thq_01 - Q A01/014/01 0 Use T.C 16s MATH 2451 thq_01 Mult...

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Unformatted text preview: Q. A01/014/01 0 Use T.C. 16s MATH 2451 thq_01 Mult Exam ble Calculus 1varia Course 15 min Duration 2016-01-21 Due Date Instructor Net ID (First Name) SW wé W W ii;€;ux;n, @[email protected]@m3§gg mu @ new @ @ © ® @J @ any .... \ i. ( “4L @ a @ mm.“ mw ® ® ® aw mm. @ ® ® ® O new 0 A: w ® ® @ mm ® ® ® ® I. \) a®@@@E®®@@©@®@@®@@®©a®e®C®®®®®®®®® ;>@5e®@@@@@@@@©@®@@®@@®@ [email protected]®@®©@@@@@®@ [email protected]®©@@@@®@®©@®®@®@@®@@@@®@®®[email protected]®®®®@ M @ Ame ® as. new @ ® @ ® ® me ® E a @ ,9 my ® ® 0 ® ® ® @ my @n ® ® sen. (Last Name) 2016-01-44 14: 15 .. A01/014/O2 . 1. (10 points) The unit circle is_;_ Wé MN IT, (Mi! Sl=flnyflrt+f=d}' Iépdarga Simiflfifiéf JF'HLT‘ (a) Show that 31 is not a subspace; of}? " ’din two points W7 (b) iAre theredany two points in 5'1 whose sum is again in 81? Find the conditions which i, 5 E 31 must satisfy so “ that 5 + b E 31, and draw a sketch of a particular 6i and b which satisfy thes§__ conditions. _:_;__ _ *7— I {a afigfli / i l ,j_ K 2“? _ \ " x «.r f; 3 5;: J, Grader Only i . O® . O. A01/014/0L o 2. [10 points) (a) Suppose points A, B and C are vert-ices of a triangle. Find E + R: + a and provide a sketch. 4 M. ' Kat Bi: Ad“ a‘,...,-.\_.L— Ad: EA A fjalifli— 17 j?“ , At“ viii-i“ 7:1 (b) Do the following lines intersect? fit) = (4, 5, #2) + (1,4, 1) t, 50:) : (3,1,—3) + (2,1,2)t=._ fir ‘ , f \ p r _. . flflfl§W%%fl§&ZJ EmfiffiefiflPfigv 151%; 3‘43 A/Efi'g : 25;} 1 157’? :3 :7 2‘31: 22%;?! 4fieag+szfigh €£g~4f§:fi2k 9 ¢ Grader Only J, .0 A01/014/04 Q 3. (10 points) In this exercise, you’ll demonstrate the Cauchy-Schwarz inequality in R” using a technique which is mo. general than the proof provided in lecture. For any two vectors 13', u? E R“, lfi-tfil S ll’fill llfiill: with equality iff 13' or 1B is a scalar multiple of the other. (a) Show the statement is true if “E or u? is 6. (b) Explain why the function gilt) = “13+ trfiHz is non—negative. éQi/c Nah} é : (c) Show that the function is a quadratic in t by multiplying-out and collecting like terms until it is in the form a t2 + M + c. (d) Use the quadratic discriminant b2 — 4 ac together with the previous two parts to conclude the inequality. The rest of the statement . . with equality iff . . .) will not be graded, but you should try to determine Why it is true (but not on this sheet of paper). - \ 7 a: ._., __:_ A)Aé§wn§mpzo_32mfcn P r :2 \v.w| :. J8: fl:-,.=li;:o [Maya/21¢? V/i’l’lll? or; Aéeamefi iii/7% ,, .. =- NW—NgeA-rf .--=;2 Névéifi ewgezs 99/} m; “medic E3 mite/$12;- N17 lfléfil’ W55" 69¢ 0N6 why- Elam" 514%a40 kfidmezfiD a Minis/H9: xii; ; ‘37? (Rb ,6; iii 2%“: e" 7-” "baits?! mu / / glammews? e '\. tar—(26770} I?weeavafllwlt twléhniwri ./ .L Grader Only .1, 0 oo O .. A01/014/05 . =4. (10 points) (a) In lecture we saw that the diagonals of a parallelogram bisect each other. Show that the diagonals of a. parallel— ogram are orthggial to each other iff the parallelogram is a rhombus. -F OA/Eab/"fll av Dlfléppfljé’ifléfiwfigfifl £1.56“; Hip/é: 3.). ngfit> : a 5’7 '1. (b) Determine the matrix A which would map the indie . parallelogram onto the unit square. HDIM’? V ’F A f5? ‘ T1 HZ, _ o é/aé at) I V 1 F. ii M Ex Jo. m l? Owe 51;! {VJ ‘f ha- I“ ’Zé“lLJ2 Ar? / r if [ f“ m M A I, f, e: x?” [fl/l e m 1414” A WI 13:11.» ; iii?) (ya/{$3 r__‘ _I__) f}. J) M in» . _. op .r J. GraderOnly N[r @@®@@[email protected]®®@® . O O (a .. A01/014/06 . 5. (10 points) (a) Let R9 : 1R2 —) R2 denote the linear transformation which rotates a vector counterclockwise around the origin by 6. Recall that the matrix of R9 is given by: [Rs] = [$33 Tigil -: Therefore RC3+¢) = Egg 0 R9, and since compulsitioh'of linear operators is given by matrix/multiplication, it must be true that [R¢][R9] = [R(9+¢)]. Use this to fact to establish the following trigonometric identities. $5? c0509 + g6) = 005(9) cos(q5) — sin(l9_) sin(q§) é; sin(6‘ + 915) = sin(6) cos(¢>) + 005(9) sin[¢)_ r’ "f: a l'; r ' - ;:"-_...__r" L .m/ :13?» _ _ — A, q , rm o > I "L ,. Hr‘. 1, r1 -P i e-m-L— new: v x __—— _ - ,- LL .n ,- ; fl , _,_ ‘ I Jfi-‘A ‘\ .. f l SKA). :1 I] \__ (b) Let a, E e R2. Show: “EH2 “Eng .. (a . S)2 = (dens, 13])2 it; “1,619 It. 1122. ,i ‘ Ariana/Le; ‘7 'fl e wens}; a« '3 item Ham“; age; He“:— tg-a L; 3f (infieéwwagm Wit/Twit; flew/:10?— egg; fr'fi‘ifif Inmate:- @fifidiwaj“@319”? iéiéiwfib b " - :a1_.bf+a17|a: iglavfr-{Cb309 10?: “51715: "2416‘? 110} :: a: 4%: 104?“'2fi40(9 l9? l9} “:Qidfirbib i u H / J. Grader Only J, O. A01/014/07 O 6. (10 points) Suppose 0:1 a: “2 A=[a,5=e1 033 kH ififl~1fifl7A+43fi +b5fi C3 0"}? cfifi Show that . J Momma 1.21.: A. (Bf-7 5-2543": _' bl (:1 b = 62 ' |det(A)| = ya. (5' x a} Vow/mg ‘ [x fléA CF" wag) 0—10 a; Hwy?) 5"?“ M 3 mi wwwwwatr .m— “1.1“ a; I ’ “I; / ’l I a adv} C I 3 b“! 6‘! L34 L4 t: 3: 195.6: :dflgkuj?’ ‘03 fl'5\’r 3 ‘9?- . E03 fig? 7 . IN ‘ {’12) '/ f’r ‘ — I {#43721 _ 37:; a7: l V, O. A01/014/08 Q 7. (10 points) Additional questions which won’t be graded, but that you should know how to solve. 1. The product in R is cancellable: [a 75 O and a m : try] :> :1: 2 y. Show that the dot product in R”, n > 1 and cross product in R3 are not cancellable. 2. Showthat [(1%fianda-e=a-gandsxe=sxg] 25:37. 3. The product in R is associative because (53 y z = :5 (y z) = m yz. Because it does not matter where the parentheses are inserted, there is no chance of ambiguity and we usually drop the parentheses. Show that the cross product is not associative by finding some vectors in R3 such that (53' x g) X :3 # i’ x [3} x ...
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