NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA1508E Linear Algebra for Engineering Homework 2 (a) Write your answers on A4 size papers. You may choose to write or type your answer digitally. (b) Write the page number on the top right corner of each page of the answer script. (c) You may use MATLAB to aid in your calculations, however, you need to show all your workings clearly. (d) If you have written your answer on an A4 paper, to submit, scan or take pictures of your work (make sure the images can be read clearly). Merge all your images into one PDF file (make sure they are in order of the page). (e) If you have written or typed your answer digitally, print it into a PDF file. (f) Name the PDF file by Student No_HW2 (e.g. A1234567Z_HW2.pdf). (g) Submit the PDF file to Canvas assignments, Homework 2. 1. Let w = 112 2 W = { u Є R¹ | u. w = 2 }, and V = {-0 for all u Є W (a) (2 points) Is W a subspace? Explain your answer. (b) (6 points) Show that V is a subspace and that T : basis for V. = is a {0·0·0}-- 4 -101- (c) (2 points) Let T be the basis of V defined in (b). Find Show your workings, however, you do not need to show your elementary row operations. 2. Let A = (a₁ a₂ aз known that a₁ = a4 a5), where a¿ is the i-th column of A, for i = 1,...,5. It is 0-0-0 (9) Suppose A is row equivalent to R = and a5 = 1 2 -12-1 01 1 -1 00 0 1 1 3 00 0 0 0 (a) (2 points) What is the rank of A? Explain how you derived your answer. (b) (5 points) Show that {0·0·0} (c) (4 points) Which the following vectors 4 1 -5 is a basis for the column space of A. -0-0-0-0 -3 = 7 V2 = 3 1 3 V3 V4= 1 belongs to the row space of A? Explain how you derive your answer. (d) (3 points) Find a basis for the nullspace of A. What is the nullity of A? 3. Given ATA = == 3 4 4 0 10-3 -3 3 (a) (2 points) Solve the homogeneous system Ax = 0. Explain how you derive your answer. (b) (2 points) Suppose (ATA) -¹AT = 3/5 -4/5 0 3/5 1/5 -1/5 3/5 0 -1/5 -2/5 -1/5 4/15 0 -8/15 -11/15/ What is A? Explain how you derive your answer. (c) (6 points) Find all possible left inverse of A. Show your workings clearly, however, you do not need to show the elementary row operations used. 4. Let T = V1 = 5 -----0--0--0-0-0-0 13 V5 = 1 2 2 (a) (3 points) Find a linearly independent subset S of T such that span(S) = span(T). (b) (3 points) Does T span the whole R4? If not, find a set T' containing I such that span(T) = R4.