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exam 1 spring 2009 solutions

# exam 1 spring 2009 solutions - Engineﬁring...

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Unformatted text preview: Engineﬁring Mathematics (ESE 31?} Exam 1 February 4, 2399 This axam cantains six. muitipie~ch0ice problems worth two poims each, nﬁne {mewfalse prebiems worth {311% paint each, ﬁve shamanswer probiemg ward} one point each, and three frag-«response proﬁems worth i4 minis aiwgethen ﬁx an exam tam} 0f 40 points. 13311: I. Mui-tigie~€hoice Cieariy ﬁil in the ova} on your answer caré which ccﬂesponds t0 the oniy cerreci response. Each is worth two points. 1. me13 (W fr?) %&;§;wmw Z 9% _ 4“ u 5% (A) 6—: (5M)? 5’ng 2"”? “at a ; ii if t 5 a» 3 +ﬁ6§w} : {3 +8“ g; 7 « 3,; , (D)1me% & 3,,gwé5 §\ (E) 11%?“3 fjﬂ 21>: :{ﬂ 1?” i ”aka (1:) i—te‘” {éwf‘fig j Kiwsf 45%;“; /1 (@5ﬂ+m%‘ M M “f mg WW “‘"’ \$5; ,4, a (D L+v%mm4 (D 1Wa*+m4 ( 3i ’ < Letﬂiﬂm {80 12:: g::1 Findﬁh’ ﬁ gél!~’£éi£”£>] Eu.) (A) e—s,__1§ ad?) 581%- (B) 8 3 : £3 w {:53 'gggzg: ”' > (C) a A w, 3 x 31mm "if; A 1 m 99 I , a, (D) 8~35+é (ya, (jig%>} £33; {I Q —Lw-% i; mg.“ 3% 33 (E) :3 5 gas : Lama»: 296% ~£J (F) mweﬂdmig 3 1 vs 1 ‘3 M ; W V 5 *3“ 3 i“ {{G) W; "H 3:3; ’” if: “‘“ it; Xi/g ) (H) .39“ was-+3 NEW "3 W 9-“? 3m? 5 M 3%,? g :1: W M g W“"‘”’““’“’” 3. Find the determines? of the feﬂewing matrix. pg 3 a; 5;; :: wgffgf5+32> ,3: M}; if} “”26, m 1 3 é f} E) m 2 W1 i (A) —-:6 E (123) ”12 (C) we (D) W4 (E) 0 (F) 4 (G) 8 (H) 12 (I) 16 www A a} x 3 matrix A has rank 2. Describe the solution set of the homogeneous system Ax m 0. W - - WWW“W““~} 4“ (A) The soiutien set is a Eine in R3. , j . _ . . Mes (B) The solution set isalme 1n R4. M’WXZF 56 (C) The soiution set is a plane in R3. (D) The sokution set is a plane in R? (E) The solution see might be a iine in K3, or it might be the empty set. sségﬁaea 5;»; IE} (F) The solutien set might be a ﬁne in 3?, er it might be the emyty set. (G) The soiutien set might be a pkane in R3, or it might be the empty set. (H) The soiution set might be a plane in R4, or it might be the empty set. (I) The solution set contains eniy the triviei seletion. (I) The soiutien set is the empty set. (Seaside! the set V of a1} matﬁoos of the form {:23 b 0 id 5 f],whofoo+fm1. Thlssotxgggga vector spoco‘ Whioh property or properties fglf? (I) closure under adéition (H) closure: under scalar muitipﬁoaﬁon {III} existence of a zero {A} (I) 0933? (B) (H) oniy (C) (HI) oniy (o) (I)and(H)on1y (E) (I) anti (111) only (F) (II) and (E13) oniy a), (11), Hand {HDE Soivo the foliowing system of equations. If there is no soiution, 561602 (A) if there are inﬁnitely- \$1 many solutions, select (8). If there is a unique solution {y} , ﬁnd the sum 3: + y w? 2 among Z y+224 choices (C) through (I). Work carefully! { m + 2y + 32' m 6 ~42: +zm8 (A) no goiution (B) inﬁnitely-many solutions ((3-16 6675:}; é’iE‘EQEcéwogz % ziékg‘i‘c- (3)4 gzg‘as’woﬁas 52% (3)4 L~=éé53§s§j oﬁgﬁ-‘n‘g F O () ; _ f g '33.“. 3 ’ Q “E A g {(C3) 2 {a} + 3&2; . < a, fagwwaoaz ; > g {:3 3 s . 5 i (W; (H) 4 i5} 357” :23 , 3:2} (I) 6 ‘ w 5"} 2: .3 i=2? xijovgé—igé j {if} i E if! g 3:}: «g» :2: :2 A} E, 5/ if} g 3:} i gt; Jig} M. f £71K} {jg-‘3‘?“ “5””? ”32% m5 Fm Ii. Tm€~False Mark “A” an V611? answez‘ card if tile statement is true; mark “B” if it is faise. Each is worth one point. . ,, {:3 '7. Letf{t}:{é gig men£:(:)):§ X: 1 — ‘ 3 g E 7' I \x‘ I "7‘6 )r‘ r " jiggié’ f: s; E _ gmrm if {/39 (5%,; 3 \$53: ﬂ’éjaté “I N g y 3 ; . ‘5 : ‘ ' ‘m ‘ E a" M {3312}; gaff“ £313, V:T£ﬁ»:;c£ 1&ngva : 5? v]; . 8. Let A and B be m X % matﬁces. Then {ABET m ATBT. J “ é) f 9. If an n x n matrix A is nonsingular, then A“1 is 21130 nonsingular. "i . >. ’ 1" W E S ’ ' J . .5? K 513\$)?“ m WQQ/Wy afj f? m» {i , 19“ Suppose u, v, and w are eiements of a vector space V, and suppose that u+v m 0 and u~§-w=0. Thenvmw. ‘i mi”- ”‘% J“; ,gﬁ M}: w/ﬁewfﬁ AV 2;; m gawm ., gym “NEW WWEAJW g j; 1}. The foiiowing set 8 of matrices is a basis for the vector 333306 V of ail symmetric: 2 x 2 matrices. 1 0 0' 1 3‘ s f s: [0 1}: [I 0] :3: (Z&% W13”? W 3% 33‘? “’é/X‘R’ﬁéf‘é’ig/ ‘ E I f :3 w‘im WW“? ”W [a 3" g; mi} giémm may 3 W w my; m ‘7‘? , “”5 \$5 2 ygyi—ﬁvﬂ.é j . 1 2 3 4 For probiems 12 and i3, refer £0 the matrix A m 4 3 2 1 . 32‘ The vecter 38 6 4 ‘23 is in the raw space of the matrix A. tr” g“ , i g ME A, ”:3 [ a”; 3 a f 3 gram f; 2 e; if ’2: J r w . . 13. The vecior if} 0 S 03 is in the WW Space ofthe maxi): A. ﬁrm gﬁaﬁzﬁﬂjzgﬁfgzgéjjﬁdengﬁ 2. s? 1%“ ’E‘hespaﬁofﬁiewc’comil G l O 1} ami i3 MB 1 2 wégisasubspaceofﬁg. 4. {9 k km“! :5. Let A be an ’12 X 22 matrix. If the rows of A are lineariy depencient, the}: A is singuiar. f .3" Part {11. Short Answer Each of the foliewing is worth one point Each answer is either right or wrong: :10 work is required, anti n0 paﬁial credit will be given. J /’ , :3 16. Finé £{e5‘cosh6t}. :35 { gawk {9%} 3: ,_ :2 5;? {22' ”M ”" g s’ , ,. _ M J {[8 \$555939 m 3’ wijgiazj’ w-1 W2 W3 “4 «5 ~61 m1 m2 _3 —4 ——5 0 m2 m2 m3 «4 G 0 m1 m2. m3 0 0 0 w1 —_2 0 a 9 0 «4 E7. Find the éeterminant of tha matrix A = @0066 MNC‘DNM QHMWQ 1 - 1 For preblems 18 and 19, consider the matrix A m 0 1 I 18. What is the dimension of the 001mm space of A? g} 2 “jaéfiif «3:.» mif- £3 jivqﬁ’iuséﬁdbé #346353; Maggi/324‘? ﬁliiéxfr’iﬁi M {L yfzﬁ/E 19. What is tha dimensien 0f the solution space of {BS sygtern Ax m 9? j "TE/i W ME. I" 3 ”W Z 3 j 29‘ Suppose A is a: :1 x 5 matrix with rank 2. E? the system Ax m B is incensistem, what is the rank of tha augmented matrix A? ( ’3 3 Fart 1‘31 Free Respﬁnse Feliew difectiens carefully, ané Show ail the steps naeﬁed :0 333% at the 3011133013. The point value for each problem is shown to its 161%. 23. Use canvoiuﬁer} is ﬁnd [3'3 (W) Van do 310?; need to simplify year answer. -.,§,,.§ x/Wi to: I“ Miw'jgf/I f . ., a .W») > . 3 33 2“ 3 , 3 (m) 3 W : \$33333% 333' \$933“? ,,. 335 M 5333 55:33?) “333332;”; \$31} 3"? 3:5”; 3 3 333333 3' 3333 3333333 3333 .33 {3' 3“ f”; {3/ r3 3" " " Aft ff) 3 1:3 .3: Na > f i’iwgﬁfﬂh‘ 3 J; (ﬁﬂﬁg: £53539 §3ﬁgﬁ “M {:zﬁﬁi 333333» f9 a?) 1 f3 .5. ’3 (if) A (:3: {£33 lac-33: 3333 , g 33;: 3 “if: L ‘M M 32:” )2 3fm 5.3335,»? 5333“; (ﬁéﬁ 2:353 3333;333ij 53 , 3 _ 3 M; M .3; ' 3 7 3; é» 5333 23% "@335? ﬁg: 33 5,3332%) Some Intagrai Formulas 33”” m W333333M‘” "‘"MW‘WW 313(27): (3} fcesgmdrm—éx 3 isin2x+C (2)3fsmwd33mérxw31i33m7xwt? .3 33333333 MW “MM- {3) fwsﬁt {17\$ :3: sin x — ésin‘gx 3— C’ (4) fsizl3x (ix 2 wees x :3 écos3\$ + C ex (5} [33300527333333 3 %;{COS;I:-§3 33133) +0 (6) I333 sinxdx: 2 {\$3an (2033:) +0 (4) 22. \$38 Lapiace transfamts to ﬁnﬁ the solutiém Ofthe feliawing {iiﬁereatia} uaﬁon. 3‘} y”+éy\$§(tm3) WW2 3“sz a3§gﬁ<€z\$§n~2g wégtgzg’g jgwg 5:;i &;\ng3 :2 2g “ﬁg/«aw {75(5)} 2:: :6 “é” wméwm E? ..... 35 5 +4 :3 555%}; (3-) 23. Consider #5118 set V of 33- eiments of K5 whese ﬁrst three components sum to zero. For example, the vector {1 2 m3 (3 8] is in V. Then V is a subspace 0f R5. (You do net need to verify this.) Find a basis for V. 95”” “” a! 5: g g ‘ , 5 Wuag ﬂaxamﬂg” 3 f. 5;" g3 “.5;ng a géj ‘5’; f .. j 7. f ; 3 t a a”: «a; £3 &__ +£5.32 i; w; 53 £3 ““2 J “M +1]; Q 556 vasi"z({3£3€353¢£§ ” ' :5” ,, ”E :ﬁfiaw; 5:; ﬁiwém 2' w»; a m if x w w W i; g; G 5:: a; z 5:3 7 4+ {5 fig 5:: 5:: C 5”; 3"- NE w £22,ng i f 5} M} £3 5’} ”if; gal} é w? 5:? 1:34)} iii} {:3 (L i 52} [25.) {3‘ G 52'} g] (12) Tahie of Lapizce Transforms ...
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