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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 selfcontained, silent, batteryoperated and pocketsized
calculator. The calculator should be nonprogrammable, have numericaldisplay
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.
 Fall '10
 Dr,Li
 Linear Algebra, Algebra

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