2. Suppose that the k-means algorithm was used to find clusters in the following matrix 1 0 (a) Without actually doing any calculations, speculate on what clusters might result from running kmeans. What would the result mean? (b) How would your answer to part a change for different numbers of clusters? (c) Describe ways that might be used to detect this situation (particularly for much larger data sets that exhibited a similar phenomenon)? 3. Suppose that M is a 8 x 3 matrix. Let A be the matrix consisting of the first and second columns of M and let B be the matrix consisting of the second and third columns of M Suppose the k-means algorithm finds 2 clusters for matrix A and produces a cluster vector a. Suppose the k-means algorithm finds 2 clusters for matrix B and produces a cluster vector b Suppose that in both cases, the k-means algorithm actually succeeds in optimizing the objective func tion. The examples below should be drawn by hand without using R. Try to choose examples so that it is casy for a human to see the clusters. (a) Give an example of actual numeric entries in M where the cluster vectors a and b are the same. State the actual cluster vectors and draw plots to illustrate. (b) Give an example of actual numeric entries in M where the cluster vectors a and b differ in just one entry. State the actual cluster vectors and draw plots to illustrate. (c) Give an example of actual numeric entries in M where the clusters found from A differ significantly than the clusters found from B. (Note that just interchanging cluster names 1 and 2 in the cluster vectors does not actually alter the points in each cluster.) State the actual cluster vectors and draw plots to illustrate. (d) For the example you gave in part (c), draw a plot of the first two columns of M labeled or colored based on the cluster vector b. Also draw a plot of the last two columns of M with the points labeled or colored based on the cluster vector a (e) For the example you gave in part (c), do either of the cluster vectors a or b actually succeed in providing meaningful clusters of the data? By what measures would they be judged to be good or judged to be badi