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Unformatted text preview: THE UNIVERSITY OF HONG KONG DECEMBER 2006/JANUARY 2007 EXAMINATION MATHEMATICS: PAPER MATH1101 LINEAR ALGEBRA I (To be taken by BSc, BBA(Acc&Fin), BEcon&Fin, BEng(CompSc) & BSc(ActuarSc) students) 28 December, 2006 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 examination scripts. Answer ALL FIVE questions NOTE: You should always give precise and adequate explanations to support your conclusions. Clarity of presentation of your argument counts. So think carefully before you write. 1. (24 points) Prove (by giving a proof) or disprove (by giving a counterexample) each of the following statements. (Note : You will not get any credit if you answer only True or False.) (a) Suppose a system of linear equations AX = B is consistent. If A and B [t], [2:] where A1 and B1 have the same number of rows, then the system AlX = have block form B1 is also consistent. (b) If A, B, C, D are square matrices of the same size, then det [g g] = det(A) det(D) — det(B)det(C). (C) If A, B are n X n diagonalizable matrices, then so is A + B. (d) For any m x n matrices A, B, the set {X€R" | AX=00rBX=0} is a subspace of IR". (e) If A, B are square matrices such that B is invertible and AT ~ B‘1 (i.e., AT is similar to B‘l), then A is also invertible and BT ~ A‘l. (f) If {X1,X2, . . . ,Xk} is an orthonormal set of vectors in IR", then it is linearly independent. 2. (14 points) Consider the matrix 1 —2 1 1 2 A: 1 a 1 2 0 2 —4 2 3 2 where a is a real number. Find a basis for each of the following subspaces. (a) rowA (b) colA (c) nullA. (Hint : Divide into cases of difierent values of a.) 3. (20 points) Let rk and fk denote the population of rabbits and foxes respec- tively in a certain region in year k. Suppose they are related as follows: Tk+1 1.2T)c — 12 fk fk+1 = 0.01 rk + 0.4 fk If there are initially 1000 rabbits and 30 foxes in the region, determine the limiting populations. 4. (22 points) Let U be a subspace of R”. (a) Suppose U is spanned by p vectors and U contains q linearly independent vectors. Show that q S p. (b) Show that if {X1,X2,...,Xp} and {Y1,Y2,...,Yfi are bases of U, then P = 9- (c) Now suppose B = {X1,X2,...,Xm} and C = {Y1,Y2,...,Ym} are two bases of U. Let k be such that 1 S k < m. Show in details that one can always enlarge the set {X 1, X 2, . . . , X k} by putting m— k vectors from 0, say, YiNYZ-z, . . . ,Yim_k, into it so that {X1,X2, . . . ,Xk,l’;1,l/,-2, . . .,Y1-m_k} is a basis of U. 5. (20 points) (a) Let A be an m x n matrix of rank T. Show that one can perform elemen— tary row operations and elementary column operations on A to make it IrO become the block matrix [0 0] . (Note : if r = 0 then I, is void, ITO 00 (b) Let A, B be matrices of the same size. Show that the following conditions and hence [ ] is just the m x n zero matrix.) mX'n. are equivalent: (I) There exist invertible matrices U7 V such that U AV = B. (II) dim(nullA) = dim(nullB). ******** EndofPaper******** ...
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