Mathematics 719 Applied Linear Algebra Assignment 2 Due: March 16, 2017 in class. Include all written answers, tables, graphs and images for each problem. Send a separate .m file for each question as appropriate. Also remember to send along any auxiliary .m files that are necessary to run your code. I should be able to run your code without any modifications so as to get the same results you include with this assignment. Make sure that the code you submit is your own. 1. We have studied three norms || ||1 , || ||2 and || ||1 for vectors and the three corresponding induced norms for matrices. In general, for a given vector x, these three norms will give dierent values. Similarly for matrices. We may wonder whether or not the a good approximation to x\" depend on the chosen norm. answer to questions like \"is x We say that the vectors xk converge to x if limk!1 ||xk x|| = 0. We may also wonder if the vectors xk converge to x in one norm but not another. Two norms || ||a , || ||b are called equivalent if there are positive constants and such that ||x||a ||x||b for all vectors x. See Chapter 5.1, 5.2 in Meyer for details (a) Show that the above three vector norms are pairwise equivalent. (See problem 5.1.8 in Meyer) (b) Show it is not possible for vectors xk to converge to x in one norm but not another. (c) Can a matrix A be ill-conditioned in one matrix norm but not another? Explain. to be a good approximation to x in one norm but not another? (d) Is it possible for x Explain. 2. We explore some properties of random matrices. Define a random matrix to be an nn matrix whose entries are independent samples from the normal distribution with mean p zero and standard deviation pn. In Matlab A = randn(n,n)/sqrt(n) will produce such a matrix. The factor of n is addded to make the limiting behaviour clean as n ! 1. not inversible (a) What is the probability that such a matrix is singular? You do not need to give a rigorous proof of you assertion, but a good convincing arguement is necessary. p (b) How does the factor of n aect the conditioning number of a random matrix? (c) What is the probability that a random matrix is ill-conditioned? Make well reasoned arguements qualitative and quantative arguements that are supported by good graphical evidence. How do your results depend on the size of the matrix