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Unformatted text preview: THE UNIVERSITY OF HONG KONG
MAY 2008 EXAMINATION MATHEMATICS: PAPER MATHllll LINEAR ALGEBRA (To be taken by BSc, BBA(ACC&Fin), BEcon&Fin, BEng(CompSc), BSc(ActuarSc),
BSC(Bioinformatics) & LLB students) 15 May, 2008 9:30am. — 12:00noon Candidates may use any self~contained, silent, battery—operated and pocketsized cal
culator. The calculator should have numericaldisplay 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 emamination 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. 2 1 1 2
1. (20%)LetMu 4 2 2 4 .
——2 m1 3 2
(3.) Find elementary matrices E1, E2,    , E19 such that the product Eh    EgElM is in reduced row echelon form.
(1)) Find a basis for its column space C(M). (c) Is it possible to ﬁnd an 4 x 3 matrix S such that SM is nonsingular? Explain. (d) Is it possible to ﬁnd an 4 x 3 matrix T such that MT is nonsingular?
Explain. (Show clearly your calculation and explanation.) 2. (15%) Let A = HMNH k
1
0
0 Mun—AM (a) If rank(A) 79 3, ﬁnd ali the possible vaine(s) of it. (Show clearly your calculation.) (b) Suppose that P and Q are non—singular matrices such that the product
FAQ is welldeﬁned. (i) Prove that dim N (A) = dim N (B) where B = FAQ.
(ii) Is dim N (ET) = dim N (A)? Explain. 3. (15%) Let X be a nonzero 4 x 4 matrix and X4 = 0. (a) Show that 0 is an eigenvalue of X and X has no other eigenvaiue.
(b) Is X diagonalizable? Explain your answer. . ((3) Let I be the identity matrix of order 4. Is I — nonsingular? Explain. If yes, ﬁnd its inverse. 4. (25%) Let P3 be the vector space of all polynomials of degree 5 2. Deﬁne
PCB) m 3 " 332, (1(a): 2+:t — 3mg, ﬁx) = 1 m» 233+ 4:32. (3.) Which of the following is a subspace of P3? Explain your anwer.
i. U m {u(a:) : u 6 P3, u(1) m 711)}.
ii. V = : ’u E P3, 0(1) =p(1)}. (b) Show that the polynomials p, q, 7" form a basis for P3. (0) Consider the ordered basis E = [p,q,7"}. Find the coordinate vector of
“113) = 1 — 333 + 83:2 with respect to E. (d) Let F = [1,1 + 33,372] be an ordered basis of P3. Find the transition matrix S from E to F. (e) Let T : P3 we P3 be the differential operator, deﬁned as
T(a + bx + cm?) = b + 20112, for any a 4— ba: + 0332 6 133.
Find the matrix representation of '1" relative to E and F. [Showr clearly your calculations.) 5_ (25%) (a) Let L : 1&4 —> R be a linear transformation, and L(_1Q) 75 0 for some g E R4.
Show that R4 = Span(§) EB ker(L). [Remark U EB V denotes the direct sum of the subspaces U and V. Le.
we write UGV for U+V ifUﬂVz (b) Let ELLE 6 IR4 and 133; = 3. Deﬁne the 4 X 4 matrix M = QQT.
Is M diagonaiiabie? Explain your answer.
What are the eigenvalues of M and their geometric multiplicities? [Remark We identify an 1 X 1 matrix and a scalar. So the 1 x 1 matrix (3) are viewed as the scalar 3.] ******** EndofPaperaahkaksakm ...
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 Fall '09
 CHAN
 Math, Linear Algebra, Row echelon form, University of Hong Kong, elementary matrices E1, PAPER MATHllll LINEAR

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