Winter 2008 - Mengi's Class - Exam 2

Winter 2008 - Mengi's Class - Exam 2 - Discussion Section...

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Unformatted text preview: Discussion Section TA: Midterm 2, Math 20F - Winter 2008 Duration: 100 minutes This is an open. book mum. Calculatmzs are not allowed. To get full credit. you should suppufi your answers. 1. Given (b) (1 point) Based on your answer to part (a), indicate whether each of the following is true or false with a brief oxplmmtion. (i) The null space of A consists of the zero vector only. m I . .. [A /J {INZE A [J l/IVPr/llule/ die (if/’3‘”; l'AJI/(fia/{A‘é => A x :0 My /,, (ii) There exists a b 6 R" such that the systmn 31.13 = b is inconsistent. F215 6 ff/Ile A is I'm/er IIAIE/ 3 Me (01/ 0/ fl 5/00 / (:9. b g [or a/fl A Z) A X to U [MW/"04 [22/ aflfl - - - - - Total (20 points) (c) (2 points) Find an LII-factorization for A. Is this LU—factorizatiml unique? If it is. explain why. If not, give another LU -l'uct.0rization for A. -I 3 -2. l 0 O ::Q_3q 'I 3 “2 -35 u o lo 'Laro-I/ID3 , , ’ F 100 /00 3E, wlere ZZ,“3I 0 / Efo ’ 0 301 /30/ / é/eIM/t 5’} J é/t’mm nr influx ("fa/“m "9 1770’“? (wave/valor) 0— ggaar’ (0 ’5 i294]? .. no} un/yu e [A ' l I ‘_/—l: V M wi/II cm [W e U 5y 3 - 7:; 3 .2 You 01/4,?! anal-lax)- _l_ (3 -4. I0 Ll/ [within/Ian 3 o 0 _ uppose , Ia . . 1e system An: = b is consistent for all I) E R“. In “I 0 o -t 3 -2 10 0 [illicit/lull?” A z 3 I Ojio ‘4 [a] fag/’flu/{M A‘33 I 0 'J 0 each part. .itlstify your answer. (a) (1 point) Determine the rank of A. 3 ~—-——-—-. A =1; '5 (awn/ml or 4],? $51? ‘0” W Me —‘> goEC/l’)= "’23 f 42“” j] --—) ell/I7 '2 rank : 3 fan/(f (b) (1 point) Determine the dimension of the null space of A. O/I'm I» 0/17?) fl 0/ ro/Uan 3 # O/nn AIM/pm): 5 :) Jim Nujflf/«lFQ (C) (2 boiuts) Is the system ATJ: = r- consistent. for all a? No i! is noé. ___L___.___...__ A T [5 5. X 3 I 3 (o/umn We ‘1 or} (anno/ Van [/3 :3 TAM exub a ( Jutl Mu! (lg/CoE/A) _ 1 . - J’ —--”"3 711w owl; a ( Jud: //m/ AX‘(,-n(ipnrrilen2‘ 3. A (QR-factorization for the matrix A is given by Q “pd—q h’ 0.8 —0.6 0 I) —3 10 a —1 4_ () 0 —().8 —().(i 0 3 —4 3 ‘ _ u 0 —().(i 0.8 0 0 —7 —7 0.6 0.8 0 0 I) 0 0 2 where QTQ = I . (a) (2 points) Without constructing the matrix A solve the linear system 1(1 ‘ 10 Am: 10 10 by exploiting the. QR~fuct0riznti01L @ :0 — O Q R X ' l o . no @ SD’VQ '_ ." .' 1 " to 9 —l "I _ _-* o 3 -c, 3 "z : - 0 O _; ‘} = 0,? 0 0'6 I0 5% 0 0 0 x9. 7% 2 vflbé 0 0 0,? lo I [I'm ' “0"? ’04; 0 m B} [Ufef [:31 :[I: 0 0'3 0 M (b) (2 points) Without. constructing A decide which of the Inatrit-vs Q, I? and./i are invert- ible? Explain your reasoning. cm I R n .1?) Q is sin/edible, 5mm sh 0150 inveriILie (01)} are 10!— Ma]. [0 veil/LIP. 4. The set. S2x'l:{[b (1]:(LJ)ER} a b is a subset of 2 x 2 symmetric matrices. Specifically if A E 8")“, it satisfies the property AT = A. Let T : R2 —> S2X2 be the linear trzuisformation defined as T([3ii)=[u+3 “3]- (a) (1 point) Write down a symmetric matrix A satisfying AT = A that does not belong to 82x2- ~ . ,t 5 mine/Hr Ina/rm aux/,4 0/, ’[prm ffirmj “joy [lie BJIchMaI / A W _ r. .' film - 8199- - l I P 6/1 9 (b) (1 point) Show that S2X2 is a vector S] ace. i O + a :9 DJ q, E é"? I Anj rel Varmfla/ (DJ/[0 7} ’U a“ 0/ “W” i l 0 i; a Valor J/M‘e' . . . . a . ' . . o r e (c) (1 poult) Give a baSlS B for 8”“. Fmd the coordinate vector [A]5 of the matrix I ere/ [3 4 A: _ valor J/DCNG, 4 3] I [S a )9 0.311 0 ,1 a; [Xe Ina/rue; 0'6 M ( ens/(’4‘; [70/ my]; [a q (41‘ a/Iu’r: I}, relative to [3. 8~ 6 i‘ “i [0 " 0 I ’ l (i) [/m 127 I; [Aden/{Z [AP famzjj 7Ae 5‘er flan/U f2 a 10W” in [A] = 5 f.— l 0 “o I 4—1.3.)- (1 Doint) Find a basis for the Kernel of T. 2 g] 23/ 0 ,J 1' ‘9 l l 0—,) Kerneflrmfv : mpg :) [A332 3] ={[g].f7’(m)=tfr ,{m} [lye aMPY bye 8: {3]} U :- Eil flag] :0} (oar/1‘ for /(emefl[7), \ .. .- 4 1w» for in: m; a-o b , Kemp/(‘7) : 6 6m 75/9”, 5. Consider the symmetric matrix 2 3 I) A = [ 3 5 1 ] 0 1 2 A],ng r satisfying the property AT = .-‘l. Symmetric matriccs posscss some remarkable properties. Onc such property is that for an n x n symmetric matrix all Vectors 1' e R" can be written f the form 1: = v" + vn whcrc v" belongs to the null space and 1),, belongs to thc column ) (3 points) F ind bases Br, 3,, and 8,. for the column, null and row space of A, respectively. A in In an étAé/a A f‘, rm _ For (ofl Me : 8L. Fri/oi w :1an arm 4 [nu/J we, [2 3 £272?! [9 xo/ve 0x20 “g 1:2 0 w l A ' ‘ ‘ I 252:2 ergorogi/f)” arm a 5m; ‘ BF - {L2 3 01/ [0 ‘1} en t 1‘ (b) (1 point.) t the union of the bases 86 and 8,, forms as basis for HP. 7A2 anion BC 0111/ 8,) 0 “ 3 ‘3 is a ham/£9 J i 3);], :2 [ht/(VJt’flD/e/Jl 1625, k wcre ut/IM - 3 ‘3 a 2 3 ~3 r 3" f2.“va 2 3 "3 r3:=f3_2r'_ 2 LP, 3 5 _2 1H 0 “2 5/2 H O 6 O I “I 0 l -I o O — in i‘c'ale) [In B U [Inearfi fao’P/aeno/Mv-[r lingf’l” £ 8: 5/002 [lime xp/VL ME/F/Jma/em" Vet/or} m R"? j;,€,zf,' f”’” 4 50"” /0’ R, 8 Maxi Z9 0 $4.01, é ill)?” ...
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Winter 2008 - Mengi's Class - Exam 2 - Discussion Section...

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