e., their entries are either 0 or 1. Give an interpretation of (zj)i, the ith entry of the j group representative.