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com05sA - DOCTORAL QUALIFYING EXAlN"llNATlON SPRING...

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Unformatted text preview: DOCTORAL QUALIFYING EXAlN-"llNATlON SPRING 2005 Ad\-'a1'1ced Calculus & Linear Algebra. N A l\-‘I E : 1D#: There are four questions from Linear Algebra and four question from Advanced Calculus. For full credit1 answer any THREE questions from Linear Algebra and any THREE questions from Advanced Calculus. Start your al'lswer on each question sheet. Attach all extra sheets you use to the appropriate sheet. Hand in all question sheets. Date: January 24, 2005 Time of Examination: 09:00 -11:UU Place of Examination: 8138 S228 1mg 1. (a) A square matrix A is nilpotent if Ak : 0 for some positive integer PC. Show that if n X n. matrices A and B are both nilpotent and AB : BA, then A 4.. B is also Ililponent. Point out where in your proof you use the fact Lhat .4 and B commute (28. that. AB 2 BA). {h} Give an example of a 2 x 2 nilpotent matrix A and a. 2 X 2 nilpo- tent matrix B such that A + B is not nilpotent. I.D.# ‘3. Let U and W be subspaces of a vector space V. Prove that U U W is a subgpaee of V if and Only if U Q W 01‘ WT Q U. 1.D.# 3. Let Iii-1’ be the subspace of R4 spanned by the vectors m =<1=1=11111 v2 =(1,2,3=2)1 U3 = (MM). lfu = (4, U, 4, 2)1 find p'r'ojfu.‘ W"): the projection of 1;. onto W. (in other words find to E 14" which minimized? "U — '-u,-‘||) 1.11%; 4. Find a real 2 x 2 synnnetric matrix with eigenvalues A] : ‘2 and A2 2 3 and with eigenvector “U1 : (1:2) belonging to A] = 2. nus; 5. Prove that. 3’2 is an irrational number. I.D.# m. DC 6. Let. the sequence {'ELH_TL:1 be defined as follows: "11.1 2 1 and uHH : V2115” for n : 1,2, 3, - - -. (a) Prove that Jim '11..” exists. h-«oo {b} Find the limit. in (a). I.D.# ?. 1f rig]; fin?) : A and 333(3) : B gé 0, prove directly that 11111 f”) : 5 (Note: “prove directly” means give an “6:6” proof). I—*'Io 9135.1 B' I.D.# 8. Test each of the following improper integrals for convergence. Indicate clearly what your conclusion is (convergent or divergent) and how you reach that conclusion. $+ V'E ...
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