09 DEC - THE UNIVERSITY OF HONG KONG DECEMBER 2009...

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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 c-oro 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 [Eh-yz] 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) = T-IXT — 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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09 DEC - THE UNIVERSITY OF HONG KONG DECEMBER 2009...

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