Fall 99 final exam solutions

# Linear Algebra with Applications

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Unformatted text preview: 1. [6 points] A matrix M is of the form 0 * 5 =1: * :1: =1: * ’ o Where as usual the *’s denote unknown and possibly different real numbers. Given that M is in row—reduced echelon form, ﬁnd all possible M, and explain why there are no other possibilities. For each Of the M that you have found, determine its rank, image, and kernel. Chen .‘5 “UH-Ll (1“:0: an: 5. M4 _ M [n Vow-wheed; dyovm. A bin“ H“ a“; "U B “A an an“. “~54 Lave-n leaning :L Began 1H“. Hon-LMIQh'IYYJ Hem! an: l- Evvxl 0M m a. column ui‘l’L n £9140": i 1M“; L” 35’5- ku 4122 2 0 ' ’ . 1.? a we," non“ ave J'HuM \ka could Ajqu 'Hv-L SECOWJ WW “W MA km“! 5‘6th 5 H“ 600ka L: lid-[0v Mal 5'? a\2-._"‘LiaL Pr MA Vow*wdﬂeeo' Icema- ﬁ '9“ * , F “ a1“: 0 . M Mui'l; lot a .‘L so,er w-leat "Rd “Fwd noh'zum l? 0. WW hoH'Z-Mo I I «‘3 woulqt L: we Mace :4 I: I". 14-th Column 23 ' GH¥1‘79£‘HU Stcﬁnat run. G? a [again-.3 . ' HUM aunt M“ 015* '. Thus Rum-LL“? “ on o a: _ I. W avi- a: f 1‘! ahaolﬂnj ii 44.4% a‘q q? WaIwﬂl #111 r; L7 Mi 2 c: Z . MW): 2 .‘MJJMJT-Wﬁium 2 {Rain . tau/(114,): SIMEP] [~95]; ' I I 3 ," 3 . w a s was” an, rm kw;st n- 'ILnna~vw~nJoi°“‘L"‘*H= - 'Z"[ I M2: [0:5 1 mums—l Meme.“ 31W“le I 0000 o N am 0 I o 1 . ' I " 0 ' ° 1 ' a“: :i “brat”! Ml- Lc a no»; Iv! Lam: a“ Waikiki-ms.) 2. Consider the matrix n—w—Im r—Irqr—a a i]- a) [4 points] Construct an orthonormal eigenbasis for A. 1‘1 _—| "' 11—A : u; 1—2 -1 — "l -I 3—2 Ask (LI-M: (1-2134 -—l — six—z) = A3“ exact) —~ ti - =l (m-Uhhsxm} _ = (yum—«3 A («as u‘rMVfa-[ws ! (ﬁg-MALI) owl “I (43......H. i). I " l I ‘ =2 luxh-I—AY: m H ' 4 Pre£(l‘I“A\ - [a o o] ) Ea - Sf - I - o o o J, O O I J; 1, ' l 0 -! ' ' _ . w! wz. We? (LI-I'PA) 2 I o l _| y E"[ = “(4.1-” O 0 0 - [I _ " SfaM § 1 i ' - “G. _ TL” iﬁslwtp’s} "‘ “WM u‘tmlvut} ‘erA’ 3 LIA". A}! up; ou‘uaeuura-QL b) [2 points] .Is A diagonalizable? Why or Why not? _ Yes, SMQ we IMAM WE‘VUJAJ M(0V{'Ldmvm1¢” uTAL-ﬁu - Le+ S: .9 1’ 17:. . _I_ I m, 3As;['c‘f§] I, 004 3 G, : 2‘ 0 - qtvbwkﬁﬁq'ﬁter m lfgnomthJ 2- i, 3;» ‘ L5; 133: a}. = —\- [ I ] I [(\$311 03 1 - «T3? Iii moi 'ohﬂh‘lﬁl ' aatha-g'rs «C» A. 4c) [4 points] Using parts (a) and (b), compute A2000. _.!_. ._ \E J: f: o .53.. 1 a: T3 .. 1 o o B:- % A3 -* 0 £ 0 _ a o "l ions 4 ' 9.000 —1' 200° 3. Let:B be the matrix ‘ B = QCCTOLJCF} II a) [3 points] Find all eigénvalues of B. 3—1.1. .051 XI- B 7- —|.3 ’).-o.‘{ HUI—Bl: [Ur-1.2K)— om + (ova-allitk—mm-0.2) + (o-‘l\[1.&,)] I: [mt-ll) +0.H€+0.51][)f—2_). +0.“, 4 I.o‘{~\ 9?— LH + I)(A‘—~.2> +2) .lgz Z - 7. b) [3 points] Does the dynamical system ' _u .: II A —IIIII H" I" m + 1) = 3m) have a point 'of stable equilibrium? Why or why not? “is is a ﬁscal-c ClYUmwu‘csl 51‘4th . _ ‘ ctull‘lﬁv‘la. “femur wl‘m {L4 I MJOLJ“ a; EILE— aﬁrmuqlwr are less Hun- d. . This Is Ito-l q“ cat! Luv-t “CW-{St ; [Anal 2‘ “Ml hull: ‘5 4 c) [4 points] Describe qualitatively the behavior of _this dynamical system if 5:10) is the unit vector OOH Wing] 10¢ lwi'Hu‘ as a Mum. WLMqL,m ‘f- “u dathvnibrs s. L wt m2 5.1m ;+; HM wt ewe Cmer‘Ll “'4 Euro. 00 -0 8m Ii,[=\ m (ALI-sh ‘l‘rajgol'avv "‘C [H M ellt‘PSC . ' 4. Consider the linear transformation T : cw —) 000 given by T0) = f” — 2f’- a.) [6 points] Find e11 real eigenvalues of T and their corresponding eigenspaces. ' L61 ’XelR “475%.” A a, M. e.~.Kaa.al.a (1.. T; m rm: H' 4c... am in a”; ' 3W“ f"—2£' = )1? f"— M“— M3 = 0 ma :5 23am“; +6 43mm: Ha Lanai .0 LL, mar alﬂécuwkal new!“ '11-?» = cw—a c” (12131? = £"-— ZP’—- H. m! u .I 'T- t“ =- 1“ maL .rln, 19 l ’ HIM . K 2“ m, In, ro-i's 7‘: 2:04; 43 a grin: “use roe-i3- an oiu‘s‘i'ihci' wales: )H'h—l - wLUM “a. Iro.{1: of “a; eLAFtPILi. 0N ans-MC“, M 361...)? + - — i 4 Law [134‘ ‘-" srcw- i EMS“ e 1 +3 g I H: A) ‘4 J Hun “use {10: Luis ‘thn ‘ms an; read} ' ' (“WW MJ u»! (my: Ex: MCI-“'3‘: Tug? “1 I}<"“,Hwn MUSQ thﬂ} clvmelu: I I (|+:\]—l"3)'e : + a EPA—E)- C J C(l'i\.]“l-A H 2' e{(mm1c n. [maﬁa-i i) eU-ﬁEHi' E ml w-c lam E)“:— iwr (T-ﬂ 3? Sf‘w‘iet. cmd'l‘k 4:, b) [2 points] Let T2 be the same linear transformation restricted to the subspace P2 of C°°, consisting of polynomials of degree at most 2. (That is, T2 : P2 -+ P2 is the transformation taking any polynomial f of degree at most 2 to f” — 2]“ Choose a basis for'Pg, and write the matrix A2 of T2 with respect to this basis. Pz has a mach/ml Law's g 1,1311} . I 0 gm = 0 mi manta. Leda». “rib”? “lilfﬁl 2 "Z 0 "Z 7. ' “Wt-x": [a] T253): 2-H“? ' 7- e 0 4‘ l ‘ o O 0 c) points] Find the image and kernel of this matrix A2. Check (part of) your work by explaining the relationship between parts (a) and (c) ' o l -l' D 1 NA”) = O o I = o o o o ' o- o o I WW: ‘F‘MHOR = W?“ t _ _ o T“. rm; [ms Cut» Magus: 5f“ =' steel-1, 2-4; 7: srm {1,53 0 l 0 Am that-m x Lama}: LurCTJ Amt; 45.14;; Lam, LN... drum o, W: linen; Cm. v“! («1" +Lul' '1“, (T) :_ ED .—. 5r“ i e 2e) 6::ch r I ' 3 5!“ la“; 4—? m at; a cam 41mm, Ls e“ 94. P1. 5.‘ r1_‘he following continuous dynamical system models the populations a:(t), y(t) of two species: 25:“: = (“y-1h: 6y _ a - wiry—3):} - We consider only the ﬁrst quadrant a: 2 0, y 2 0 since we are modelling populations. a) [2 points] Sketch the nullclines of this system, and ﬁnd any equilibrium points. " ‘ . 12-0 ' b) [4 points] Use linear approximation to ﬁnd the J acobian J of the system at any equilibrium points from part (a). 39-"? £(xl‘3) : 03*“ x' . it i‘W’ ("u—3): 9’: - 93 - 5-; : (a l -- ‘X 94 = 9 = — 9x L; 5% 31-23 3 9 'I BI a I V n {215% _ Sikh) EEC M x a 9 9 a: -lo 53 5% a.) an GM ‘1 - c) [4 points] Analyze the stability of each equilibrium point by computing the eigenvalues of J. 30)] La: eigenvalue: ' “'i ) ‘3 sﬁue H {s Amaimi. Be” on Maui-I've mo‘ rial, s. P f: a r1}n+o£ six“. eﬁhilgbh‘ym (mm: 3 < (Lg)1 , go ) i has thvmiuu 3 ss‘hce H is [aw-Ur Water. 30% m Posi‘l'iw: anal Mai, 51: G. .‘s m“- a fu‘h-i a-c AU: zﬁv'ufil-w‘um, } ‘ ' (méelze<(§) o a #- I. as Jam“. 0 - i: 0).: 3‘1“- )‘_'Z' OCR] L {mu : (34.10“) MA Lume- [mu-s uYMthlu-Ls Z )._.l _ I W141; ovu. [Mdkuc-GMJ am M3 We, K h me a raMi «C'si'aue ab. 9 d) [2 points] Sketch (approximately) the phase plane of this system, including behavior near equilibrium points and approximate direction of the ﬂow lines in the regions separatedby the nullclines. 7 10 6. Consider the temperature T(3:,t) of a metal bar extending from a: = 0 to m = 7r._ The temperature satisﬁes the heat equation a_T 32:" 31::“5585' and the ends are held at a constant temperature of zero, i.e. T(0, t) = O and T(7r,t) = O for all t Z O. a) [3 points] Show that the functions e‘mztsinmm) satisfy the diﬁerential equation and the initial conditions for all positive integers n. 1 - mt . Lgl- Tn(z,tl= e F 'Mnx. 2 “MM 9.1:: __ *ﬂnl. eh/‘aé Jaw-MK _ ¢£ - J4}: : n- e [m m M‘X X 2 z . LE: _“1.eH/uu+.miﬂx x = .. “w .- .H How“ 25% m); El? — “fun?- 2 [A max "'/un.(---|r\2‘-e/M flown?!) a = _O. ._ 2+ I" .. A‘59’ 8 MM 0 —- O "" “1£ , b —- Ila “GIN Elma: 8’” AM" WW " o 5 ‘9 11 b) [7 points] Suppose now that'the initial temperature of the bar is given by , T(x, 0) = 0(3) = sin3 3:. Determine T(93,t) for all a: E {0,7r] and all times t 2 0. [Hint Use Euler’s formula eiy = cosy + isin'y if you are not sure about the relevant trigonometric identities.] Examine the behavior of T as t —> 00. m 74.31; . Wt Inf“; a “Lei-1'“: dc ‘Hu. ﬁfe: Tctxff) :- Z L)“. e . m max . ' “=1 , n g ' m I I mad-val («Adam W“ "‘4' Tbsp) ’3 Z. by. 74".“ “x h:\ ~ 3 TCx,0)_= AMA. " " F; I hi 56 we. 4L Cwei' axPnTSI‘bm .5 "Hut W‘u'ff 9“ M k . we lure gt a £5034 IMP Laval..qum op 4L.“ gucg‘onr ﬁu‘l (3:510? ‘ 'HM. Mﬁ‘r‘ai raw-43““. {helm sum“ Max 1} a night and farm u I Is 3‘! . gauge“ 0min:ng Las‘c +ﬂhjaﬂ°wtklsc ﬁanc‘l‘l‘m’l F' 5"“: H El... 3 _ Euiu‘s For-“4*: e?)Ex -= e '13“) = mo 37: +1445“ 37: . . r 3 33"“ = (enjs ':—-" (co-é ‘K + £45.0- 7c) 7 . 2 - = c.ng + 3:. we K WK 2 n '5 --% WK» X“‘£¢u'm 'X -1 "2 (m3x~3mx°w K) . '3 1 . + 1(3mxmxr—‘m Mg'i-oL-Vla “awaken, for“! Yigidci ' 3 2 . - a c K“ K m3K -- 3W7‘m J‘W‘vg '—-» 3(|—-,¢;..‘x Wit—M K ' .3 ‘2‘ sycax-r Lia/WK . i . Tm: Bﬁer.w3¢5 MAM“ .M'ng = "meé—ngz {2’33me Ham 1 F. w N5 Mmbui- ...
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