LINApp2006 - THE UNIVERSITY OF HON G KONG DECEMBER 2005...

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Unformatted text preview: THE UNIVERSITY OF HON G KONG DECEMBER 2005 / JANUARY 2006 EXAMINATION MATHEMATICS: PAPER MATHllOl LINEAR ALGEBRA I (To be taken by BSc, BBA, BBA(Acc&Fin), BEcon&Fin, BEng(CivE), BJ, BSc(ActuarSc) & BSC(CSIS) students) 19 December, 2005 9:30am. — 12:00noon NOTE: You should always give precise and adequate explanations to support your conclusions. A mere yes, no or numerical answer will not earn any marks. Arguments must be well presented and clearly written. Think carefully before you write. Answer ANY FIVE questions (each question carries 20 marks) 1. Let A1: = b be a system of m linear equations in n unknowns where n > m and let C = [A | b] be the augmented matrix. Prove or disprove each of the following statements. (a) If rank 0 = rank A, then the system has infinitely many solutions. (b) If rank C = m + 1 then the system is inconsistent. (c) If the system is soluble for b = E1, E2,- - - ,Em, the m columns of the m x m identity matrix, then rank A = m. 2. Let M be an invertible matrix. (a) If M is upper triangular, show that M ‘1 is also upper triangular. (b) If the 1"“ column of M is E, where E1,E2, ~ -- are the columns of the identity matrix, show that the 2"" column of M "'1 is also equal to E,. (0) Let C be the following 4 x 4 invertible matrix: poor—Ira comb—4A *-x-x-* **** where * denote some unknown numbers. Can you determine explicitly the second column of 0—1? If yes, please do so. If no, explain why. . (a) If the square matrix A has all its eigenvalues not equal to 1, Show that I —— A is an invertible matrix. 0 2 ——2 (b) Diagonalize the matrix X = —1 3 —-2 and hence compute X100. -1 2 —1 . Let 111, yz, ~ - ~ £1,142, - - - , etc. be vectors in R1“. Prove or disprove each of the following statements. (a) If {y1,y2,~- ,yn} is linearly independent and 3g 55 span{y1,y2,~-- ,yn}, then {111,141,22, - - - , 3”} is also linearly independent. (b) If {21,142, - - - , pm} is not linearly independent, then span{y,2,23, ' ' ' syn} = span{y1,y2,- ' ' ayn}' (c) If ghyz, 23 are linearly independent, so are the three vectors: yl +22, —y2+ 9.3, 21—22‘l'113 .Letf:X—>Yandg:Y——>Xbemappings. (a) Prove that, for any subsets A and B of X, f(A n B) Q f(A) fl f(B). Give an example to show that f (A {'1 B) may be a proper subset of f (A) m f (B) (b) If g o f is injective, what can you say about f and g? (No need to prove your conclusion.) (c) Prove or disprove the statement: If Y is a subset of X and g o f is bijective then both f and g are bijective. 6. (a) Let A and U be matrices. Show that row(U A) g row A, with equality holds if U is invertible. Does the converse of this hold? Why? (b) If A is an n x n matrix and b is n x 1 such that the system Ax = b is inconsistent, what can you say about the solution set of the system Ax = 0? Please justify your conclusion. 7. (a) Given that the statement p -—> q is false, determine the truth value of ((~ 10) /\ (~ (1)) —+ q- (b) If R is a relation defined in a non—empty set A, show that the relation R‘1 o R is symmetric. (c) Let N be the set of all natural numbers and let IP = {2,3,5,7,11,13,--} be the set of all prime numbers. Let R = (N, 1?, G') be the relation in which the graph G is given by G = {(n,p) 6 N x 11" : n is divisible by p}. (For example, (15,3), (100,5) 6 G but (18,7) ¢ G, (18,6) ¢ G'.) (i) Determine (with explanations) which of the following belong to G-1 o G' and which do not: (10,4), (36,21), (45,4)- (ii) Determine whether the relation 5 = R—1 o R is reflexive or transitive. *******END******* ...
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