first_final_practice_coverpagefinal_practice1aFIRST PRACTICE FINAL EXAMINATION Math 21b, Fall 2007 MWF10 Evan Bullock MWF11 Leila Khatami MWF12 Yum-Tong Siu 1 20 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 10 12 10 13 10 Total 140 • Start by writing your name in the above box and check your section in the box to the left. • Try to answer each question on the same page as the question is asked. If needed, use the back or the next empty page for work. If you need additional paper, 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 for the grader can not be given credit. • No notes, books, calculators, computers, or other electronic aids can be allowed. • You have 180 minutes time to complete your work.Problem 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 f′′′(x) + f′′(x) + f(x) = sin(x) form a linear subspace ofall smooth functions.4)T FThe initial value problem f′′′(x)+ f′′(x)+ f (x) = sin(x), f(0) = 0, f′(0) = 0has exactly one solution.5)T FEvery real 3 × 3 matrix having λ = 1 + i as an eigenvalue is diago na lizableover the 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∞(R) has dimension 100.10)T FT f(x) = sin(x)f(x) + f(0) +Rx−1f(y) dy is a linear transformation onC∞(R).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 ofR3.13)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 determinant 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∞(R) given by T (f)(t) = t + f(t) is linear.18)T F~0 is a stable equilibrium for t he 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 obj ects 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) x + y = 3, 7x + 3y = 4 , 8x + 5y = 10.f)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.vi) A system of linear equations.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 equationf′′(t) − 2f′(t) + f(t) = 4e3t.Find the unique solution given the initial conditions f(0) = 1 and f′(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 fo r 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 f or the rotation inR3by 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?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, −1).Problem 13) (10 points)We analyze the nonlinear system o f differential equations˙x = x2− y2+ 3˙y = −x + 2y − 3a) Find the nullclines.b) There is one equilibrium point. Find it.c) Find the eigenvalues of the Jacobean at the equilibrium.d) Is the equilibrium stable?
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