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lab11_sol_f_08 - Math 115 — Lab 11 — Fall 2008 Topics 0...

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Unformatted text preview: Math 115 — Lab 11 — Fall 2008. Topics: 0 orthogonal diagonalization7 0 vector spaces, 0 subspaces of vector spaces, You are to provide full solutions to the following problems. You are allowed, and encouraged to collaborate with your classmates, use your notes and textbook and ask the TA for guidance. Direct copying of solutions is not encouraged, nor is it allowed or ethical. Last name: First name: Student number: Math 115 — Lab 1 - Fall 2008. Student number: 1. In each case, determine if the matrix is orthogonal. If the matrix is not orthogonal, normalize the rows or the columns to transform it to an orthogonal one if possible. c056 —sin0 0 a) < sigfi 0059 (l) ) OY‘l’lIOflOflCLQ ’ be ((L‘AS‘ ,-~ (056‘ ' —5 an El 0 \(— C - s-ne C — A ' l — '5 ’ - " ‘ O , 1 936 C3 0 D l —1 2 2 2 2 —1 “1%qu ’ — , an bU‘aJASQ »"\ ’Hw (Olumnb ‘FCWY‘ NoT Or-Hwogon 03); or+h090na9 Set , fixed my n01 umfi vectors. 1 2| ,2 ., __ , v _ 3 "(if “:3 C: - 7- \ _ , nu“, 5 l \i] I szlrwé] / (3;: [a ”Li“ _ \JC" 3 , / ”I " 7 [ /3 213 2/3] 6!‘H‘090n¢\9 :93 ”V3 2/3 1/3 2/; "V3 2 1 1 c) 1 —2 0 no.‘. 0 r—thagrj‘l‘CLC, be Cam/7‘ -§oxm om vakmpml 3’67 (Swim/a row5) ‘ 1 ,’ _ . W2 (flan m1 main/101136 (0W5 Of COIMMV\§ 0; ‘l’hIS modTIX ‘l'C 9 6T ‘9 Qolumrms ale floJV 0m orthogonal? wwd'rix- 2. Determine if each of the following matrices is orthogonally diagonalizable or not. In each case provide your reasoning. If a matrix A is orthogonally diagonalizable, find an orthogonal matrix P such that P‘IAP is diagonal. 1 0 —1 . a) (g 3 S > Not 83mmeTnC :) Ed Principal axis FM 14' is ‘in arihogoflkaeia lagmflfik 3 0 0 , b) (0 2 2 >‘A 33mmel-vi'c -——> althogonaflo 0 2 5 éiqs‘gomwa‘gqtfle 1—3 0 o C(‘1I=JJ( _ .f’ ‘—~6 ,-_ /:.'- A 9 1’3 2w z (31 ”(I )(l’ 0 r4) 2;; (’ASfriValwcS - l o 9 Al _-: ~23, X2 : 0 X3 : l l ’9 2 Norm/19‘5”" M 94'39/nvé’mr5, we 36+ F a '( “ 0 3 -2 . 4:, a o F , .g: l/ o S S 2/ 9 I C) -2473 0 J: 2 i “S :3 0 GS“ 0 2/ (ha y: \ 6 a1 lmgonch V5" op—Ap—n ox—xr—t MOO c>< Wagmwdor associodeé 4% Ape :3 #1:.2] Eusic en‘jU’wecfors assoq'ufei +0 A) ,2 . 1 9 . X2 :[ 1 x3 : a] ) CH”): “1'02 2 9‘50. ! o {Kl 7 X2 , X3} )5 Or-Pnojarmp. RF: '4: 0' ”47 '47 0 o Nom‘uflnzfl ,K‘) x2 WK?) +0 559+ P ‘ 3. Are the following sets vector spaces with indicated operators? If not7 why? a) The set of polynomials of degree 2 4 together with 0 and ordinary addition and scalar multiplication of polynomials. Mi a WHO? 5PM , £7ch 14¢: and 364 are elements Of My set Bur (14+1)’ 14 : QC halo «419166 (L4 => Nof i" W 397, b) The set of polynomials of degree equal to 4 together with O and ordinary addition and scalar multiplication of polynomials. (14+1)v(7(q) hem dogma i 8 Auto/Q not c) The set of all the 2x2 matrices whose determinants are zero with ordinary ad— dition and scalar multiplication of matrices. loo“ 4. Let M272 denote the vector space of 2 X 2 real matrices with matrix addition and usual scalar multiplication as vector space operators. a) Is U: {<8 (1)):a,b€R} asubspace OfMg’g? No/ beam-44 [fb fl] ad [07. 192] are m U, 10 ’ [31" W“: “i .3 [3“ “3“} 7““ b) Is V = {A : A 6 M23, A2 : A} a subspace of M23? N0 - flo'i C iOSQA U “691 SCAQq/r (Milli [P “(when c)IsW: (3 12> abcER}asubspaceofM22 T613 _ ~ . , a 102 5 H92. (alb‘) g (a; in) L; W ;> (10 “I. )C-W \‘D C ‘3 C2 N (IT(2 5. Bonus Question: Let A be a symmetric n x 71 matrix. Show that A is orthogonal if and only if all the eigenvalues of A are either 1 or —1. a i: (=7) A ,« @Hhvgonag <=> H=A v t Bu’r Pr is ngmdrnc .2) A :A («a—)' Li A 192 O 53mmd—n‘c Mdnx «AMA GAQBVWCUWO '1' l . 9 Synmm'nt 3’) {S P Oflhxz) ._ ST' 3) A 5 PD r5l View #‘J D _- OHi’mjoflcaO / .2) A Oerjm'tQ _ . 6 b90ma flu PFQAU’CC \{Q 07409011 J) migfnzas r; orfhocpndx ( 115,03 ‘Hu {mg Hwt A ., Ofihacjamtfl <,=‘—> A";A‘t) ...
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