FIRST PRACTICE EXAM FINAL Math 21b, Fall 2004Name:MWF10 Janet ChenMWF11 Oliver KnillTTh10 Oliver Knill• Start by writing your name in the above box andcheck your section in the box to the left.• Try to answer each question on the same page asthe question is asked. If needed, use the back or thenext empty page for work. If you need additionalpaper, write your name on it.• Do not detach pages from this exam packet or un-staple the packet.• Please write neatly. Answers which are illegible forthe grader can not be given credit.• No notes, books, calculators, computers, or otherelectronic aids can be allowed.• You have 180 minutes time to complete your work.1 202 103 104 105 106 107 108 109 1010 1011 1012 10Total: 130Problem 1) (20 points) True or False? No justifications are needed.1)T FIf A is a symmetric matrix such that A5= 0, then A = 0.2)T FIf A and B are 3 × 3 symmetric matrices, then AB is symmetric.3)T FThe solutions of f000(x) + f00(x) + f(x) = sin(x) form a linear subspace ofall smooth functions.4)T FThe initial value problem f000(x)+f00(x)+f(x) = sin(x), f(0) = 0, f0(0) = 0has exactly one solution.5)T FEvery real 3 × 3 matrix having 1 + i as an eigenvalue is diagonalizable overthe complex numbers.6)T FIf A is a nonzero diagonalizable 4 × 4 matrix, then A4is nonzero.7)T FThere exists a real 2 × 2 matrix A such that A2=−1 00 −1.8)T FThere exist invertible 2 × 2 matrices A and B such that det(A + B) =det(A) + det(B).9)T FThe kernel of the differential operator D100on C∞() has dimension 100.10)T FT f(x) = sin(x)f(x) + f(0) +Rx−1f(y) dy is a linear transformation onC∞().11)T FIf S−1AS = B, then tr(A)/tr(B) = det(A)/ det(B).12)T FIf a 3 × 3 matrix A is invertible, then its rows form a basis of3.13)T FA 4 × 4 orthogonal matrix has always a real eigenvalue.14)T FIf A is orthogonal and B satisfies B2= 1 then AB has determimant 1 or−1.15)T FIfddt~x = A~x has an asymptotically stable origin thenddt~x = −A~x has anasymptotically stable origin.16)T FIfddtx = Ax has an asymptotically stable origin, then the differential equa-tionddtx = Ax + (x · x)x has an asymptotically stable origin.17)T FThe transformation on C∞() given by T (f)(t) = t + f(t) is linear.18)T F~0 is a stable equilibrium for the discrete dynamical systemx(n + 1)y(n + 1)=1 −11 1x(n)y(n).19)T FIf A is an arbitrary 4 × 4 matrix, then A and ATare similar.20)T FIf A is an invertible 4 × 4 matrix, then the unique least squares solution toAx = b is A−1b.TotalProblem 2) (10 points)Match the following objects with the correct description. Every equation matches exactly onedescription.a)ddtx = 3x − 5y,ddty = 2x − 3yb) ft= fxx+ fyy.c) D2f(x) + Df(x) − f(x) = sin(x)d)ddtx = 3x3− 5y,ddty = x2+ y2+ 2e)ddtx + 3x = 0.i) An inhomogenous linear ordinary differential equation.ii) A partial differential equation.iii) A linear ordinary differential equation with two variables.iv) A homogeneous one-dimensional first order linear ordinary differential equation.v) A nonlinear ordinary differential equation.Problem 3) (10 points)Define A =1 −2 3 −4−5 6 −7 89 −10 11 −12.a) Find rref(A), the reduced row echelon form of A.b) Find a bases for ker(A) and im(A).c) Find an orthonormal basis for ker(A).d) Verify that ~v ∈ ker(A), where ~v =0242.e) Express ~v in terms of your orthonormal basis for ker(A).Problem 4) (10 points)Find all solutions to the differential equationf00(t) − 2f0(t) + f(t) = 4e3t.Find the unique solution given the initial conditions f(0) = 1 and f0(0) = 1.Problem 5) (10 points)a) Let f(x) be the function which is 1 on [π/3, 2π/3] and zero elsewhere on the interval [0, π].Write f as a Fourier sin-series.b) Find the solution to the heat equation Tt= µTxxwith T (x, 0) = f(x).c) Find the solution to the wave equation Ttt= c2Txxwith T (x, 0) = f(x) and for whichTt(x, 0) = 0 holds for all x.Problem 6) (10 points)Find a single 3 × 3 matrix A for which all of the following properties are true.a) The kernel of A is the line spanned by the vector111.b)101is an eigenvector for A.c)12−1is in the image of A.Problem 7) (10 points)a) Find all solutions to the differential equation (D2− 3D + 2)f = 60e7x.b) Find all solutions to the differential equation (D2− 2D + 1)f = x.c) Find all solutions to the differential equation (D2+ 1)f = x2.Problem 8) (10 points)Find the matrix for the rotation in3by 90◦about the line spanned by221, in a clockwisedirection as viewed when facing the origin from the point221. You get full credit if youleave the result written as a product of matrices or their inverses.Problem 9) (10 points)a) Find the eigenvalues of the matrix A =1/2 −1/21/2 1/2.b) Is~0 a stable equilibrium point for the linear systemd~xdt= A~x ?c) Describe, how the solution curves ofd~xdt= A~x look like.d) Is~0 a stable equilibrium for the discrete dynamical system xn+1= Axn?Problem 10) (10 points)Does the systemd~xdt= B~xwithB =0 −1 −9 −9 −80 0 0 −1 −95 0 5 0 −51 9 0 0 01 9 9 8 0have a stable origin?Hint No lengthy computations are needed. Especially, no eigenvalues have to be computed. Ifλ1, ..., λ5are the eigenvalues, can you say something about their sum?Problem 11) (10 points)A 4 × 4 matrix A is called symplectic if AJAT= J, where J =0 0 1 00 0 0 1−1 0 0 00 −1 0 0.a) Verify that J itself is symplectic.b) Show that if A is symplectic, then A is invertible and A−1is symplectic.c) Check that if both A and B are symplectic, then AB is symplectic.d) Show that for a symplectic matrix A, one has det(A) = 1 or det(A) = −1.Problem 12) (10 points)Find the ellipse f(x, y) = ax2+by2−1 = 0 which best fits the data (2, 2), (−1, 1), (−1, −1), (2,
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