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Unformatted text preview: THE UNIVERSITY OF HONG KONG
DECEMBER 2009 EXAMINATION
'MATHEMATICS': PAPER MATHllll LINEAR ALGEBRA (To be takenby BSc, BEA(Law), BBA(Acc& Fin), BSc(QFin), BEconiEtFin, BEcon,
BSocSc 85 BEng(CompSc) studes) 18 December, 2009 r 9:30a.m. — 12:00n00n
Only approved calculators as announced by the Examinations Secretary can be used
in this examination. It is candidates’s responsibility to ensure that their calculator ' operates satisfactorily, and candidates must record the name and type of the calculator
I _used on the frontpage of the examination script. Answer ALL SIX 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.  (25%) . (a) Consider a. linear system whose augmented matrix is of the form . 1 2 1 _ 1
—1 4 3 2 .
2 —2 a 3 '1. For what values of a will the systembe consistent? ii. Could the system have more than one solution? Explain. B='( )a Where b is a, real number. (b) Find the adjoint of B if coro
0H0
MCG (c) Evaluate the value of a if the vectors X, Y, Z in R2“ are linearly indepen .x=(s we, 2:), 1)
1=(:): wt) H31» ﬁnd vectors 1_u_1 and 22 so that S will be the transition matrix from [Ehyz] dent; where '(d) Given lé t0 [wisﬂzl ' (e) Find rank(N) and rank(N2) if 'where rank(N) denotes the rank of N. 2. (20%) Let A be a 4 x 4 matrix. After applying the followingpperations in order (1) —R1 + R4.
(2) —R2 + R3,
(3) —4R2 + R4,
(4) £33, '(5) 4333 + 34, (a) Write donrn the elementary matrices E4 and E5 corresponding to the above operations and r (b) Find an 4 x 4 matrix E such that EA 2 U. (c) Find a basis for the nuIISpace N (A) of A.
((1) Evaluate the dimension and ﬁnd a basis for the row space r(A) of A. (e) Evaluate the dimension and ﬁnd a basis for the column space C(A) of A.
(Show your calculation clearly with explanation.) . (10 %) Let F be a nonsingular 4 x 4 matrix whose last two rows are (2 0 0 9)
and (1 2 1 8), 1e. _****
****
F=2009
‘1218 Suppose P is an 4 x 3 matrix and FF = U where
I 3 1 —1 0 O O 0 Find, with explanation, a. basis for the nullspace N (PT) of PT where PT denotes the transpose of P. . (10%) Let W be a subspace of R“, deﬁned as (1+5 0
W— a_5 .a,6E]R 0 Is it possible to ﬁnd three linearly independent vectors 31421 y4 E R4 such that Explain your answer. 5. (20%) Let V be the vector space of all 3 x 3 matrices under the usual matrix operations. (a) Let W be the subset of V consisting of matrices whose entries on the diagonal are equal to 0. i.e. A 6W if and only if A is of the form Is W a subspace? If yes, ﬁnd a basis for W.
.(b) Let ' y where a > b > c 5 0, and deﬁne a linear transformation dag» : W a W as follows. (W is deﬁned as in part (a).) For X E W, deﬁne
¢T(X) = TIXT — X. Let B be any ordered basis of W. Show that the matrix representation of
Q51" with respect to B is always nonsingular. 3 4 3
6. (15%) Let M: —1 ——2 —3 .
 1 —2 —1
(a) Evaluate all the eigenvalues of M. [Remark 2 is an eigenvalue of M (b) Evaluate the algebraic multiplicity and geometric multiplicity of each eigen value. (c) Is M diagonalizable? Explain. *akaraukauncEndofPaper******** 4 ...
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