third_final_practice_coverpagefinal_practice3aTHIRD 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 FThe matrix2 −cc 3is always invertible for c ∈R.2)T FThe solutions of f′′′+ f′′+ 17f = etform a linear subspace of C∞(R)3)T FThe solutions of1 32 4~x =14form a linear subspace ofR2.4)T FIf A and B are 5 × 5 matrices, then rank(A + B) = rank(A) + rank(B).5)T FSimilar matrices have the same rank.6)T FIf A is a matrix which has orthonormal columns, then det(AAT) =det(ATA).7)T FIf A is a 2 ×2 matrix with det(A) < 1, then the discrete dynamical system~x(t + 1) = A~x(t) has a stable origin.8)T FIf A, B are n ×n matrices, then det(2A + 3B) = 2ndet(A) + 3ndet(B).9)T FIf A and B ar e 2×2 matrices with the same trace and the same determinant,then A and B have the same eigenvalues.10)T FIf A = QR is the QR decomposition obtained by Gram-Schmidt orthogo-nalization, then A and R have the same eigenvalues.11)T F If ~x∗is the least-squares solution of A~x =~b then~b is orthogonal to A~x∗.12)T FIf two matrices are symmetric and have the same eigenvalues (with the samealgebraic multiplicities), then they are similar.13)T FEvery system of linear equations has a least square solution.14)T FIf a real 2 × 2 matrix A has i as an eigenvalue, it is orthogonal.15)T FIf A = BCD, where A, B, C, D are all 3×3 matrices and A is not invertible,then one of the matrices B, C, D are not invertible.16)T FIf every eigenvalue λ of a matrix A satisfies Re(λ) < 1, then~0 is an asymp-totically stable equilibrium of the discrete dynamical system ~x(t + 1) =A~x(t).17)T FThere is an n ×n matrix A which has an eigenvalue λ of geometric multi-plicity 0.18)T FIf f(x) = 3 cos(7x) + 4 sin(2004x) + 2, then1πRπ−πf(x)2dx = 29.19)T F7 is an eigenvalue of T (f) = f′′+ 7f′+ 77f on the space X = C∞(R) ofsmooth functions on the real lineR.20)T FIf B =13,37and ~x =14then [~x]B=−21.Problem 2) (10 points)Pick the five of the dynamical system 1) - 9) which correspond to the phase portraits.1.d~xdt=0 00 1~x2.d~xdt=1 02 3~x3.d~xdt=1 10 −1~x4.d~xdt=1 00 0~x5.d~xdt=−1 0−2 −2~x6.d~xdt=1 10 1~x7.d~xdt=2 00 2~x8.d~xdt=1 7−7 1~x9.d~xdt=−1 7−7 −1~xProblem 3) (10 points)To match the dynamical systems to the left with the description to the right, fill in a)-e) in t heboxes. No justifications are necessary.a)ddtx = sin(xy) ,ddty = x2+ yb) ft= fxxxxc) ~x(t + 1) = A~x(t)d)ddt~x = A~xe) f′′+ f′+ f = sin(t)Partial differentia l equationLinear system of ordinary differential equationsnonlinear differential equationinhomogeneous linear differential equationdiscrete dynamical system.To match the matrices to the left with the description to the right, distribute a)-e) in the boxes.a)1/2√3/2√3/2 −1/2b)1/√2 −1/√21/√2 1/√2c)1/√2 1/√21/√2 1/√2d)1 0−4 1e)0 −33 0skew symmetric matrixrotationreflectionprojectionshearProblem 4) (10 points)Let A =3 1 0 00 4 0 00 0 5 00 0 2 5.a) (3 points) Find all eigenvalues of A with their algebraic multiplicities.b) ( 3 po ints) Find the geometric multiplicities of each eigenvalue.c) (2 points) Is A diag onalizable?d) ( 2 po ints) What is the determinant of A3?Problem 5) (10 points)Let A =1 1 02 1 01 1 1.a) Find A−1.b) So lve Axyz=100.c) Find the matrix of T (~x) =2 0 00 1 00 0 3~x with respect to the basis B = {121,111,001}.Problem 6) (10 points)You have only to solve 5 from the following 6 problems to have full credit. But yo u can attemptall of them.a) (2 points) Find a 3 × 3 matrix A of ra nk 1 with no zero entries.b) ( 2 po ints) Find a matrix A which has123in the image of A.c) (2 points) Find a 3 × 3 matrix A whose kernel is 2-dimensional.d) (2 points) Find a 2 × 2 matrix A with different eigenvalues such that A2− 3A + 2I2is thezero matrix.e) (2 points) Find a 2 × 2 matrix A for which A−1and AThave the same eigenvectors.f) (2 points) Find a 3 × 3 matrix A such that every vector inR3is an eigenvector of A witheigenvalue 3.Problem 7) (10 points)Let A =1 2 −3 31 1 0 11 1 −1 10 −2 3 −4.a) (4 points) Find a basis of ker(A).b) ( 2 po ints) Find the rank of A.c) (1 point) Is there a vector~b such that A~x =~b has no solution?d) ( 1 po int) Is there a vector~b such that A~x =~b has exactly one solution?e) (1 point) Is there a vector~b such that A~x =~b has infinitely many solutions?f) (1 point) Find det(A).Problem 8) (10 points)Let V be the plane spanned by122and012. Find the matrix of reflection a t the planeV .Problem 9) (10 points)a) (4 points) Find all the solutions of the differential equation f′+ 2f = e−2t.b) ( 4 po ints) Find all the solutions of the differential equation f′′+ 4f′+ 4f = e−2t.c) (2 points) Find the kernel of T (f) = f′′+ 4f′+ 4f.Problem 10) (10 points)We analyze the nonlinear dynamical systemddtx = yddty = = x3− x − ya) ( 2 points) Draw the nullclines, and indicate the direction of the field along the nullclines andinside the regions determined by the nullclines.b) ( 2 po ints) Find all the equilibrium points.c) (4 points) Analyze the stability of all the equilibrium pointsd) ( 2 po ints) Which of the phase portraits A,B,C below belong to the above system?A B CProblem 11) (10 points)Find the Fourier series of the function f(x) = cos(x) + sin(2x) + x defined on [−π, π]. Show allcomputation steps.Problem 12) (10 points)a) Find the solution of the
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