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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 Iii1’ 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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 Fall '08
 Feinberg,E

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