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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. — 12:00n00n Candidates may use any selfcontained, silent, batteryoperated and pocketsized cal
culator. The calculator should have numerical—display facilities only and should be used only for the purposes of calculation. It is the candidate ’5 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. 2 1 12
1. (20%)LetM= 4 2 2 4 .
2 ~13 2 (a) Find elementary matrices E1, E2, . '  , Ek such that the product Ek    E2E1M is in reduced row echelon form.
(b) 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 = Hus4H
mel—I
corIa: (a) If rank(A) 75 3, ﬁnd all the possible value(s) of k. (Show clearly your calculation.) (b) Suppose that P and Q are nonsingular matrices such that the product
FAQ is welldeﬁned. (i) Prove that dim N (A) = dim N (B) where B = PAQ.
(ii) Is dim N (BT) = dim N (A)? Explain. 3. (15%) Let X be a nonzero 4 X 4 matrix and X‘1 = 0. (a) Show that O is an eigenvalue of X and X has no other eigenvalue. (b) Is X diagonalizable? Explain your answer. (c) Let I be the identity matrix of order 4. Is I — X nonsingular? Explain. If yes, ﬁnd its inverse. 4. (25%) Let P3 be the vector space of all polynomials of degree 3 2. Deﬁne
29(x) = 1 — $2, q(x) = 2+ :r — 3:132, r(:c) = 1 — 2m+4m2. (a) Which of the following is a subspace of P3? Explain your anwer.
i. U = {u(ar): u E P3, u(1)= r(1)}.
ii. V = {v(:c): v 6 P3, 11(1) = 17(1)}. (b) Show that the polynomials p, q, r form a basis for P3. (c) Consider the ordered basis E = [p, q,r]. Find the coordinate vector of
t(:r) = 1 — 3x + 8x2 with respect to E. (d) Let F = [1,1 + 25,552] be an ordered basis of P3. Find the transition
matrix S from E to F. (e) Let T : P3 ——> P3 be the differential operator, deﬁned as
T(a + bx + cm2) = b + 2cm, for any a + bar + c332 6 P3.
Find the matrix representation of T relative to E and F. [Show clearly your calculations] 5. (25%) (a) Let L : R4 —> R be a linear transformation, and Mg) gé O for some g E R4. Show that R4 = Span(_:§) EB ker(L). [Remark U 69 V denotes the direct sum of the subspaces U and V. i.e.
wewrite UGBVfor U+VifUﬂV= {Q}.] T (b) Let 14,2 6 R4 and gTy = 3. Deﬁne the 4 X 4 matrix M ﬂy . Is M diagonaliable? 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.] ******** EndofPaper******** ...
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