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University of British Columbia: Math 307
Final examination Date: April 25, 2009
Time: 8:30 am
Name (print): Student ID Number: Signature:
Instructions:
- No notes or books are allowed. You will be provided with a sheet of MATLAB/Octave commands.
- No calculators are permitted.
- Read the questions carefully and make sure you provide all the information that is asked for in the question.
- Show all your work. Answers without any explanation or without the correct accompanying work could receive no credit, even if they are correct.
- Answer the questions in the space provided. Continue on the back of the page if necessary.
This examination consists of 16 pages (including this cover sheet). Check to ensure that it is complete.
Rules governing formal examinations
- Each candidate must be prepared to produce, upon request, a UBCcard for identification;
- Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions;
- No candidate shall be permitted to enter the examination room after the expiration of one-half hour from the scheduled starting time, or to leave during the first half hour of the examination;
- Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination and shall be liable to disciplinary action: - Having at the place of writing any books, papers or memoranda, cal- culators, computers, sound or image players/recorders/transmitters (including telephones), or other memory aid devices, other than those authorized by the examiners; - Speaking or communicating with other candidates; - Purposely exposing written papers to the view of other candidates or imaging devices. The plea of accident or forgetfulness shall not be received.
- Candidates must not destroy or mutilate any examination material; must hand in all examination papers; and must not take any examination material from the examination room without permission of the invigilator.
- Candidates must follow any additional examination rules or directions com- municated by the instructor or invigilator.
Qu. Mark Maximum 1 14 2 14 3 14 5 14 6 14 7 15 8 15 9 15 Total 100
[14] 1. (a) Calculate the 1- and 2-norms of the following vectors:
a = (2, − 3 , 1) and b = (2 + 3i, 2 − 3 i).
(b) Define the matrix norm for a matrix A.
[Continued on next page]
[14] 2. (a) A cubic spline f (x) interpolating three points (x 1 , y 1 ), (x 2 , y 2 ) and (x 3 , y 3 ) can be written as f (x) =
p 1 (x) if x 1 ≤ x ≤ x 2 p 2 (x) if x 2 ≤ x ≤ x 3 where p 1 (x) and p 2 (x) are polynomials. What conditions do these polynomials have to satisfy?
(b) Write down a matrix equation that you would need to solve in order to find the cubic spline passing through the points (1, 2), (2, 4), and (3, 3).
[14] 3. Let
v 1 =
, v 2 =
, v 3 =
(a) Write down a 4 × 3 matrix A such that the equation Ac = 0 has a non-zero solution c if and only if v 1 , v 2 and v 3 are linearly dependent.
(b) Explain how you could use MATLAB/Octave to determine whether v 1 , v 2 and v 3 are linearly dependent. What output would tell you that the vectors are independent?
[Continued on next page]
[14] 4. (a) Draw and label the connections between the nodes in the diagram
so that the incidence matrix of the resulting graph is given by
D =
(b) Suppose the resulting graph represents a resistor network with all resistances equal to
- Write down down the Laplacian matrix L.
[Continued on next page]
(c) If the entries of the vector v =
v 1 v 2 v 3 v 4 v 5 v 6
represent the voltages at the nodes of the graph,
what do the entries of Lv represent?
(d) A battery is attached to nodes 1 and 2 with voltages v 1 = 0 and v 2 = 1. Write down the MATLAB/Octave commands that compute the voltages v 3 ,... , v 6 for the rest of the network. Assume that the matrix L has already been defined in MATLAB/Octave.
(b) Suppose φ 0 (x), φ 1 (x), φ 2 (x),... is an infinite orthonormal basis of functions on the in- terval [0, 1] and we have the expansion
eiπx^ =
∑^ ∞
n=
dnφn(x).
Suppose, in addition, that φ 0 is the constant function φ 0 (x) = 1. Is it true that d 0 = c 0 , where c 0 is the Fourier coefficient above? Give a reason.
For questions 6, 7 & 8 choose and answer any TWO of the questions. Note that only two of the questions will be marked. Indicate which two you would like to have marked by circling the appropriate question numbers.
[15] 6. (a) Define the algebraic multiplicity and geometric multiplicity for an eigenvector λ. Under what conditions is a matrix diagonalizable?
(b) Given a non-singular matrix A, and an “initial guess” vector x, write out the steps involved in the algorithm for the power method for calculating an eigenvector corre- sponding to the smallest (in magnitude) eigenvalue.
[Continued on next page]
[15] 7. Given the recurrence relation xn+1 = 3xn − 2 xn− 1 with the initial condition x 0 = a and x 1 = b,
(a) Solve the recurrence relation. (Give a general scalar expression for xn in terms of n, a, and b).
[Continued on next page]
(b) For what values of a and b will the sequence x 0 , x 1 , x 2 ,... converge to a finite limit?
(c) Suppose you are told that the eigenvalues of the stochastic matrix are
λ 1 = 1, λ 2 = 1, λ 3 =
3 ,^ λ^4 =^ −
and the corresponding eigenvalues are
v 1 =
v 2 =
, v 3 , v 4 ,
and that the eigenvectors form a basis of R^4. Write down a matrix equation that you would solve in order to find the unique set of coefficients {c 1 , c 2 , c 3 , c 4 } to express the initial state of the game, x 0 =
[
]T
, in terms of the above basis of eigenvectors. Just write the equation symbolically using vi, ci, and x 0.
(d) Given that when the initial state x 0 is written in terms of the eigenvectors, vi, the coefficients (c 1 , c 2 , c 3 , c 4 ) = (0. 43 , 0. 57 , 0. 28 , 0 .93), determine the probability of winning the game. Explain your solution by writing a expression for the long-time behaviour of the game in terms of ci, λi, and vi.
