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07aut-m2sols - Problem 1(10 pts a Write the definition of...

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Unformatted text preview: Problem 1. (10 pts.) a) Write the definition of when /\ is an eigenvalue of A. \ —) ‘ I There is CL VlOv1~ Zero Vader v Sada flawl- ~— ‘ -) Aw}: 'A'v b) Let A be the following 3 X 3 matrix: * 2 * A: 1 —3 0 7k ——1 * Where * denotes the entries of A that are unknown. 4 Assuming that the vector 3 = 4 :l is an eigenvector of A, find the corresponding eigen— 1 value. ~36 4 1 ~37 Av: 442?: fife; .i i; g .5 i ‘u= V” L ’ .2 Problem 2. (10 pts.) Let A be the following matrix: 0 —2—1 A: 2 4 1 —2—2 1 Find a basis of the eigenspace corresponding to the eigenvalue /\ = 2. .. ‘ Z " 2 '- i + 2 + Z 1 A—Zl : [_1 2 ( ~ 0 o o Z --2 ‘l O O 0 field“: 2,3. x1 Mi x3 on.) ‘OO-POLU-«E‘LQSJ be 8,042: ~( J/L N < A ' 2.1:) 1' S’VC‘KW 0' l ‘0 s Problem 3. (10 ptS.) Find the inverse of the following matrix: ' “ loo ”1 ~(l :00 j -3 0*] 0&0 n. 0.40 ‘,0‘§~)O 0 2t Got 0 2| OMJ 50 L I00 ~2~3-, A O‘O‘i—io 00! ”22‘ -v “‘2 ‘3 ‘47 30'. A = “ '| o , mummy” Problem 4. (10 pts.) Let T : R3 —> R3 be the linear map: a: m+2y T . y = ——:E+y+z z —2a:—y+z a) Find the derivative DT of the map T. .. l 2 O D‘ ' -| i \ *7— ‘l \ b) Find the inverse of T or show that this inverse does not exist. 30 4544 {amt-I6, §OEJHHi exi‘J‘t, Problem 5. (10 pts.) A 3 x 3 matrix M has two linearly independent eigenvectors corresponding to eigenvalue /\ = 4 and one eigenvector corresponding to eigenvalue A = ——1. a) Is M diagonalizable? Explain your answer. CocreJPOnolLy-za ‘(o e-VOhc (f «are. indepenalem‘l OV‘ e’vfici‘Or-S Corm?0u5~£vx (0 Q‘VW ~( c 5° : her-J at “4W :4 (agenmtza E eiaewveoiors at. 50.325 CowsIS'l'CwS 94 e- vem‘ocs ‘ b) Find det M. do,“ F1 : (”0&va 9;} e‘VQ’Q‘AGJ C mifi WMZfiaegj/j ': will“ <4) : .. {g Problem 6. Find an equation of the tangent plane to the graph of the function f(a:,y) = x3312 — yza: + x2 +1 at the point (1,1,2). V 1C): -‘ 3¥zaz+az-r2x Norma/C Med-or: ‘C 3 ' a : b a X + 26x \ l -4 eAmeszcmme n Problem 7. (10 pts.) Let Q(:I:, y) = 2:2 — 2amy + y2 a) For what values of the parameter a is the quadratic form Q positive definite? Subc‘CoM I: L2] Comvie-HOM 94 “ac Sjuare: ow); (x-efi W)- at £3;- l~c3>o he no} Le. 44am W 50LW+£OM 12: Q Lg”. & Corr-eJVOM ouzf MAG/{DEC [‘1‘ .(k.] 9;“ characfu; PRC. [aoldmow‘cvf ((-A)2~Q2 bluue WOOL: Our: 9.: flex. Far Q 40 Le. with: QegCnli-L “1%) a“)! Fax} Co; Ei>c~)—=I b) For What values of the constant a does the function Q(.’L‘, y) satisfy the following expression: 82¢? _ @ away — 8:52 @x9 =- §;<~qu*26)= ~Zq So q=-I‘i Problem 8. (10 pts.) Let a? : R —> R2 be defined by: E 35(t)= [ Zeost] sin t {17 y ]) = 2:2 + y2. Find the derivative of the composition andszzaRbegivenbyf([ ch : F22C°JL ~2$qu¥ % ELZSCH‘LJG‘ [ C04+ ] 3 ~8~093+ Hw++25§h¥w+z é ‘ '3$zm2_+.' Sou/Hr,“ 1? NM H3293): H may») * 4%“ m = 3wfi+ ti Problem 9. (10 pts.) Let P be the :cz— plane in R3. a) Let T1 be the linear transformation T1 : R3 ——-> R3 which reflects every vector across the plane P. What are the eigenvectors and eigenvalues of T1? Boob 94 X2 ' Plume L [:1] [fl __ #13“, «€er out [6“; a \ K} . ‘l’L‘ §QM . {.e. “a «a a—Veo'l‘ou Cornxgmguar 4° A=l Baal at Law, Mandate,— ‘L’ Xz—flwe: [S] , TM; mellow“ 8641 flzfldfi 4L4, PKG-Mi Le. ll» 34 {Nectar correufwmébp 5‘“ a-VOJM: 3" “l, b) Let T2 be the linear transformation T2 : R3 ,-> R3 which orthogonally projects every vector to the plane P. What are the eigenvectors and eigenvalues of T2? As 1,475“: [if] if?) €orrero-nol 41> §=t \ V9. , 3 6f .53 Li M in} 1 e“ o 39‘ Problem 10. (10 pts.) Let f : R2 —~> R be the function defined by ‘M a?- kfi 3 3% x-k‘x + x Z : k ’flc Xq‘r kqx“ {*Efi 10 ...
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