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09 MAY

# 09 MAY - THE UNIVERSITY OF HONG KONG MAY 2009 EXAMINATION...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG MAY 2009 EXAMINATION MATHEMATICS: PAPER MATHllll LINEAR ALGEBRA (To be taken by BSC, BBA(IS), BEcon, BEcon&Fin, BEd(LangEd)LE), BFin, BSc(ActuarSc) & BSocSc students) 21 May, 2009 9:30am. — 12:00noon Candidates may use any self-contained, silent, battery-operated and pocket-sized calcu- lator 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. Answer ALL FIVE questions NOTE: You should always give precise and adequate explanations to support your conclusions. Arguments must be well presented and clearly written. A mere yes, no or numerical answer will not earn you any marks. Think carefully before you write. 1 —2 0 1 1. (18%) LetA= —1 2 2 1 . —1 2 0 —1 (a) Find a basis for the column space of A. (b) Evaluate the nullity of A. (Nullity means the dimension of the nullspace.) (c) Is it possible to ﬁnd a nonsingular matrix E such that 11 EA: 1 2 ? 21 i—‘HM l—IOH 2. (17 %) Let 1 3 3 2 1 —4 Q = _1 a Q = 1 a Q '_ 5 2 1 —4 (a) Deﬁne S = {g E R4 : ng = QTQ}. Is 8 a subspace of R4? 1 (b) Let g = i where h e R. h Determine all values of h such that g, Q, g are linearly independent. (0) Is it possible to ﬁnd a vector 4 E R4 so that g, Q, Q, _d_ form a basis for R4? 3. (30%) Prove or disprove each of the following statements. (a) Let A be an 3 x 3 matrix and adj A denote the adjoint of A. If detA + det(adj A) = 0, then A must be singular. (b) Let EDE’Qsz E R4. Given that i. {\$1, \$2} is linearly independent, ii. {711, g2} is linearly independent, iii- Spanalsz) ﬂ Span/4&2) = {9}- Then g1,gv_2,gl,y2 form a basis for R4. 1 2 (c) Let A = i (1) for some a, b, c, d 6 IR. Then ATA is always nonsingular. c d (d) The number 15 is an eigenvalue of the matrix 1 2 3 4 5 2 3 4 5 1 3 4 5 l 2 4 5 l 2 3 5 l 2 3 4 2 1 2 (e) Let H: 0: 1 [3 ,for some a,ﬂ€R. —1 —1 —1 If H has an eigenvalue 1 Whose geometric multiplicity is 2, then H must be singular. 4. (20%) Let P3 be the vector space of all polynomials of degree < 3. Deﬁne the linear transformation T : P3 —> P3 by T(a+b:c+c:cz) =(a—l—b)+(b+c)ac+(c+a)ac2 for any polynomial a+ bx + c372 6 P3 (i.e. a, b, c E R and x is the indeterminate). Consider the two ordered bases for P3: B= [1+x2,1+m,x+x2] and E: [m2,a:,l]. (a) Let p(x) = 3 + 3:1: + 2m2. Evaluate the coordinate vector of p(a:) with respect to B . (b) Evaluate the transition matrix from B to E. (c) Find the matrix representation of T relative to B and E. (d) Is it possible to ﬁnd a basis F such that the matrix representation of T with respect to F (i.e. relative to F and F) has a zero row? ). (a) Find an invertible matrix S and a real number a such that 0100 SMS“1=010. 001 (b) Does there exist an invertible matrix B so that M = B2? 5. (15%) Let M = ( HI—‘N I—IMH tor—Ira ******** EndofPaper******** ...
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09 MAY - THE UNIVERSITY OF HONG KONG MAY 2009 EXAMINATION...

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