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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 selfcontained, silent, batteryoperated and pocketsized 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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