FINAL EXAM Math 21b, Fall 2003Name:MWF10 Izzet CoskunMWF11 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 10Total: 120Problem 1) (20 points) True or False? No justifications are needed.T FAll symmetric real matrices are diagonalizable.T FThere exists a 3 × 3 real symmetric matrix whose Jordan-normal form isi 0 00 −i 00 0 3.T FIf A is any matrix, then both AATand ATA are orthogonally diagonalizable.T FAll orthogonal projections are diagonalizable.T FIf the regression line y = ax + b obtained by fitting some data{(x1, y1), ..., (xm, ym)} happens to contain all datapoints, then the corre-sponding least square solution of A~x =~b is an actual solution of A~x =~b.T Fdet(1 2 2 22 1 2 22 2 1 22 2 2 1) = −7.T FThere exists a symmetric 2 × 2 matrix A such that A"12#="36#andA"11#="22#.T FThe kernel of the operator (D−2)5is spanned by e2t, te2t, t2e2t, t3e2t, t4e2t.T FLet A be a 2 × 2 matrix. The systemdxdt= Ax is asymptotically stable ifand only if the eigenvalues of A have negative real parts.T FLet A be a 2× 2 matrix. The discrete dynamical system A(t + 1) = Ax(t) isasymptotically stable if and only if the eigenvalues of A have negative realparts.T FThe subset of X = C∞(R), the set of smooth functions of the real line,defined by Y = {f ∈ C∞(R) : f(0) = 1} is a linear subspace of X.T FThe subset of C∞(R) defined by Y = {f ∈ C∞(R) : f(0) = f00(2)} is alinear subspace of X.T FThe operator T (f) = (D2+ 12tD + 17)f defines a linear map from C∞(R)to C∞(R).T FThe operator T (f) = (D2+12t3D+17t2)f defines a linear map from C∞(R)to C∞(R).T FIf A is 2 × 2 matrix with det(A) < 0, then the systemdxdt= Ax has 0 as astable equilibrium.T FIn the Fourier series expansion of the function t+1 on [−π, π], the coefficientsanbelonging to cos(nt) are zero for all n ≥ 1.T FIf a 2 × 2 matrix A has the eigenvalues −2, −1, then the orbits of systemx(t) 7→ x(t + 1) = Ax(t) stay bounded.T FIf a 2 × 2 matrix A has an eigenvalues −2, −1, then the orbits of the systemddtx(t) = Ax(t) stay bounded.Problem 2) (10 points)Match the following differential equations with the correct description. Every equation matchesexactly one description. No justifications are necessary.a)˙x = 3x − 5y˙y = 2x − 3yb)˙x = −4y + 2x2+ 2x3˙y = 4y(1 − x2)c)˙x = −x + 2y − y2˙y = 3x − y − xy − y2d)˙x = 3x − 5y˙y = x2+ y2+ 2e)˙x = 2y(x − y) − x˙y = y(x − y) − yFill in 1),...,5) here.a) b) c) d) e)1) The equation has a stable equilibrium at x = 1, y = 1.2) The equation has an unstable equilibrium at x = 1, y = 1.3) The equation has a non-constant solution which stays on the line x = y.4) The equation has a closed periodic orbit.5) The equation has no equilibria.Problem 3) (10 points)Let A =0 0 0 1 11 0 1 1 01 1 1 −1 −20 1 0 −1 −1.a) Find a basis for the kernel of A.b) Find a basis for the image of A.Problem 4) (10 points)Let A =0 0 0 10 0 1 00 1 0 01 0 0 0.a) Find all possibly complex eigenvalues of A with their algebraic multiplicities.b) Does A have a possibly complex eigenbasis? If so, find one.c) Is A diagonalizable? Why or why not?d) Let T be the linear transformation defined by T (v) = Av. Describe T geometrically.Problem 5) (10 points)Find the function f(x) = a + b cos(x) which best fits the data(x1, y1) = (0, 1)(x2, y2) = (π/2, −1)(x3, y3) = (π, 1)(x4, y4) = (2π, 1)Problem 6) (10 points)a) Find the solution of the differential equation f0(t) + 3f(t) = e−2t, f(0) = 0.b) Find the general solution of f00(t) + 4f0(t) + 3f(t) = 1.with f(0) = 1/3, f(1) = 1/3 + 1/e3− 1/e.c) Find the solution of f00(t) = −4f(t) with f(0) = 1, f0(0) = 2.Problem 7) (10 points)a) (7 points) Find a 4 × 4 matrix A with entries 0, +1 and −1 for which the determinant ismaximal.b) (3 points) Find the QR decomposition of A.Problem 8) (10 points)Define f = sinh(x) =ex−e−x2on C∞([−π, π]) be a function on the interval [−π, π]. Find asolution T (t, x) of the heat equation˙T = Txxwhich satisfies T (0, x) = f(x).Hint.Rsinh(x) sin(nx) dx =cosh(x) sin(nx)−n cos(nx) sinh(x)1+n2. You can leave terms like sinh(π).Problem 9) (10 points)a) Find the Fourier series of | sin(x/2)| on C([−π, π]).Hint.Rsin(x/2) cos(nx) dx =− cos((12+n)x)2n+1+cos((12−n)x)2n−1.b) FindP∞n=1(−1)n14n2−1.Hint. Evaluate f(x) at π.Problem 10) (10 points)An ecological system consists of two species whose populations at time t are given by x(t) andy(t). The evolution of the system is described by the equationddt"xy#="x(x − y + 1)y(x + y − 3)#.a) Find all equilibrium points and nullclines of this system in x ≥ 0, y ≥ 0.b) Sketch the vector field of this system in the first quadrant x ≥ 0, y ≥ 0 indicating the directionof the vector field along the nullclines and inside the regions determined by the nullclines.c) Are there any stable equilibrium points? Justify your answers.d) If both species start with positive populations, can either become extinct? Explain.Problem 11) (10 points)Consider the linear differential equation˙x = ax + y˙y = ay˙z = −z .a) Write the system in the formddt~x = A~x, where A is a matrix.b) For which parameters a is the system
View Full Document
Unlocking...