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Unformatted text preview: THE UNIVERSITY OF HONG KONG MAY 2008 EXAMINATION MATHEMATICS: PAPER MATHllll LINEAR ALGEBRA (To be taken by BSc, BBA(Acc&Fin), BEcon&Fin, BEng(CompSc), BSc(ActuarSc), BSc(Bioinformatics) & LLB students) 15 May, 2008 9:30am. — 12:00n00n Candidates may use any self-contained, silent, battery-operated and pocket-sized cal- culator. The calculator should have numerical—display facilities only and should be used only for the purposes of calculation. It is the candidate ’5 responsibility to ensure that his/her calculator operates satisfactorily. Candidates must record the name and type of their calculators on the front page of their examination scripts. Answer ALL FIVE questions ‘ Note: You should always give precise and adequate explanations to support your conclusions. Arguments must be well presented and clearly written. A mere yes, no or numerical answer will not earn you any marks. Think carefully before you write. 2 1 12 1. (20%)LetM= 4 2 2 4 . -2 ~13 2 (a) Find elementary matrices E1, E2, . ' - , Ek such that the product Ek - - - E2E1M is in reduced row echelon form. (b) Find a basis for its column space C(M) (c) Is it possible to find an 4 x 3 matrix S such that SM is nonsingular? Explain. (d) Is it possible to find an 4 X 3 matrix T such that MT is nonsingular? Explain. (Show clearly your calculation and explanation.) 2. (15%) Let A = Hus-4H mel—I cor-Ia:- (a) If rank(A) 75 3, find all the possible value(s) of k. (Show clearly your calculation.) (b) Suppose that P and Q are non-singular matrices such that the product FAQ is well-defined. (i) Prove that dim N (A) = dim N (B) where B = PAQ. (ii) Is dim N (BT) = dim N (A)? Explain. 3. (15%) Let X be a nonzero 4 X 4 matrix and X‘1 = 0. (a) Show that O is an eigenvalue of X and X has no other eigenvalue. (b) Is X diagonalizable? Explain your answer. (c) Let I be the identity matrix of order 4. Is I — X nonsingular? Explain. If yes, find its inverse. 4. (25%) Let P3 be the vector space of all polynomials of degree 3 2. Define 29(x) = 1 — $2, q(x) = 2+ :r — 3:132, r(:c) = 1 — 2m+4m2. (a) Which of the following is a subspace of P3? Explain your anwer. i. U = {u(ar): u E P3, u(1)= r(1)}. ii. V = {v(:c): v 6 P3, 11(1) = 17(1)}. (b) Show that the polynomials p, q, r form a basis for P3. (c) Consider the ordered basis E = [p, q,r]. Find the coordinate vector of t(:r) = 1 -— 3x + 8x2 with respect to E. (d) Let F = [1,1 + 25,552] be an ordered basis of P3. Find the transition matrix S from E to F. (e) Let T : P3 ——> P3 be the differential operator, defined as T(a + bx + cm2) = b + 2cm, for any a + bar + c332 6 P3. Find the matrix representation of T relative to E and F. [Show clearly your calculations] 5. (25%) (a) Let L : R4 —> R be a linear transformation, and Mg) gé O for some g E R4. Show that R4 = Span(_:§) EB ker(L). [Remark U 69 V denotes the direct sum of the subspaces U and V. i.e. wewrite UGBVfor U+VifUflV= {Q}.] T (b) Let 14,2 6 R4 and gTy = 3. Define the 4 X 4 matrix M fly . Is M diagonaliable? Explain your answer. What are the eigenvalues of M and their geometric multiplicities? [Remark We identify an 1 x 1 matrix and a scalar. So the 1 x 1 matrix (3) are viewed as the scalar 3.] ******** EndofPaper******** ...
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