thq_02 - O A02/002/01 Use T.c 16s MATH 2451 thqr02 Exam ble...

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Unformatted text preview: O. A02/002/01 . Use T.c. 16s MATH 2451 thqr02 Exam ble CalcUlus. 2016—01-26- McCary ' 1var1a Mult Course Due Date Instructor W W 318/ fiwéw @@@@@@§:; u®®®w . V®@@@@®©® \ fi[email protected]@@@@©[email protected]@®®fi@®©®®®@®@®®@®@®@®a r e) - {.mi ; ‘x../ "x \_z W J3 .I\1l @ mm @ @ @ Q @ ® ® ® ® ® fl. «a; Re © ® ® ® ® ® . yea®®©®®©®©®®®©® es“ @@@@[email protected]@p mmweeeeeeeeeeee see? . . , [email protected]®@©@@®@[email protected]@fi 0 Net ID (First Name) ®@®®®@ nh®®©®®®®®®® [email protected]©@@@@@@®@@®@@eeeeeeeeesfimeseeeeeese [email protected]©@©@@@gfi assesseseeeeeeeseeeseee SMBQG was wé®® 0 (Last Name) 10 2016—01-19 14: Cftfivffi-J! €234" I xi say-N bib-tgjf/JV'AO:§ O. AO2/OO2/02 0 1_. (1 points) Let c, .5 E R be nonzero, ,r Find the real eigenvalues or Show there are no . ",1 I? , 7 , f pom-116% a eaten-Mg; :cmrerq _ L 7" j, n. 03—27kfl» +/\7+6?::O b” I M c + 5 -: 27m. ~39?" 7- "?” I: Ill/f A 'fi“: “3 C +4 «a AL?“ Afi/ )1: (HDLS 7i=d—/gg (b) Find the real eigenvalues or Show there are none. Ac [6 1 3-“6 @an13 l; a; /r-; i; b “(limit-h 1 + u; ,4;- “ r 8+ / w / 63%;, RH" 3F .- :r' t :- if: (c) Find the real eigenvalues or Shaw there are none. I‘d—r. A [E i] i .Mfl-NF a“) 5 2:0 g /\ iGraderOnlyJ, O 009 ©eeeeeeeeeo O a .. A02/002/0L . 5. (10 points) Evaluate. They all end-up as “nice” expressions. You’ll see all of these later! (3.) Easy. 1 det [003(9) —T 3mm] m (305 Q raga-Jrémiérg‘gwe} : Y(40§79 l'grdjfxg ) sin(6) 1" 005(9) “’"' VGD§~Q+ r6?! N26, :2 w (b) Madame- A Mr... *- + cos(l9) —r sin(9) O 1...! I, W j 1 f + —r det sin(l9) T 005(9) 0 —r l # Vgl “’6 I. 2:? (‘J-zizi‘iy -' QN'E} l"‘i"'£‘~fi'{7l “- f— + -; .-O” - - 0m 1 - . l ‘ r . . u "r I. “I NJ. A I”; I ’- —r r pm_ - awe #kawl ragengw- ‘1 ‘ L” A. I _ If \i Aim. WIN. (0 Her figure it out somewhere else and then neatly copy it here. Jr“ * [5mm (305(9) —p sin(¢) sin(6'] p cos[q15) 005(6)] _ det p sin(¢) cos(6_] p cos(qb] sin(6) r,4/ _ ewe 0 —[email protected] 1 l _ . ' ’ ' ‘J /’ g‘ ‘ l“ . fl-f‘ ' ’ .- r" 4,- , f , {In [I L n ‘ I 1 fl v.4; _; it {Hum} ._ ' dflgzngawe pwélifl: «J +~’_....xg,szil2 ; NC?) fit.” it." -i ;-_: 609 /_ §_ 4 / / fl,..~ H, 1» - __ “WJPSWU 5:; E“ EN {) 4N ‘ a’ 9 «4.1”.» P: I “j “ v "’u" I ./I h R! If "I; -‘ - l ‘ 7-. Hi- / I Ii 5- ‘_:\\ 7&5cl7/(ipérwips’wéfijfleflnécbgrwdf— pSWCfiaflfl l? } ’— , H K - *3 ‘ J "\ \1 I f ‘ — :‘hxl‘ll : ._ p" " .‘3 VA, ' ‘. ‘ ," / .- #" J“- : l H; '4’;~...¢":) 1. __. f; am Fifi \ ‘ , '1 " rs ‘1 *3. :Coéchf’if)?’6\:\]7€}§§w¢ __7 p?éfl$;{6 gtmqg) ‘ ‘9‘ H Wfi H l {g / K / _ r _ __ t + ~42 mam we)“ 3. -7 f flfl “- N‘ ~— "73/33 will do rug/4) (“U i _ ' (fig-'7} aid l M74) Jré‘nflll J, Grader Only J, o @@@@@®@®®®® O O. A02/oo2/04 Q r 1‘} 1t 01 (3.) Plot the parallelogram determined by Mt 61 and Mt Eg for the given value of t. ‘ l I F a _ O l ’ J 3. (10 points) Consider the matrix M: = [ ], known as a horizontal shear. / .—. .. ‘ _ " I _ .--:‘ t— .sz D: E: Dig/LEII—J r . a /., _ It is very important to notice that det M3 = 1 which implies that Mt is area—preserving (all of your parallelograms should have the same area, which is interesting because you horizontally shear the unit square out to t = 1042 without changing its area). f @ onsider the matrix A = [63, 5], where ('1' and 5 are given in the plot. ' i. Determine the matrix R which will rigidly rotate the the indicated parallelogram such that the image of a: will sit on the positive r—axis. That is, the columns of RA = [R 5, R3] will determine the new, rotated parallelogram. ii. Determine the matrix S which will horizontally shear RA into a rectangle. That is, columns of S RA will moi/“‘1‘ Sé,determme a rectangle. ' If - _m____: JC’- WEE/)1 '0 ll /‘ x /— (Jr Procedures such as this are related to matrix factorizations which are varied and important. The purpose here is to make sure you understand the associated geometry of what is happening and not just the (important, but boring) steps of something like an L U factorization. ,L Grader Only J, 0 ea xa<’[email protected]@@©®@@® . 0. A02/002/05 Q 43}— 15;?" 4. (10 points) The shaded regions are ellipses centered at the origin. (a) Find the matrix R which performs the following (mapping, and compute det R. 4be% if/ (at/[Lizj” i ‘ 2Q} ~ 3311! C3) + %} I "77/13 =:i AngJe Effigy; _I Igo dose 3-519 if'flw‘a :! ;._.—— (b) Find the matrix L whichperfornls the following mapping, and compute det L. / r— ' ,a— , ‘ " Ji—t‘ X \. J: I_»_ ’7: _: 12$ a 6k [0? 2: v I i 3 C3 ’|.:j{j' i‘“73 % i,» 3 =1; IFS-u Li , x“ L} (’J\J i I } J: , ' «j £1” E “x 1 s. i . _ Z 3 f _ 3 1.13 . . V— i 9 _ v ' anc,11a’L< r f. . (c) Using the previous parts, find the matrix M which performs the following mapping, and compute det M- W1! may: 61,1. .2 h‘ii‘ q = » - A ,fL—izs-J ,, ;_ t} . El (d) What is the area of the tilted ellipse?_ f .. In, x I; P in; AV ‘ 7" a H 1 841,? (HI J, Grader Only J, i/ x I, O 06:) O \. Q. A02/002/06'. This pagé won’t be graded. ...
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