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LINApp2005 - THE UNIVERSITY OF HONG KONG DECEMBER...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DECEMBER 2004/JANUARY 2005 EXAMINATION MATHEMATICS: PAPER MATHllOl LINEAR ALGEBRA I (To be taken by BA, BSc, BSc(CSIS) & BSc(ActuarSc) students) 13 December, 2004 7:00pm. — 9:30pm. 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’s 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. NOTE: You should always give precise and adequate explanations to support your conclusions. A mere yes, no or numerical answer will not earn you any marks. Clarity of presentation of your argument counts. So think carefully before you write. There are altogether SIX questions, each carrying 20 marks. You may answer any FIVE of them. Unwanted attempts must be crossed out clearly and completely. 1. (a) Let f : X ——> Y be a mapping. Show that the following three state- ments are equivalent. (i) f is injective. (ii) f(A n B) = f(A) fl f(B) for any subsets A and B of X. (iii) f"1(f(A)) = A for any subset A of X. (b) Let So 2 d), the empty set, and define inductively Sk+1 = Sk U {5k} for k = 0, 1, ~ - -. Show that each element a: of SC is also a subset of Sk. 2. In Rmx”, the set of all m x n matrices over R, define the relation 8 by ASB if and only if there exists an invertible nxn matrix V such that AV 2 B. /Q.2(a) ..... (a) Show that 8 is an equivalence relation. (b) Let A be a matrix in Rm“ and B E A/S. Prove or disprove each of the following. (i) If A is symmetric then B is also symmetric. (ii) If A is diagonalizable then B is also diagonalizable. (iii) If the system AX = (1 1 1 - - - 1)T is consistent then the system BX = (2 2 2 2)T is also consistent. 3. Let A be an n X n invertible matrix. Let g and g be n x 1 column matrices and a E R. (a) Compute the product A‘1 Q A g ”ETA—1 1 pT O! (of two (n + 1) x (n + 1) matrices). Here _Q is the n x 1 zero matrix. (b) Determine the reduced row-echelon form of the matrix A q R - tr al for different values of a. For which value of or is R invertible? ((3) Consider the (n + 1) x (n + 1) system of linear equations ti 3.] Lil = til where g and Q are n x 1 column matrices. Find a value of bn+1 ( in terms of g, A, Q) such that this system is consistent irrespective of the value of oz. For this value of bu“, write down explicitly the solution of this system. 4. Prove or disprove each of the following. (a) If C and D are both n x n diagonalizable matrices, so is CD. /Q.4(b) ..... (b) IfAis2X3andBis3X2thenBA7éI3. (c) If the homogeneous system EX = 0 has a unique solution in X, then for any F the system EX = F always has a unique solution. (d) If A is a matrix such that A3 2 A2 then A2 = A. 5. (a) State and prove Cramer’s rule. (b) Let A = (aij) be an nx n matrix and let Cij (A) denote the (i, j)— cofactor of A. Prove that for 1 _<_ i 54$ k g 71, 2:1 aijij(A) = 0. State clearly all the properties about determinant you use. ' (c) If det B = 1 , show that (i) det(aij) = 1, (ii) adj(adj(B)) = B. —1 —2 6. (a) Diagonalize the matrix M = [ 4 5 matrix R such that R2 = M. J. Hence find M100 and find a r—Im ONO 0 (b) Why the matrix Q = [ 0 J is not diagonalizable? ——1 O (c) Prove or disprove: if both X and Y are of the same size and diagonaliz- able, then XY = YX. ******END****** ...
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