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# 107 CH 5,6,9 Answers - CHAPTER 5 LINEAR TRANSFORMATIONS AND...

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Unformatted text preview: CHAPTER 5 LINEAR TRANSFORMATIONS AND EIGENVALUES AND EIGENVECTORS EXercises 5.1. pp. 243-245 2;'=Is-not . 4. Is 6. Is 8. Is not 10. Is 12. Is not *2 1 —l. 1 —1 3 ml 16. - 23 . 1 _4 18. 2 3 --1 —2 _ 5 _2 —5 3 7 —5 —3 5 2x + y "1/2 x/2+y/2+z/2 20‘ (a) —5/2 (b) —x/2 + 11/2 + z/2 —-7/2 —x/2 — y/2 + z/2 22. (a) x2 (b) ax2 + bx + c 1 O 0 24. No basis for kernel; basis for range: 0 , I , 0 0 0 l —8 5 l 755 0 1 0 26. Basis for kernel: 1 , 0 ; basis for range: . 0 , 1 0 1 ——1 2 23. Functions whose area above the x-axis from a to 1) equals its area below the x-axis from a to b. 32. For vectors in R2 or R3, IN) is a vector in the same direction as 1: that is longer than 1) if k > 1 and shorter than u if 0 < k < 1 and —-v is the vector in the opposite direction of v. Exercises 5.2 pp. 252-253 2. 3.5.43, 4' —-2x—l3y 6. 5x+5y 3y —3x + 6y x — 3y 8. ax +b 10. 640x+192a+64b 12. e“ cosx,e"‘sinx 14. cosx,sinx,xcosx,xsinx Exercises 5.3, pp. 267-269 5 3 . m1 .4] 1 29' 1s 9 27 13 (d)[ —45 —2s ] (e)[ —13 J (D [ "41 j (3)[ —14 ] 1 1 1 1 —1 1 4. (a) 3 15 —16 (b) 3 13 —14 CHAPTER 5 LINEAR TRANSFORMATIONS AND EIGENVALUES AND EIGENVECTORS 1 1 o o (d) 0 2 O (c) (g) 35 1 0 —1 31 0 0 0 1 o 2 —1 -2 2 6.(a) 2 o 0 (b) 010. (c) 0 1 o 0 1 0 1 1 1 1 1 —1 —4 2 —8 —9 0 (d) 2 0 4 (e) 2 (f) 2 (g)2x+2 2 —1 4 5 o 4 1 o o o 1 0 -—1 0 o o 1 0 0 1 o —1 Ma) 0 1 0 0 (b) 0 1 o 1 0 o o 1 1 0 1 0 1/2 0 0 1/2 1 0 0 0 o 12 1/2 0 0 1 o o (c) / ' (d) —1/2 0 ‘0 1/2 0 o 1 0 0 —1/2 1/2 0 o o o —1 5/2 5/2 5/2 5/2 1 3 (c) 3/2 (t) 3/2 (9|: 2. 4] 1/2 41/2 2 —1 1 0 7 b A 10. (a)[ _3 3 2 3] ()[4] (0)7w1+4w2 14. Rearrange the row and column entries of [T]£ in the same way to get [T]; 1 1 —1 —1 2 1 0 2 —1 2 1 2 3 1 —1 0 1 1 0 2 18.(a) 2 3 1 3 0 (b) 10 1 o 2 1 —3 o —1 1 1 1 1 0 2 1 —3 0 o .2 1 0 1 2 —1 0 —1 o 1 o -—2- —7 -—7 0 o_ —1 1 o —4 —1o —11 (c) 4/7 3/7 —3/7 —3/7 2/7 (d) —2/7 —3/7 2/7 ——3/7_ 3/7 4/7 —3/7 2/7 33/7 39/7 58/7 —2/7 2/7 5/7 —2/7 -1/7 29/7 47/7 62/7 6/7 -29/7 —31/7 16 CHAPTER 5 LINEAR TRANSFORMATIONS AND EIGENVALUES AND EIGENVECTORS W 4 23 5 3 21 213 (e) 4/7 (ﬂ -2 (g) -6 “17/7 _13 15 —23/7 —16 16 0 —1 1 0 1 1 0 o 1 o 0 1 0 0 1 1 20'(a)_ 0 o o —1 (b) 1 “1 o 0 o 0-1 0 _0 0 1 —1 1/2 0 1/2 0 1/2 4/2 —1 0 1 2 o —1 2 0 1 2 —1 2 o —1 01/2 0 1/2 1 0 1/2 —1/2 0 1/2 0 —1/2 0 1 1/2 —1/2 2 3 () 1 (ﬂ 0 (g)3sinx+3xsinx+xcos B I -5/2 1/2 1/2 —1/2 . Exercises 5.4, pp. 277-278 1 2.2;132: 2 4.2,-2;E2: 2 ,E_2: 1 ' 1 1 1 1 7 i , ~1 2+J§§ 6 1 6. (5+‘/33)/2,(5*—~/3.3)/2;136%,?”2 ;[ / /1 J ; _ —1/2—J33/6 1 E(5-~/3_3)/2' 1 1 1 -1 - ' 8. 1,2131: 0 ,E2 ' —l 3 o 1 _ —2 o —1 10. —2, 2; E_2 : —1 ,E2 : 1 , o 1 0 1 1“ __2 ' 12. LE]: —1 I CHAPTER 5 . LINEAR TRANSFORMATIONS AND EIGENVALUES AND EIGENVECTORS 17 1 «5+3 —J§+3 14.1,~/§,—~/§;E1: 0 ,Eg: J2 ,E_ 2: — 2 - 0 1 1 ' 1—' 2 1 ' 16.2i,—2i;E2,-:|: ‘/1],E_2,.:[ “/3] 1 ‘1 1 18-331+i:1_i;E3: 0 1E1+I _l+i :E—l—i '"l—‘l 0 1 1 1 0 0 0 _ 1 0 Bil-(11.612, uduIEdg: 0 ,Edzi 0 , ’19:!" 0 0 _ 0 1 5 —2 24. 1/3, -114; 51,3: —-2 ,E..1/4: 8/3 1 1 28. 32/3 + 31/3 + 2, -32/3/2 .— 31/3/2 + 2 + J36?” — 31/3)i/2, __32/3/2 _ 31/3/2 + 2 _ «5(32/3 _ 31/3)i/2; 24/29 + 11(32/3)/29 + 11(31/3)/29 — 4(32/3 + 31/3 + 2)2/29 E32/3+31/3+2 : 1 18/29 + 32/3/29 + 31/3/29 — 3(32/3 + 31/3 + 2)2/29 5—3213/2-3Ila/2+2+J§(3213—3”3)i/2 : [24/29 — 11(32/3)/58 — 11(31/3)/58 + 1145(32/3 — 31/3)1/58 —4(—32/3/2 — 31/3/2 + 2 + «582/3 — 31/3)i/2)2/29 1 18/29 — 31V3 [58 4 31/3/58 + «582/3 — 31/3)i/58 — 3(—32/3/2 — 31/3/2 + 2 + «562/3 4 31/3)1/2)2)/29]T E—azﬂ/2—3ll3/2+2—J§(32l3—31ﬂ)i/2 : ' [24/29 —— 11(32/3)/58 — 11(31/3)/58 — 11J§(32/3 — 31/3)1/58 —.4(—32/3/2 w 31/3/2 + 2 - J56” — 31/3)i/2)2/29 1 18/29 — 32/3/58 — 31/3/58 — ﬂew — 31/3)i/58 — 3(—32/3/2 — 31/3/2 + 2 — 45(32/3 — 31/3)1' /2)2)/29]T 30. 5.522333394, 2388333022 + 55233066751", 2388333022 -— .5523806675i; _ 6603645192 E's-529333394 : —.3232764206 3799452233 18 CHAPTER 5 LINEAR TRANSFORMATIONS AND EIGENVALUES AND 'EIGENVECTORS ' W 41175607093 —'- .9773859383i tE_2388333022+552213066751 : —5.879753598 + .2093978539i —3.441595042 + .17112223381' ——1.175_607093 + 97738593831" E.2333333022__5523305575§ .' “5379753598 - .20939735391' --3.441595042 — .17112223331' 1 0 1 —3 7 1 32. -1, 2; E_1 : 2 , —2 , E2: 1 1 l 1 0 1 1 ‘ i Exercises 5.5. pp. 286-287 - I! 2'. Not diagonalizable 4. Diagonalizable, [ 3 (2) ],P = [ i i ] . . (5 + mm. 0 _ ' . b1 , , . 6 Dxagonahza c l: 0 (5— mm . 1 P _ —1/2 + «/33/6 —1/2 — J33/6 ' f "' 1 - 1 8. Not diagonalizable —2 o 0 —2 0 —1 i 10. Diagonalizable, 0 2 0 , P = —1 l 0 0 0 2 1 o 1 12. Not diagonalizable . - . ' 1 0 0 1 «5 + 3 —JE + 3 - 14. Diagonalizable, 0 Ji 0 ,P = 0 J5 “/5 0 0 -Ji 0 . 1 1 ' . . 2i 0 1 —1'/2 1 +1 /2 ' 1 9 P = 16 plagonahzablc l: 0 —2i ] [ 1 l ] 3 o 0 ' _ 1 _ 1 1 ,18. Diagonaliable, 0 1+1 0 , P = 0 —-1+i —1—i 0 0' l—i 0 1 1 5 1') o. 19 S m T C E V N E B E ’ D .|.....IIIIJ a ANn _l||_ S 00001 000004 , , a _ m _|l|||_1:|llllll__.llllll. E 000 0 w 1 00011 000004 000004 000014 W 01000 00010000040 _ _ w m 1000001000000400000040000140000040 G _ I 0000010000 . E . 001400 001400 001400 _|....||L m , 00000004000 _ _ _ _|_FII.|..|I.|L A 004000004000 004000 m 00001 ,. 130000 _ . _ ﬂ _|l_ W 00010 000.11300000130000130000130000 A .hr|.||_ M 0000000010 300000 300000 300000 R TIIIIII._|I.|.ILFIIE.I.|__.IIII.I,IL 10000 W 00000000004 _|..lJ_Iailllll||__|lllilll.lJ W 0000010000 _ . A , 000004 000004 000014 m _|.|_, 00000000040 _ _ _ _||_rll...i|.|L nn 000040000140 000040 M. 00001 , 000400 _ _ . _||I..|J W 0001000011 001400001400001400 .004000 0000000010 _ 004000 004000 004000 % 00000 00000 030000 _ _ _ E 00000 00000 300000 030000 030000 030000 P FilliL rlzllL 4A..” . 00000 . 300000 300000 300000 4 _|I|1.|ElZ|.||.__III.I]1L c n. rIIIIIL 2 ‘ 20 ' CHAPTER 5 LINEAR TRANSFORMATIONS AND EIGENVALUES AND EIGENVECTORS 3 0 O O 0 0 3 1 0 0 0 0 O 3 O 0 0 0 0 .3 0 O 0 0 O 0 —4 I 0 0 0 0 —4 1 O 0 0 0 0 —4 1 0 ’ 0 0' 0 —4 l 0 O 0 O 0 -4 1 O 0 0 0 —4 I 0 0 0 O 0 —4 0 0 0 0 0 —4 —2 0 0 2 l 2 0 O 26. 0 2 0 28.[0 2] 30 0 l l O 0 2 0 0 l 1 I 0 0 O O —1'—1 —5 —2 —3 —4 01 0_0 O O —2 0 —6 —6 —4 —6 O D 2 l 0 O 0 —2 —6 -3 —4 --5 . ,P= 40 0 O O 2 0 0 0 —2 m5 —3 —3 —5 O O 0 0 2 1 O —2 —~5 —2 ~3 —4 0 0 0 0 0 2 —1 --2 —5 -—2 —3 —4 Exercises 5.6. pp. 291-292 1 1 2. (a) 0, -1; V0 = l , V_1 : 0 (b) No! diagonalizable 1 1 O 0 O . (c) O —l l 0 0 —1 4. (a) 1, 3, 2; V1 :x2/3 +x,'V3 : 2x + 1, V2 : x (b)Diagona1izable 1 0 0 (c) 0 3 0 0 0 2 6. (a) -1; 1L; : sin x, cosx (b) Diagonalizable —1 0 (c4 0 4] 8.0 12. -3, 2; V_3 :x3 _— x2 + x + 1, V2 :x3/5 + #5 + 6x/5 + 1; not diagonalizable CHAPTER 6 SYSTEMS 0F DIFFERENTIAL EQUATIONS 21 W Exercises 6.1, pp. 301-302 6- Y1 = C1 y2 = C283): 8 311 = ale—3x y2 = 9264" 33 = c3122" 3’4 = 6465’r 10. y] = 1 ‘ Y2 = _e3x § 12. y1 = 253* y; = e‘z’|f y3 = —e2" Y4 = 0 14. y1 = c1 — cosx y2 = C2831 _ e2: 16. y; = ole—3" + #31: 3’2 = 4228—2" )3 =C3ezx+x2+x y4 = 64:25" — 5*} sin 3x -— 33-;- cos 3:: 2e2x+4 -—e"-1 24- —2.x —2e +2 3e3x+1 26 2e‘x+16 —4ex—1 ' ——2e"+8 —2e"+1 Exercises 6.2, pp. 311 2. 321 = clez" + cue—2" y: = %cle2‘x 4- 626—2): 4_ y1 = ﬂeas—afﬁx + ”Jaws—3):: 1 3—3/3? c 1 316—155): + 3+1/3_3€23§(5+J3_3)x y2 = 6. y] = clez" + an?” yz = czez" + £1364" _ y3 = —clez" -- liege—2x 8. y1 = cle" + 623% + Gas—‘5" — 3 232 — yz=ﬁfﬁczeﬁx—#63e ﬁx . 22 ' ' CHAPTER 6 SYSTEMS OF DIFFERENTIAL EQUATIONS M 3’3 = 337ch3‘5': + ﬁr—ﬁqe‘ﬂ 10. y1 = Cl sin2x + (:2 cost y2 = @q + écz) c‘oszx +(%c1—§c2)sin2x 12. y; = c1e3" + c223“ cosx + ege" sinx ‘y2 = —c2(e" cosx + 2’ sin x) +C3(e"" cosx —- ex sinx) y3 = czar cosx + Cgex sin): 14. y; = 2‘22)“ -— 2e”2" y2 = e23 — Zea—2" ' 16. 321 = 2e" + (¥ —-1)e~/5~= — (% + _1)e-~/ir l—aée‘ﬂx + ¥e"‘/§x I 13. 321: 23"“ —ex cosx —exsinx y; =2excosx y; = —excosx —— e" sinx 20. 371 = Clesx + 6.2er y; = %cle5" + Cgeﬁ" 22. y1 = cle‘s‘ + czes’ y2 = ”Brae—5‘ + %Cz€5t 24. y1 = cle‘ + 62:24" + age—5" m=mf y; = me" + cze’z" + 2632—5Jr 26. y1 = cle‘”r sinx + Cge" cosx 3’2 = éqe“ sinx -" %Cle—x cosx + ércze—x sinx + gcze‘ﬁ‘ cos x )3 = 6383': 28. x = 3e‘ + 2.25! y : —3e2t + 665' z = —32§e’ —13€5’ + g?‘ Exercises 6.3, pp. 314-315 . 21 .P= '; ya = (CI + czxkz" y2 = 661 - 31c: + %czx)e2‘ CHAPTER 6 SYSTEMS OF DIFFERENTIAL EQUATIONS 23 W —1 1 1 4. P = -1 0 1 ; 1 0 0 321 = me" + Cgez" + ngex )72 = Cge" + 6282': ya = -C2e2" 2 0 1 6. P = 1 1 0 ; -1 0 0 = (c1 + 62x2 + 03x)ex = (%c'11—16‘2 + 62x + %szz + %C3 + %c‘3x)e‘ __ (—--— C] + '2-62 — 162162 —— 5632:)?” -—I —-3 18 1 -—l Gsi—315 7 E32 3,13: 4;! a3 18 —2 4—33 83 86 .3— —3 13 1 - 3—6 #3 13 1 E y] = clez" + one“ + C3xez" + C4x2e 2“ y2 = clez‘ + 4:233:2—Jlr + C312” + age?“ + 8C4ez" + 264x32‘ + C4x2e 2"+ 7C5e x )3 = me” + c3xe2" + 2C4e2" + C4}: 2432‘ — 2653 x y; = £1.92" + 2623“ + 63x2” + c'43c2e2Jr + cse“ 282" + 656“ 315 = (2132" + (‘26—): + ngez" + C4): 10. y] = 4x82“: 372 = (—1 + 2x»:2x 12. y] = (10x + My?r y2 2 (1+ 13:: +4x2)e". = (4 — 5x — 4x2)e" 18. For each eigenvalue 1 of A, we must have Rea) < 0. Exercises 6.4, pp. 318 2. y;=c1e%(5“/3—)‘+cge%(5+‘/3_3)x— 121+2x_ 82x ”=3___,/—3qe:(5—f"3')x+ 3___+J3_3€2€:(5+J373)x+14_5_ 3 x__12x 4. y; =(:1sii12v1r+r.-2cc1v52x+3 3 —3x Y2 = ('E'C] + £62) 60322: + (gal — écz) sin2x - 3x 6- y1 = 6163‘ + cze" cosx + C3e" sinx + Tlﬁe-zx 24 ' CHAPTER 6 SYSTEMS OF DIFFERENTIAL EQUATIONS yz— — "egg?" cosx + e" sin x) + C303“ cosx — eJr sin x) — \$78—25: y3— = czex cosx + C362" sinx +T—10e'2x 3- 3’1 =(C1+C‘2x)82x + I —2x2e 2” 3,2 = (-é—q _ 4ch + %c2x)e2~‘r + % + %x + (1 — 5:2)er 10- 3’1 ‘—“ (61+ 02x2 + 031%" +11+ 6x + ée‘x + %xe_" y2 = (£61 - écz + 62x + %62x2 + éc3 + %C3x)e" + 2 — éxe—x y3 = (—%c1+%cz '- %czx2 -— %c3x)eJr — 5 — 2x — ée'x — éxe'x 12. y} = 69+13J373‘e ‘(5—_J_3)x + 6___9—13J3“3'e §(5+«/3_3)x_ _21+2x _ 332x 5,2:__5J_+_____3_7e %(5—J3T3)x+5__J31_3—378 §(5+ﬁ)x+; 15 —%x-—§e2*‘_ 14.311: e3"— \$6“ cosx+.] inexsinx+11—oe 2" y2 = gel’icosx + 3e: sinx — 39’2”” y3 = - %excosx +1-%e’sinx+-1—10e’2” 16. x = ~go— 5t+ 1499' + ﬁgem y— — 25 7+ 3t—14—93‘ + \$33? 2" _ %I-+11—7t— 209 et+ ng3e3t_ 227010735: Exercises 6.5, pp. 322 2. v; = cle’r + cue“ + C3e“4" —4x U2 = clex — C26” —~ 4639 v3 = 6‘13“ + Cze-x + 16033—41 y=vi.y’=v2.y”=va 4. v1 = (:1 + (\$21578Jr ‘ v2 = —8cze y = v1, 3” = v2 6. v1 = (c1 + (:25: + C3x2)e'2‘ U2 = (—261 + 62 - 262x + 2C3x — 2c3x2)e—zx = (451 "' 4C2 + 4¢2x + 263 — 8C3): + 4C3xz)e—2‘ y = v1.3” = v2. y” = v3 8. v1 = cie'z" + czes“ — gen 02 = ——2c1e_2" + 36233“ — gem” y = v1, 3” = v2 12- vi = c1'+ czx + care?“ + C4e‘x v2 = (:2 + 263:?” -— C43” 1’3 = "-61 — 02x + 3636” CHAPTER 6 SYSTEMS OF DIFFERENTIAL EQUATIONS 25 W v4 = -—62 + 66382]: V1=vi,yi=v2,y2=v3,y§=v4 Exercises 6.6, pp. 331-334 2. an: ﬁe???" +132 qz=—%e hi; 4- «Mi: = "—‘2q1+——'q2 d _r12 r2.1 +712 Etta-W41- v2 —42 (b)r1,1+r21 — ri,2=r2,1+r2,2 _ 12° 265eﬂiﬂ45+4265ﬁ _ 152—0. /—526 e—gﬁmaHJ—m ): (c) 611—— -3- ‘12 = (50— égJZWkWFMm)’ + (50 +5 §§J2T)e’ ’mmﬂ/Z—fﬁ)’ 6. i; = gee-2‘ F ge—t i2 = —%e"2‘ + %e_' 8. i1: '75 + e 3’(E 2 sin llot— 75cos 11—0!) .12 = gee—é! \$111 110! 10. x '= 3‘ +9“ y = 26’ + 2e" 7, = 3e‘ + 3e" 12. Assuming that there is no friction, the matrix for this system is: 0 I 0 0 0 0 O 1 £2; 0 —£22- 0 The initial conditions are 121(0) = 1, 122(0) = 0, 03(0) = 1, v4(0) =' 0 14. x1 = —% sin 3: — 1% sin w‘é—ét x2 = slams: — %§sin§t 16. x1_ = ——3~/' e-%'(21 sin g: + zﬁ sin gt) xg- .- ég-{ée %1 (—42 sin gt + ﬁsin 1233” 18. x1- — gscosx/it—gz ozsinft+ 39c052ft+22 3 —-—35in2ft+ 191 sint x2=§§cos~J§tw Tﬁsinft— 3—0 lcostt—zzi gsin2ﬁ1+3° -1—]sin1 22.70(e4 +e- 4—e2—e 02) 24.x=O y = -— +— __9_: _ —9t Z_loe+]10e 26 CHAPTER 6 SYSTEMS 0F DIFFERENTIAL EQUATIONS I . Exercises 6.7, pp. 342-343 2. (a) (0. 0), (1. 0), (0, 2), (é, %) 0 ._ _ (b)A(0,0)=[é 2],A(1,0)=[ (I) J] . -1 o h _i A(o,2)=[_6 s2]:A(:%)=[_ ﬁg] 2 r — —t (c)(0,o)[ “2 ],(1,0)['C‘e (+62," , NIH cze —cze_ (0 2)[ —%c‘e—Iz]’(%‘,%)[ Cleﬂﬁﬂﬁﬂze—gﬁﬂt ] I I lull-UNIF- cle‘“‘ + cze_ - —\/§c1e%(‘/§_m + ﬁcze—5(ﬁ+l)’ (d) '(0, 0) unsiable, (1, 0) stable, (0, 2) stable, G, %) unstable 4- (a) (0: 0), (2: 2), (2! —2) 0 2 —2 (b)A(O,O)=[0 0],A(2,2)=[ 4 _:].A(2,*2)=[i 2] I c + c2t lcze—Zt _l 82, (C) (0’ 0)|: 11 :l! (2'! 2)[ 618—241 + 2: :l! (29 _2)[ 2:201 4t ] ECZ 028" (d) (0, O) unstable, (2, 2) stable, (2, —2) unstable 6. (a) (Jkn, him) where k E Z+ U {0} _ 0 (-D" (b)A(«/I_CET_,—kﬂ')— [ 2J5 ] 1 018%(l—vl+(—1)*8JE)1+Cze§a+ 1+(—1)*3JE): (C) 5-3120 — 1/ 1 + (—1)k3«/ic?)cle%U-Wﬁ +L'7‘Lku + 1 + (—1)*s~/E>cze%“+mﬁ (d) unstable for each k e 3" U {0} 8. (a) (o, 0), (2, 2) s o o —% (b)A(0,0)= 0 0 vA(2:2)= 1 1 § —5 (c) (o 0) ““3"!“ (2 2) “WI COS §t + '62 sin \$0 - ’ C2 ’ ’ %e‘5'(c1 cos gt + ﬁe] sin %t + c2 sin gt —— ﬁcz cos \$1) (d) (0, 0) unstable, (2, 2) stable :0. (a) (o, 0). (g, 2) l o (b)A(0,0)=[g 0 ],A(§.2)=[ MID-i o I | I--| wl—n |__....__I viii w CHAPTER 6 SYSTEMS OF DIFFERENTIAL EQUATIONS 14. 16. 18. l 1 e1’ 2"? c cos it + c2s'n1t (c) (0.0) .“c 41.2) 3 ( ‘ 2 ‘ ) 1 _ 2 5e I‘(c1 cos— %t + Cl sin it + C2 sin —t— C2 cos 5‘) (d) (0, 0) unstable, (g, 2) stable x = T%(88t e-—8t) y = %(88l +e—St) The equilibrium solutions are (0, 0) and (%, ﬁ). —a 0 a 0 7 A(0,0)=[ O y]'A(%’F)=|:—ﬁ a_8 O —¢f (0 (”[0] me :|(5’ﬂ)%|:y aclcosrt+czsinrta y] C231" cur“ L.._...._..l Both solutions are unstable. —alnly| +l3y— ~ yln|x|— 6x+C The equilibrium solutions are (0, D), (0, Q ~), (°' 0) and (—-—- W ”W; w ﬁv r -ﬂv my a: 0 - 0 —o: —— A(0,0)=[0 5].A(o,§)=[ 322 :6].A(%,0)m[ 0 2342], l) A(y_______6-av cut—ﬁll): 1 13(1)!” “— ya) 1’03”) "'" Y6) w-ﬁv wa— ﬁv w-ﬁ" ”(s5 — c111,) 9038 — am.) at cle (0,0)[ my, ] __ _ v— 5 6 yb‘ :51; nine—”L; (05 E (Iv—Z51, , Cle e—5l+62€ v Eli—«E (9. o)[ cle-M‘We I ‘3’ 5—1: au—aﬁwﬂélge t or? The solution for the equilibrium 91:23: , ng—iyf; or a similar program. The solution' IS quite ungainly. The equilibrium solution at (0, 0) is unstable. The stability of the solutions at the other three equilibria depend upon the y____8-av au— ,88) ) can be found using Maple magnitudes of a, [3, y, 8, ,u, and v. ' 27 34 ' - ' ' CHAPTER 9 INNER PRODUCT SPACES W Exercises 9.1, pp. 419-421 2. 0, JE, JET, up 4. —6, Jﬁ, JS—l, cos-1(_6/~/i?8'1) 6. 0, ﬁ, ﬁ, 31/2 8. (b) —13, @, Jﬁ, cos-‘(—13/J3T9) 10. diagww—l, m, . . . , m 12. (b) 1/7, 1N6, 1/(2JE), cos-1(4Ji/7) Exercises 9.2, pp. 429-430 2. (b) —v1/~/§ + (1N5 + 23/3122 + (1N5 — 13/3018 + (1N5 + 5/3)v4 3N??? —2/«/ﬁ " 1N5 1N6 —1/J§ 4'[—2/~/ﬁ]’[—3/~/ﬁ] 6.[ o, 2N3, 1N5 I/Ji —1/J6 1N3 0 1/3 2/3 '—2/3 3 1/3 *3/9 2/9 —2/9 ' 2/3 ’ 2/9 ’ —5/9 ’ —4/9 —2/3 "2/9 —4/9 ~5/9 10. 1/45, «Ex/J5, 3J1‘0x2/4 — «55/4, sﬁx3/(2ﬁ) — sﬁx/(zﬂ) 12 1N5 «5/15 ' o ’ —J§/3 1 0 0 20. . 0 , —3/5 , “4/5 0 4/5 —3/5 Exercises 9.3, p: ‘439 «5/2 Ji/z - 1——6 [mrH 1] ...
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107 CH 5,6,9 Answers - CHAPTER 5 LINEAR TRANSFORMATIONS AND...

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