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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. — 12:00noon 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 ’3 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 emamination 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 1 2 1. (20%)LetMu 4 2 2 4 . ——2 m1 3 2 (3.) Find elementary matrices E1, E2, - - - , E19 such that the product Eh - - - EgElM is in reduced row echelon form. (1)) 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 = HMNH k 1 0 0 Mun—AM (a) If rank(A) 79 3, find ali the possible vaine(s) of it. (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 = FAQ. (ii) Is dim N (ET) = dim N (A)? Explain. 3. (15%) Let X be a nonzero 4 x 4 matrix and X4 = 0. (a) Show that 0 is an eigenvalue of X and X has no other eigenvaiue. (b) Is X diagonalizable? Explain your answer. . ((3) Let I be the identity matrix of order 4. Is I — nonsingular? Explain. If yes, find its inverse. 4. (25%) Let P3 be the vector space of all polynomials of degree 5 2. Define PCB) m 3 " 332, (1(a): 2+:t — 3mg, fix) = 1 m» 233+ 4:32. (3.) Which of the following is a subspace of P3? Explain your anwer. i. U m {u(a:) : u 6 P3, u(1) m 711)}. ii. V = : ’u E P3, 0(1) =p(1)}. (b) Show that the polynomials p, q, 7" form a basis for P3. (0) Consider the ordered basis E = [p,q,7"}. Find the coordinate vector of “113) = 1 — 333 + 83:2 with respect to E. (d) Let F = [1,1 + 33,372] be an ordered basis of P3. Find the transition matrix S from E to F. (e) Let T : P3 we P3 be the differential operator, defined as T(a + bx + cm?) = b + 20112, for any a 4— ba: + 0332 6 133. Find the matrix representation of '1" relative to E and F. [Showr clearly your calculations.) 5_ (25%) (a) Let L : 1&4 —> R be a linear transformation, and L(_1Q) 75 0 for some g E R4. Show that R4 = Span(§) EB ker(L). [Remark U EB V denotes the direct sum of the subspaces U and V. Le. we write UGV for U+V ifUflVz (b) Let ELLE 6 IR4 and 133; = 3. Define the 4 X 4 matrix M = QQT. Is M diagonaiiabie? 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.] ******** EndofPaperaahkaksakm ...
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