Final Exam Practice 3, May 8, 2018 Math 21b, Spring Name:

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1 Final Exam Practice 3, May 8, 8 Math b, Spring 8 Name: MWF 9 Oliver Knill MWF Jeremy Hahn MWF Hunter Spink MWF Matt Demers MWF Yu-Wen Hsu MWF Ben Knudsen MWF Sander Kupers MWF Hakim Walker TTH Ana Balibanu TTH Morgan Opie TTH Rosalie Belanger-Rioux TTH :3 Philip Engel TTH :3 Alison Miller 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 unstaple the packet. Please write neatly and except for problems -3, give details. 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 8 minutes time to complete your work Total: 5

2 Problem ) ( points) True or False? No justifications are needed. ) T F The matrix A = [ 4 ] is symmetric. Direct check. ) T F The vector v = is always in the kernel of a matrix A. By definition, v = is never an eigenvector 3) T F For any matrices A, the kernel of A is a subspace of the image of A. This is already false for the zero matrix 4) T F If a matrix A is diagonalizable then the matrix A T is diagonalizable. A = S BS implies A T = (S T ) B T S T. 5) T F The Fourier series of the function f(x) = sin (x) only involves cosine terms and the constant term. The function is even. 6) T F The space of smooth functions satisfying the equation f(x) = f( x) forms a linear space. The function f = satisfies the equation but f = not.

3 7) T F The algebraic multiplicity of an eigenvalue of A is equal to the nullity of A. The geometric multiplicity is the nullity. 8) T F The sum A + B of two real matrices A, B which both have eigenvalues λ = i, i is a matrix with eigenvalues i, i. Take a matrix A and then A. 9) T F A discrete dynamical system x(t + ) = Ax(t) with a matrix A is asymptotically stable if det(a) <. We need det(a) <, as for discrete dynamical systems, the eigenvalues should be smaller than in norm. ) T F If x (t) = Ax(t) and x(t + ) = Ax(t) are both asymptotically stable, then all real eigenvalues λ satisfy < λ <. Yes, this assures it ) T F The sum of two orthogonal projections is an orthogonal projection. Take a projection P and form P + P. This is no more a projection. ) T F The function f(x, t) = e 6t sin(3x) solves the heat equation f t = f xx. Yes, just check this 3) T F If a system of linear equations A x = c with matrix A has infinitely many solutions, then there exists b such that A x = b has no solution. 3

4 There is a nontrivial kernel. The image is therefore not the entire space. 4) T F The equilibrium point (, ) of the nonlinear system x = x, y = y is asymptotically stable. It is not stable because the eigenvalues are zero. 5) T F All real symmetric matrices are diagonalizable over the real numbers. This is the spectral theorem 6) T F 3 sin(5x) 7 sin(x) = 5 +, where the length f of f is defined by the inner product f, g = f(x)g(x) dx. π π π This is Parseval s equality. But the result is ) T F The sum of the eigenvalues of a symmetric matrix is. Take a matrix with eigenvalues,. 8) T F The differential equation x (t) + 9x(t) = sin(3t) with x() =, x () = has solutions for which x(t) becomes arbitrary large. This is a resonance case. 9) T F For any invertible matrix, A is similar to A. The eigenvalues are /λ. 4

5 ) T F If an n n matrix has the property that the sum of all matrix entries are zero, then the matrix is not invertible. A counter example is the matrix [ ]. 5

6 Problem ) ( points) No justifications needed a) (6 points) We are given a real 3 3 matrix A and define B = A 3 + A. Statement If A is invertible, then B is invertible If A is diagonalizable, then B is diagonalizable If A is symmetric, then B is symmetric If A has a zero determinant, then B has a zero determinant If A has zero traces, then B has zero trace If A has an eigenvalue, then B has an eigenvalue Always true Statement Always true If A is invertible, then B is invertible If A is diagonalizable, then B is diagonalizable * If A is symmetric, then B is symmetric * If A has a zero determinant, then B has a zero determinant * If A has zero traces, then B has zero trace If A has an eigenvalue, then B has an eigenvalue * b) (4 points) Match the differential equations with possible solution graphs. Enter A-D Differential equation f (t) + f(t) = sin(t) f (t) = sin(t) f (t) f(t) = sin(t) f (t) + f(t) = sin(t) A) 5 5 B) 5 5 6

7 C) -3 D) DABC Problem 3) ( points) No justifications needed a) (6 points) Assume T is a transformation on C, the linear space of smooth functions on the real line. Which of the following transformations are linear? Transformation T f(x) = 3f(5x ) Check if linear T f(x) = f(4 x) T f(x) = sin(x)f (x) T f(x) = f(5)f(x) T f(x) = f(5)x T f(x) = + f(5f(x)) 7

8 Transformation T f(x) = 3f(5x ) T f(x) = f(4 x) T f(x) = sin(x)f (x) T f(x) = f(5)f(x) T f(x) = f(5)x Check if linear X X X X T f(x) = + f(5f(x)) b) (4 points) Match the differential equation d x(t) = Ax(t) with the phase portraits. There is dt an exact match. Matrix [ ] Enter a) - d) A = [ ] A = [ ] A = [ ] A = a) b) c) d) c)b)d)a) 8

9 Problem 4) ( points) a) (4 points) Find all the solutions to the following system of linear equations: x + y + z + u + v + w = 6 x y + z u + v w = x + y + z + u v w = b) (3 points) The system in a) can be written in matrix form as A x = b. Find a basis for the kernel of A. c) (3 points) Find a basis for the image of A. a) Row reduce the augmented matrix to get the pivot columns and free variables: r s t 3. Write this again as a system using the free variables to get The solution is x y z u v w = 3 + r + s + t. b) A basis for the kernel is given by the vectors which come with the free variables: B =,,. c) A basis for the image is given by the pivot columns B =,. Problem 5) ( points) 9

10 3 Using the least square method, find the hyperbola xy + ax + by = which best fits the data points (x, y): {(, ), (, ), (, ), (, ), (, ) } write down the equations: This leads to the matrix and the vector b = [ 7 A T A) = 8 is xy (/3)x (5/3)y =. a = a + b = a + b = a b = a + b = A =. We get the solution with x = (AT A) A T b. We have ] /5 and A T b = [ 3 ] [ /3. Now x = 5/3 ]. The best hyperbola Problem 6) ( points)

11 Let A = a) (3 points) Find a matrix B which is diagonal and similar to A. b) (3 points) Find a matrix S such that B = S AS is that diagonal matrix obtained in a). c) ( points) Solve the discrete dynamical system v(t + ) = A v(t), for which the initial condition is v() =. d) ( points) Is the system defined in c) stable? a) Since the eigenvalues are 5,, 3, we have the similar diagonal matrix B = 5. 3 b) S contains the eigenvectors of A S =. c) Since = +, we have v(t) = 5 t + 3 t. The system is not stable. Problem 7) ( points)

12 The following matrix is called the Laplacian of the star graph 3 A =. a) ( points) You are told that B = { Find the eigenvalues of A. 3,,, } is an eigenbasis of A. b) ( points) Find the characteristic polynomial of A. c) (3 points) Write down the solution to v (t) = A v(t) with initial condition v() = Side remark: You just have solved the heat equation on the star graph. d) (3 points) Write down the solution to v(t + ) = A v(t) with initial condition v() = 3. 3.

13 a) The eigenvalues are 4,,,. b) The characteristic polynomial is ( 4 λ)( λ)( λ)( λ). c) Since 3 = 3/4 3 + (3/4) v(t) = ( 3/4)e 4t 3 3, we have + (3/4)e t. d) v(t) = ( 3/4)( 4) t 3 + (3/4)t. Problem 8) ( points) Let A = [ ]. a) ( points) Compute A and A. b) ( points) Find the eigenvalues of A T + A. c) ( points) Find the QR decomposition of A. d) ( points) Find the characteristic polynomial f A (λ) of A. e) ( points) What are the algebraic and geometric multiplicities of the eigenvalues of A? Is A diagonalizable? [ ] [ ] 4 8 / / a) A =, A 4 =. / [ ] 4 b) A + A T =. λ = 6,. 4 c) The QR decomposition is A = I A. d) The characteristic polynomial is λ 4λ + 4. e) The algebraic multiplicity of is, the geometric multiplicity. The matrix is not diagonalizable. 3

14 Problem 9) ( points) Remember the song: Laplace, Row Reduce Partitions, Triangular Eigenvectors, Eigenvalues Patterns you can use! a) ( points) Find the determinant of the following matrix: b) ( points) Find the determinant of the following GCD matrix 3. 4 c) ( points) Find the determinant of the following matrix d) ( points) Find the determinant of the following matrix e) ( points) Find the determinant of the matrix 4

15 Follow the song! a) 8 b) 4 c) d) 5 4 = 65 e) 9. Problem ) ( points) Find the general solution to the following differential equations: a) ( points) b) ( points) c) (3 points) d) (3 points) f (t) + f(t) = t + f (t) + 9f(t) = t + f (t) + f (t) + f(t) = t f (t) f(t) = t + e t a) C e t + t t + 3. b) C cos(3t) + C sin(3t) + t /9 + 7/8. c) C e t + c te t + t 4t + 6 d) C e t + c e t + te t / t. Problem ) ( points) We analyze the following nonlinear dynamical system d dt x = y x d dt y = x y a) ( points) Find the equations of the null-clines. b) ( points) Find all the equilibrium points. c) (3 points) Analyze the stability of the equilibrium points. 5

16 d) (3 points) Which of the four phase portraits A,B,C,D below belongs to the above system? Make sure that also here, you justify your answer, as always. A B C D a) The nullclines are y = x and y = x. b) The equilibrium points are (, ), (, ), (, ). c) The Jacobean matrix is J = and J(, ) = ]. The points (, ) and (, ) are not sta- [ ] [ ] x. at the critical points we have J(, ) = x y [ ] [ and J(, ) = ble. The point (, ) is stable. d) Comparing the stability at the three points shows that A is the only choice. Problem ) ( points) 6

17 .5 a) (7 points) Find the Fourier series of the function π < x < π f(x) = π < x < π. π < x < π The graph of the function f on [ π, π] is displayed to the right b) (3 points) Given the Fourier coefficients a, a n, b n in the previous problem, find the sum a n + b n. n= n= -.5 a) The function is odd. It therefore has a sin series. we have b n = π π π/ sin(nx) dx = (cos(nπ/) cos(nπ)). nπ The series is n= b n sin(nx). b) The Parcival identity tells that the sum is f = π π π/ dx =. Problem 3) ( points) a) (5 points) The partial differential equation u t = u xx + u is a modification of the heat equation on the interval [, π]. Solve it with the initial condition u(x, ) = sin(x) + 4 sin(7x) + sin(3x). b) (5 points) The partial differential equation u tt = u xx + u is a modification of the wave equation on the interval [, π]. Solve it with the initial conditions u(x, ) = sin(5x), u t (x, ) = 3 sin(5x) + 6 sin(8x). P.S. As usual, we only consider sin-fourier expansions in x. 7

18 a) The eigenvalues of the operator T f = f xx + u on the right hand side are λ = n + for the eigenvector sin(nx). This reduces the heat equation to u = λu and a factor e λt in each case. The solution is u(x, t) = sin(x)e ( )t + 4 sin(7x)e ( 7 )t + sin(3x)e ( 3 )t. b) We have the same eigenvalue but since we want to reduce the equation u tt = λu to a harmonic oscillator, we have u tt + c u = and c = λ. For the initial position component we have to include cos(ct) and for the initial velocity factor, we have to include sin(ct)/c. The solution is u(x, t) = sin(5x) cos( 5 t) + 3 sin(5x) sin( 5 t) sin(8x) sin( 8 t) 8. Problem 4) ( points) a) (7 points) The pseudo determinant of a matrix is the product of the nonzero eigenvalues of a matrix. Find the Pseudo determinant of b) (3 points) We know that det(ab) = det(a)det(b). This identity is no more true for pseudo determinants. Find a counter example. Hint: You can find diagonal examples. a) The eigenvalues are with multiplicity 9. The largest eigenvalue is 9, the trace. b) Take A = Diag(,, ) and B = Diag(,, ). Then AB = Diag(,, ) has pseudo determinant, while A, B both have pseudo determinant. 8

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