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Unformatted text preview: THE UNIVERSITY OF HONG KONG DECEMBER 2007 EXAMINATION MATHEMATICS: PAPER MATHllll LINEAR ALGEBRA (To be taken by BSc, BBA(Acc&Fin), BEcon&Fin, BEng(CompSc), BEng(LESCM), BFin, BSc(ActuarSc) & BSocSc students) 18 December 2007 9:30am. - 12:00noon Candidates may use any self-contained, silent, battery-operated and pocket-sized 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. Note: You should always give precise and adequate explanations to support your conclusions. Clarity of presentation of your argument counts. So think carefully before you write. Section A. Answer ALL SIX questions. 1. (5%) Let A and B be 2 x 3 matrices. The matrix A is reduced to B through the following elementary row operations in order: (1) R1 H R2 (interchange the first two rows), (2) ——2R1 + R2 (multiply the first row by —2 and add it to the second row), (3) 2R2 + R1 (add twice of the second row to the first row), (4) —R2 (multiply the second row by —1). Find a matrix E such that EA = B. (Show your calculation.) —2 —1 1 —3 2. (10%)LetA= 2 1 1 1 . 4 2 1 3 (a) Find a basis for its row space r(A). (b) Find a basis for its column space c(A). (c) Find a basis for its null space N (Show your calculation.) 3. (10%) Let T : R3 —> R2 be the linear transformation defined by T((x y z)T) = (a: + y 22 — w)T. (a) Find the standard matrix representation of T. (b) Let a1=(1 0 —1)T, u2=(1 1 1)T, a3=(1 0 of. Show that yl, g2, g3 form a basis for R3. (0) Find the matrix representation A of T relative to the ordered bases E = [ybymgg] and F = [yhyz] Where _'u_)1 = (0 1)T and £112 = (1 0)T. i.e. [T(y)]p = AME for every 9 E R3. u 3 —-8 1 —3 (Show clearly your calculation.) 4. (5%) Let A = ( Compute A2007 by diagonalizing A. 5. (10%) Let U, V be subspaces of a vector space W. Show that if U n V = {Q}, then dim(U + V) = dimU + dimV. 6. (20%) Prove or disapprove the following by giving a proof or a counterexample: (a) If g1, g2, g3, L, are linearly dependent vectors, then £1 E Span(_:1_:2, g3, g4). (b) If A is row equivalent to B , then they have the same column space, i.e. C(A) = 6(3). (0) If A is a square matrix, then the nullity of A equals the nullity of AT, i.e. dim N (A) = dim N (AT). (d) There is exactly one linear transformation T : R3 -—¥ R2 such that T((0 1 1)T) = (1 2V, T((1 —1 0)T) = (2 0)? T((1 0 if") = (3 2)T. Section B. Answer ANY TWO of the three questions. 1. (20%) Let A be a square matrix of order 71. Suppose rank(A2) = rank(A). Show that C(A) n N (A) = {Q}, where C(A) is the column space of A and N (A) is the null space of A. 2. (20%) Let L 2 R3 ——> R3 be a linear transformation such that L(§1) = (1 3 2V, LE2) = (2 6 4)T, Meg) = (1 3 2)T where g1, g2, g3 denote the standard basis for R3. (The 2th entry of Q is 1.) (a) Let M be the standard matrix representation of L. Is M similar to A 100 A2000? 000 (b) Can we find two ordered bases F and G for R3 such that A is the matrix where Explain your answer. representation of L relative to these bases? Explain your answer. 3. (20%) (a) Let A be an eigenvalue of a matrix A. Show that the geometric multiplic— ity of A does not exceed its algebraic multiplicity. (i.e. dim N (A — AI) 3 d,\ where the characteristic polynomial of A is p(:r) = (:5 — A)d"q(x) with (IO) 75 0-) (b) Let A be an 4 x 4 matrix and rank(A) = 2. Suppose both 1 and —1 are eigenvalues of A. Evaluate with explanation the characteristic polynomial of A. *******Endofpaper ******* ...
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