LINApp2000

# LINApp2000 - THE UNIVERSITY OF HONG KONG B.SC.: YEAR I,...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG B.SC.: YEAR I, III, III EXAMINATION B.SOC., B.SC.(ACTUAR.SC.): YEAR I EXAMINATION B.SC.(CSIS): YEAR I EXAMINATION MATHEMATICS: PAPER MATH1101 LINEAR ALGEBRA I (To be taken by BSc I, II, III; BSOcSci I; BSc(Aetua.rSc) I & BSc(CSIS) I students) 22 December, 2000 9:30a.rn. - 12:00noon Candidates may use any self-contained, silent, battery-operated and pocket-sized calculator. The calculator should be non-programmable, 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 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 Clarity of presentation will be taken into account. 1. (16 points) Let f : X —-> Y be a mapping. Show that f is injective if and only if f‘1(f(W)) = W for any subset W of X. 2. (20 points) Let M be the set of all 3 x 4 matrices. Let R = {(A,B) E M x M: A is row equivalent to B}. (a) Show that (M, M, R) is an equivalence relation in M. 1 0 -3 1 0 3 1 —14 (b) Are 2 1 -2 1 and 1 4 -—1 -17 row equivalent? 1 —1 -1 —2 — 0 —-1 -l 4 Why? MATHEMATICS: PAPER MATH1101 Linear Algebra I 3. (24 points) (a) Given an n x 72 matrix A, deﬁne the minors, cofactors and determinant of A. (b) If B is an n x n matrix with two identical rows, what is the value of det(B)? Prove your assertion. (c) For an n X 72 matrix A, deﬁne adj A. Compute A - adj A. (Prove all formulas you use.) 1 —1 -1 (d) If adj A = ( 0 —2 1 ) , ﬁnd A . [Hintz consider adj(adj —l 4 —-1 4. (20 points) Let S and T be arbitrary subspaces of a given vector space V. Prove or disprove each of the following. (a) S n T is a subspace of V. (b) S' U T is a subspace of V. 5. (20 points) Let A be a 4 x 3 matrix. Suppose the two 4 X 3 systems 0 1 1 —1 Ag — 0 and Ag — _1 l 1 both have inﬁnitely many solutions. How many solutions the system 3 4 Ag; — _1 1 has? Justify your conclusion please! [Hintz if g1, g2, g3 are solutions to the above three systems, consider AB, where B is the 3 x 3 matrix (51 2:2 ******* End ******* 2 ...
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## This note was uploaded on 12/20/2011 for the course MATH 1111 taught by Professor Dr,li during the Fall '10 term at HKU.

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LINApp2000 - THE UNIVERSITY OF HONG KONG B.SC.: YEAR I,...

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