Customer purchase history matrix. A store keeps track of its sales of products from K different product categories to N customers over some time period, like one month. (While it doesn't matter for this problem, K might be on the order of 1000 and N might be 100000.) The data is stored in an N x K matrix C, with Cij being the total dollar purchases of product j by customer i. All the entries of C are nonnegative. The matrix C is typically sparse, i.e., many of its entries are zero. (a) What is C1? (b) What is CT1? (c) Give a short matrix-vector expression for the total dollar amount of all purchases, by all customers. (d) What does it mean if (CCT) kl = 0? Your answer should be simple English. (e) Suppose you run k-means on the rows of C, with k = 100. How would you interpret the centroids 21,..., 2100?