M294F09WS_LinearSpacesSol

# M294F09WS_LinearSpacesSol - MATH 2940 WORKSHEET LINEAR...

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Unformatted text preview: MATH 2940 WORKSHEET LINEAR SPACES AND ORTHOGONAL PROJECTIONS (CH. 4,5) a a WW/ are"er a? a / 2 _,, ,1 570 [17: 5,1011] M7J425‘34/‘Lj fn/M‘mj 1. Find the orthogonal projection of 9% onto t e subspace of R4 spanned by ﬂ 1/ a/mk #4 IMg/am “‘3 *5 VL’H 2 L3—2 2 2 [197469 f (/déylrﬁm’ ﬁré 1 and 0 12 W [2/1' 2/ / [ﬂy/1:0 0 1 [‘3 [ii—2' zja/n I -—> _> A “a ' of; : vim,” dm/ #2, = “View —'> //47;//= V WV? 2 ./2‘+2.‘+/1+o":;/7:5’ // «3/1 = Wan/a ~_ t/ﬂ)‘ {Waves/12M :3 ,..«M4.M«.WW-W~~ . ‘ . Z we m; m] = {raft am + W m: [WWW‘WEMWWWW 2. Consider vectors 171, 172, and 173 in ’72“. We are told that 17} - 773' is the entry aij of matrix A. WV\W» 3 5 11 "3M 5 9 20 . -, w [11 20 49] :2 ‘27, 1. Find||172||. .. a .v ‘— 3 a _ g/‘ r ., t E [7/ '1'; 1/ ,, Fig/3471’ - / “7- ~M_ 2. Find projg2(171), expressed as a scalar multiple of 172. Freya/17;): m»- “W” W J) '5' r-v» \\ - t " “'3 [7/6 H = W/Hm/ = ’4/3’ If WaarWWnn/éalﬁ ﬂrgfﬁw/QJ'éﬁ P'Wz/WF/[f'l/yﬂ) 3. Find projvwl), where V = spanwzﬁg). Express your answer as a linear combination of 172 02 andiig. , 7 av, ' .. ’ -‘ 3 i we W 4%. mall/m 13’ 4 AW», whaA'Ma/ri‘ﬁw WM “>3 y .‘ ~-> ‘* , “’2 T714941 am“ [’7'] /I,L_y4; V! 3 5‘ 2" “yﬂrﬂvfwz enemy/3r If‘Jvr f 3 3 a, . é, 0,: a _ a , .,_3 52 fa?»al72~-c’sf<2/v17:=0~> «arwweataso as“ 4cm 20% 0 l: 3:325 - A ' ﬁn _ I : ([3. £22,432). p: :0 a - (lggizgqg ﬂay»sz W3 0 a; v, 4. Find the angle. enclosed by vectors 172 an 173. a) 5.) . V i V 20 (can? - 2:3 2 w—t = Z? ////vg// ﬁe ? 2/ ,zetzwtwcéiaf ézél 20] €2]3[\$ £0 EZ'DQH—JW'gr / w #4 [as I] (2 ~ "We ”’ 52L. W‘Z‘tﬂ': ‘9 63/” w: [I “Mi-gm *Zﬁ ? H :3 /. .Hg‘f'z’é'” :— l r :‘Zé'rifftigu 7 HIV -/_ Cs 7 W V n. lﬂﬂ‘Hj f37)’— 29,. a"? .. I/L. I”: :1. him] the nmtrix 0f the given linear transforiilation T with respect to the given basis, 8. Determine whether 01' not T is an isomorphism. If not, ﬁnd a basis for the kernel and image ul' '1'. and thus determine the rank of T. (a) 7(2) 2 from C to C. B = {1,73} MAE 714.62 rX . = r ’ K. www.125— Zl? linﬂfpﬂiﬁ" L’: Q‘I‘I'Zti ’ »r* wﬁﬁ:e7 0 w/ 23 ' ,4: [3 0/, Wm [M g i‘ﬂ'ﬂ_"///F%§/Jﬂ w/iL/r/brW/Zé' M Wrwri’ 19’; £1 '1 I [4) ’1] "/"ﬂ‘ .._—-a 1. 7’13” 6m mm” 11/3717 M 97 [7717.ng Rena“, ‘ l (b) T(f(t)) : f(#1.) from/Pg to P2. 13 == {1,if,t2 / f D ' ._ _ I I . . ' 0 f4 ’ [ﬁx/ﬂag [TI/[ejﬁ [WI {2}]21] fa . r 00 g/ . ‘ ' 7- j ; fol 'VYWM ﬁp T-/£¥/3’k " W1) : «Z ’ 41””? ‘5 +3 ’5‘ “#77023 72%;”? T‘W/ez): 247%],6. we): #4:! Wilt W1 elﬁf/Jg =5] ’ . T/WWﬁﬁ— N wide” [[email protected]] "- c T IV! = A4 1 2 — 1 2 M from U2X2 to U2“2 the set of all 2 X 2 upper triangular 0 1 0 1 _ matrices with entries in R). “([5 i’lll 5H3) ill) e T/[gfjjr [\$7][j?]~ UN; 3’] * [Jiffy]: ; mm 52+ em roéT/[o’fﬂfzr [5/ w/ an]: [30’][;7]’[lfl[3\$]: [2f JHE ’0} [35] , a. « ms] i'éféjj 5» [rum/J73: [5’] ’ ' "L z ‘ o I r N '0 ‘4 #7/ 53]): [:11me5 J» was]: [2 3] V [a 3M» 7.”.[gfj-iw5é @153]. so [T/[o’ﬂ/]g:[g] ﬂ AM ram/[1 1) 5% 7/ AM "M j: pig/kg ﬁt Mam [Md/T zw/ cfav/‘cctxz’M/‘r’wgf, M ﬁt? W [ﬁlm/[:67 3i .4 Mavé Am We \ '__.——-—~.~.M.M..mu. """""""" “ ...
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## This note was uploaded on 12/25/2009 for the course MATH 1920 taught by Professor Pantano during the Spring '06 term at Cornell University (Engineering School).

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M294F09WS_LinearSpacesSol - MATH 2940 WORKSHEET LINEAR...

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