MATH401 Exam 3 Spring 2004

MATH401 Exam 3 Spring 2004 - Prof Dancis MATH 401 Exam 3...

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Unformatted text preview: Prof. Dancis MATH 401 Exam 3 April 30, 2004 Heed these instructions: Use efficient methods for calculations. Calculations should be labelled and easily understood. Laptops and hand calculators are not permitted. Each time that the theorem about solutions of (all) homogeneous linear equations is used (implicitly or explicitly) in your calculations during this test, indicate this by stating “theorem about homo. lin eq., at the point that the theorem is used. Same for theorem about solutiOns of (all) non-homogeneous linear equations Use this info to shortcut calculations. Credit will be deducted for not using this info. If M:(“ b c d (‘5 51) (i)=<a+b>(i) (Z 3) (31)=<a-b><31) 1.(a) State the spectral theorem about real symmetric or hermitian matrices. 4 points ), then A1” = detIM (:10 :lb), whenever .det .M ;A 0. (c) Prove that every square stochastic matrix has one as an eigenvalue. State (without proof) the two other propositions that you use. 8 points (d) Prove that the product of two 7 x 7 unitary matrices is a unitary matrix. 6 points (e) Given that matrix, lbs diagonalizable and given that matrix, N is _s_i_1_r_1ilar to matrix, I”; prove that matrix, N is also diagonalizable and has the same eigenvalues as matrix, (ll/I. State (without proof) the other propositions/theorems that you use. 7 points (f) Prove that the complex eigendata for real matrices occurs in complex conjugate pairs. 6' points (g) Let .M, P and D be constant 7 x 7 matrices; Al and P are invertible. Let v = v(t) E R7 x R, be a vector valued function of time. Let W : w(t) = Pv(t) and Ill—1K = P_1DP(1,71.(11\I\"/(t)+ Kv(t) = 0. Show that + Dw(t) = 0. 10 points (h) Given that AI is a 3 x3—matrix, with top row (0. 3, 4). Suppose that v : (:r. y. z)T, with 1‘ 2 y 2 z > 0, is an eigerwector of AI associated with eigenvalue A. Show that, A g 7. 8 points OVER 2. Find a series solution, for u(:r,t), on the domain 0 g :r 3 system of equations: i, and t 2 0, to this a.” = 9m and 11.1(0, t) = 0 = 114%. t) (Reminder: uI = 2—2) Do NOT skip steps in your calculations. 1.9 points State the eigenvalue problem in a box. 2 points Is zero an eigenvalue? Why or why not? You may assume that there are NO negative eigenvalues. 3 points 3. DO NOT SOLVE this system of equations: u” = 7m + 8sin 2x and u(0, t) = 7 and u(7r, t) = 9 and u(m, 0) = —2 sin 22 + 7 + 7sin 9x + 5sin12x on the domain 0 g :r 3 71', and t 2 0. (a) Write this system in linear equation form. 2 points (b) Write down the system of associated homogeneous equations. 3 points (c) Given that two functions, u(:c,t) and v(:r,t) satisfy the equations: um 2: 7m + 8sin 2:1: and 11$; = 711; + 8sin2cc. Show that w(:r, t) = u(:c, t) — v(x, t) satisfies the associated homogeneous equation. 2 points ((1) Suppose that. some of the given data for u(:z,t) was obtained by measurements or tolerances. Call u($,f.), the calculated solution. Let -v(.r.,t) be the real solution. Specifically, suppose that 11" 1 2 |u(0, t) —- v(0,t)| g — and t) — v(7r,t)| S —, V90 10 10 ‘ 3 and lu(:r,0) — v(a;,0)| S m, VOSIS7T Find a good bound on t) — 'u($, t)], VQO and US$377. State any theorem or corollary that you use. 10 points 4. Given vn : fly/112,14, Vnez+. Show/prove that 114 = 111400. 4 points 5. Graph y — 5x = i(5y — x)2 in the my—plane. State the change—of —eoordinate matrix. 7 points How many hours per week do you study for this course? 1 points Total 100 points [\3 C005, 0 's cum-*1: 3.— go rows cup Sr svw i01— Ez‘f NAM ad Jaw W5 «no e474,- Mwsm I} am eguiv‘v/M-O‘ /;_’L )Q( 5’ anal QFS L?" U\ “A N \afi‘fimfim ms 4 g M -| \ ‘ U " u N f“ N? J x, 4 -\ -\ W" “‘9” (run) ; 04%) U,\ ,N ' u» er<NU3* \“AvS NUx ‘ ' ls 0~ Uw‘kv w. aaso aw?- % ' e> 51 A : than M‘H V ‘\>.c\\‘€k LEM?” ; e “W = a” PA ~ ,o Maw-fish ozfla]bfl2= " ’ 2F RR r (PR3 DER): r a \ I v > 7 N r E \ \' ' - I \| ,_ ‘k ,, \ ‘ {v .- a “Page vk / (3\ momma: M"K.=v“b\> M§@+\<v@=o M" K vl’e‘>= mefl /\/\~\v\ var): ?-\DW&) 3: We} NW «(4): o 1%»: ,9“wa o few), :54): 0 wk 33mm (3 / «a a W \m e N‘ => X/KM‘N” “5 7x #0 baawse. “Hm ws +0 X(’>‘)"o mm“ '5 Milt) JL, I “T 4,—- ‘Tkrv, b £K.ckP,4@% L" q zét fl' VSA:V*)aCe,fi \ CeliC, e. .- 1; ’h 2 (Am =A><®Tct> =De. 3mm”) Vhe N.,. \mmmo. D :_ AQ_ L903 = M (“803 “awed +4 flame”) -3 ">7 3% (\z'x) Q08 H60 : 0: UV)” ;t?u*-+€sm(7—"§‘ Y LL(V3 LO : “(Ofbx L3LV) =0 2 “(1‘be l I l y i LL‘ (V) : O : \A (mflB‘ ’ZSin(z’Xj-2?T:?s{h(qi) E X g i i 8‘) W (73‘?) '” RCWWB 'V<*It) >24“ +ng’5 » 7Lth 1; 7' (ub’vt 605306,. \AOMD. 219. “ML :0 :‘ \f/x’x 7K int-kw =qrvt 7/9923} E 21W: =/¢Vt= 60 N (“its R? (u‘t ‘U‘Q ‘0 \/ Wm flown e g) buxst “Va/Vs (.5 Max'va anommce, '4) \/H : /Vlv3 V3; Mvz Prof. Dancis l\/IATH 401 Exam 3 Some Answers April 30, 2004 1.(a) State the spectral theorem about real symmetric or hermitian matrices. S : O_1DO, where D is a real diagonal matrix and O is a real orthogonal matrix. QR S = O‘lDO : OTDO, where D is a real diagonal matrix. Crucial that you state that D is a real (diagonal) matrix, and either state or indicate that O is a real orthogonal matrix. Or Theorem V1129 (c) Prove that every square stochastic matrix has one as an eigenvalue. State (without proof) the two other propositions that you use. This is Proposition V1.29. (d) Prove that the product of two 7 X 7 unitary matrices is a unitary matrix. Given: A and B are unitary matrices, that is A‘1 : 14* and B—1 = To prove: AB is a unitary matrix, that is (143)“1 2 (AB)*. Proof. (.AB)‘1 : 3414—1 : B*A* : (AB)*. \/ " ,1 ,‘ ‘L (e) Given that matrix, 114 is diagonalizable and given that matrix, N is similar to matrix, M'; prove that matrix, N is also diagonalizable and has the same eigenvalues as matrix, MI State (without proof) the other propositions/theorems that you use. This is Proposition V1.3.14. (f) Prove that the complex eigendata for real matrices occurs in complex conjugate pairs. This is Proposition V1.73. (g) Let 111, P and D be constant 7 X 7 matrices; JW and P are invertible. Let v : v(t) E R7 X R, be a vector valued function of time. Let w : w(t) : Pv(t) andll/[_1K = P“1DP and MW) + Kv(t) : 0. Show that + Dw(t) : 0. 10 pu‘L'rLtS Pay? Mm) + Kv(t) = 0 => 0 : wt) +A/[_1Kv(t) = {/(t) + PAlDPv. Multiply by P on the left yields: 0 : PV(t) + DPv : we) + Dw(t) y. (h) Given that [W is a 3 ><3—matrix, with top row (0, 3,4). Suppose that v : y, fly], with at 2 y 2 z > 0, is an eigenvector of NI associated with eigenvalue A. Show that A37. This is similar to Exercise V1.25, which was done in class as board work. The eigenvalue/ vector equation is 11/1120 2 A120. Look at the top coordinate of this equation: 3y+4z=mfixz33+4ig3+4=7 ¢ .71; I 2. Find a series solution, for u(1‘,t), on the domain 0 S 1' S i, and t 2 0, to this system of equations: u“. = 9m and u$(0, t) : 0 =u1.(fi,t) (Reminder: um = Do NOT skip steps in your calculations. State the eigenvalue problem in a box. 1s zero an eigenvalue? Why or why not? You may assume that there are NO negative eigenvalues. This is similar to Example 11.7.2. Some selected lines, which are modifications of Example 11.7.2 follow: since the only way for g(:r) : h(t) is for both functions to be constant. The equations for X are the eigenvalue problem: (a?) X”(.r) + /\X(1:) = 0 is a Basic Spring Block Equation. Its general solution is: L(X(:c)) : —X”(:r) : /\X with boundcfij conditions : X’(G) : O z ‘ : A cos fizr + Bsin \/:\—CL', VA and B X/(IL‘) : —A\/:\—sin Vim + Bficos \/X..T7 VA and 3 Plug in the boundary condition: 0 = X’(0) : —A\/Xsin0 + B\/Xcos0 2 BA :9 B = o. This simplifies X’(:L') to: X’(.7;) : vAfisin Plug in the boundary condition: 1 1 l . 0:X’(1): —A\/XsinZ\/X:> /\:0 or sinlfizo. 1 1 _ sing”: O :9 EVA = MT :3» /\n :— 4R7T7Vnez+ The eigenfunctions are X.n(a;) = cos4mr1',\‘/neg+. To find Tn(t): Tm) : —%AnTn(t) : —(4n7r)2Tn(t). q 2 —(~inr\21 Tn(t) = e 9 This is an Exponential Decay Equation. ——(4nt)2! un(.r,t) :2 Xn(:L')Tn(t) = e 9 cos 471:1, Vn=1,2,3,.... All linear combinations of solutions, to a system of homogeneous linear equations, are more solutions. um t) = 2:11 um, t) = 2:1 ‘ Can ,\ = 0? The equation, X”(a;) + /\X(r£) = 0 will simplify to X”(:L') = 0 => X(a:) = C3: + D,VC and D. This, together with the bondary conditions will result in a non—zero eigenvector Xo(:E) : 1. So yes, here, /\ = 0, with To(t) = (30‘ : 1. Hence U0($,t) = X0($)To(t) = 1 Taking the linear combinations of no and u, above, yields: COS 47171-93: Vconstants an, n=l,2,3,--- é° —(4'n7.-)2L ane 9 COS 4n'7rxy Vconstants an, n=0,1,'2,3, 3. DO NOT SOLVE this system of equations: um : Tut + 88in 2:1? and MD, : 7 and u(7r) t) : 9 and u(:z:, 0) : —2sin 21' + 7 + 7sin91r, + 5 sin 12:7; on the domain 0 S :1: g 7?, and t 2 O. (a) Write this system in linear equation form. Linear—equations have the form7 L(v) : we, where wo “input” variable v. is not a function of the L1(u) : um. — 7m, : 8 sin 21‘, Lg(u) : u(0, t) = '7, L3(u) : u(7r, t) : 9 L4(U) : u(a;, 0) = #2 sin 21* + ‘7 + Tsin .92: + 5 sin 121:. (b) Write down the system of associated homogeneous equations. L100) : rum. 7 71m : O, Lg(w) : 10(07 t) : 07 L3(’U}) : 10(711, t) : O. L4(w) : 10(23, 0) : 0. (0) Given that two functions. u(:L', t) and v(;1:,t) satisfy the equations: uxI : Tut + 8 sin 2:6 and UM : 7m + 88l1l217. Show that t) : u(:L', t) — 11(2), t) satisfies the associated homogeneous equation. in“. ~ 7w: : [11(10) 2 L1(u) — L1(v) : 85in21‘ —— 8sin21L' 2: 0 That is7 the difference between two solutions of a non—homogeneous linear equation is a solution to the associated homogeneous equation. (d) Suppose that some of the given data for was obtained by nmasurements or tolerances. Call u(:t,t), the calculated solution. Let v(1‘,t) be the real solution. Specifically, suppose that l 2 |u(0,t) —'U(O.,t)| S E and fa(7r,’t) — U(7r,t)l 3 E, V/tio duo) (on/3v . LL‘ —/ "\ —— \ v./r:»r an u , u , _ 10, O#w_ Find a good bound on that you use. Similar to Example 11.8.7. Answer. Set w(x,t) : u(:,t,t) — 'U(:c,t). Then w(:L',t) satisfies these equations and inequalities: Lfiw) : wag,” — 7w; : l 2 |w(0,t)l S E and l’iu(7t,t)l _<_’ 15, V20 dl < 0)! f 3 v ', w r' \ — 1. 7 an L, _ 10, 03 g The top equation is a heat equation, without a heat source. Hence the Maximal Principle and its corollary apply to 10(1‘, t), The corollary say that when w(:L', t) satisfies a heat equation, wit/tout a heat source, (10” : 6211.135) then upper and lower bounds of w(.r,t), on the Sides and “bottom” of the domain are also upper and lower bounds of w($, t) on the entire semi—infinte strip of a domain. Here ‘13—0 ’1’. w(;t,t) ii %, for all points (at) on the sides and “bottom’7 of the domain. Hence the corollary implies that —130 w(_:c,t for all points (:c,t) on the entire semi—infinte strip of the domain. The Maximal Principle and its corollary do NOT apply to a heat equation, with a heat source, like um. I Tut + 8sin 2.95. \ / ,, l 4. Given an : 11/11)”; 1, Vnezir. Show/ prove that '04 = 114/4120. This is Proposition v1.4.1 5. Graph y — 5:1; : ::(5y — fl])2 in the xy—plane. State the change—of -coordinate matrix. 1 - - 1‘1 —1 5 ’1' _‘ Answer. Set yl : y — 01' and :51 = by — :L‘. Then : , . bo the yr —0 l y . . . —l " _ r change—of —coordmate matrix 18 <_5 y # 5:5 : i<oy — ZL'JZ becomes yl : :17, whose graph is tangent to the fol—axis at the origin.__ The $1—axis is the line 3} = 5:5; and The yl—axis is the line 53/ : x. When the graph is moved to the zy—plane, it remains tangent to the ccl—axis at the origin. ...
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MATH401 Exam 3 Spring 2004 - Prof Dancis MATH 401 Exam 3...

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