Math 54 - Summer 1999 - Pramanik - Midterm 1

# Math 54 - Summer 1999 - Pramanik - Midterm 1 - Math .54,...

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Unformatted text preview: Math .54, Summer ’99 Midterm Exam Time eﬂuwed: I hear. Intruder : Melahﬂne Premenjk. Student ID number: _ EH STﬁﬂCTIﬂES 1. PLEASE DD NUT TURN THIS PAGE GIVE-R UNTIL WSTRUETED TD DD SCI. 2. Fill in yeur name and ether details. 3- Please ehme your mark. Selutjene shearing enljr the ﬁnei! newer withe-ut the interme diete steps will receive no credit- ‘r’ en me} use the reverse side er each page fer mug}: were. 4. The ﬁgure in hreeleete fellewing eeeh questien [er pert ef questinn] deuelm the number ef peints envied In that questien {er part or queetjenj. 1. Answer “True” or “False”. Give reasons for your answer to get credit. [5 x 4 = 20 points] (a) The set {1,\$,:c2,a:3} is an orthonormal basis for P3 with inner product (17:4) = aobo + albl + a2b2 + a3b3, where, 2' 5 PW) = 00 + aim + a2\$2 + a3z3, and, q(:c) = b0 + blx + b2\$2 + (232:3. C(Xh” Cpmx H32)" Vs? Tm. 1+ deﬁne; an MW Wand beams: Pt Marinas w/ Miliﬁﬁ'iﬁgf‘l‘alf \ggmxm ii 1%.:th MWM §¢ F33 ‘ x {as h - 4~ \ .\' 3’10 c" l; x ‘ i [i { mid/X ‘y‘ him! » K): ‘3?“ i ll” ls citilmmrm \36 A K I H my" at? swh i\ r , ‘i “I? r , 5M «REA Aw: Kr» ; i ’Y (b) The set {1,\$,1‘2,.T3} is an orthonormal basis for P3 with inner product (p, (1) =/_ p(r)q(\$) d1- 1 ~ Mil “j: ‘ [065.4035 Hm: l8 E's-r“ :4 \Mer" Esmx,,,, X @‘illvﬂﬂnlg ii. a? w/ Fr“ 1 3» hp 06 HY: ﬂailiiuiiw . " Mil/32 M 903‘} @k (c) If A and B are diagonal nxn matrices, then det(A + B)7=rdet(A)r+rdet(B). ,., V a "‘ 13"” m EX ; r I : A” mm ‘ 2‘?“ 2‘; av: ‘3 fiat i“ x" (e) The functions f1 = x3 and f2(:r) = x2|\$| form a linearly independednt set on (—oo,oo). ‘ ‘ ‘ 2 4w becoux 0 {a casvaémmze ‘m "QM. twee/aw”: v a ‘ ' J 3 wﬁmkm CAM Beﬁcukﬁad m W3 WWW , 2. (a) [5 points] Find a basis for the subspace 131 W: x: 222 111+12—3z3=0 \$3 ' \ m v) \/ \ “MM v; J k X g \m ‘ M9 g‘wémfé bfmggg 2M wad? fig “3 WM (Amiga M; g“ C314 ({cms‘ﬂwy‘ 1 a“ , G» V (b) [15 pOints] Let V = < -2 ) . Find w" E W such that, i W \ a 1531; “V - WWI s Hv — WM (“(-33 ﬁvwﬁ . w \ "2 for all w E H . mm mvw; w \ \v-ﬂf‘ *“M‘ {3 .4 I ‘r' v M ﬁrm/l .6 l1v~~w H; 2M :5; (slim #mwmm {/9 WI: 3. (a) [5 points] Does the formula 12 - q = p(0)q(0) +p(1)q(1) 0 . a deﬁne an Inner product on P2 ? Explam. \g 7 4.: 06 “RM maxa (ex, : x,‘ ‘ if “an 1 x\ :H. gym mm (b) [5 points] Answer the same question as in (a) when p - q = p(0)q(0) + p(1)q(1) + p(2)4(2)- 7 4. [20 points] For the 4x4 matrix A given below, ﬁnd an orthogonal matrix Q, and a diagonal matrix A, such that Q"1AQ = A. HOP-4o OHOH X 1* “ﬂ. :3 a “x f} 1 0 1 0 HOI—‘O 5. [8 points] A matrix A is called m'lpotent if A" that a nonzero nilpotent matrix is not diagonal = 0 for some positive integrer k. Show izable. 6. [7 points] Let u 6 R" be such that uTu = 1. Let A denote the nxn matrix A = I — 2uuT. Prove that u is an eigenvector of A. What is the associated eigenvalue 7 10 7. [15 points] Compute the Wronskian of two tions without solving the equation. solutions of the following differential equa- t2y’L t(t + 2)y' + (t + 2);, = 0 ' .435ch (4 f W W )LAZ‘V a a Sr "54% if} * "l *Zlm‘ —— {+2 .4, (>2): é P f l +24% _, 7‘lele ' CG 1, C; m .2 ~ " Cl all m (g 2:» C “‘ >33! {Lcl‘ ...
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## Math 54 - Summer 1999 - Pramanik - Midterm 1 - Math .54,...

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