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The University of British Columbia Final Examination - December 18, 2009 Mathematics 221
Circle one: Section 101 MWF 8-
Section 102 MWF 10-
Section 103 MWF 1-
Closed book examination Time: 2.5 hours
Last Name First Signature
Student Number
Special Instructions:
No notes or calculators are allowed. Answer all 12 questions on the sheets provided - use the backs of the sheets and blank sheets at the end of the test if necessary.
Rules governing 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, dishon- est practices shall be immediately dismissed from the examination and shall be liable to disciplinary action. (a) Having at the place of writing any books, papers or memoranda, calculators, computers, sound or image play- ers/recorders/transmitters (including telephones), or other mem- ory aid devices, other than those authorized by the examiners. (b) Speaking or communicating with other candidates. (c) Purposely exposing written papers to the view of other can- didates or imaging devices. The plea of accident or forgetfulness shall not be received.
- Candidates must not destroy or mutilate any examination mate- rial; must hand in all examination papers; and must not take any examination material from the examination room without permis- sion of the invigilator.
- Candidates must follow any additional examination rules or di- rections communicated by the instructor or invigilator.
Total 120
Page 1 of 18 pages
Problem 1. Consider the system of equations:
x 1 + x 2 − x 3 = 2 x 1 + 2 x 2 + x 3 = 3 x 1 + x 2 + (c^2 − 5)x 3 = c
Find all values of c such that the system has:
a. no solutions b. a unique solution c. infinitely many solutions
In case c. write the general solution in the parametric vector form.
Problem 3. Consider the traffic flow diagram:
100
X1 X
100 X 350
a. Find a system of 3 equations in 3 variables that describes this model. b. The system is clearly inconsistent because the total infolw does not equal total outflow. Find all least squares solutions to the system.
Problem 4. Let W be the subspace of R^3 spanned by w~ 1 and w~ 2 , where
w ~ 1 =
(^) , w~ 2 =
Let T : R^3 → R^3 be the reflection across W. Find the standard matrix of T.
W.
T(X)
X
Blank page.
Problem 6. Let
A =
Find an invertible matrix P and a diagonal matrix D, such that A = P DP −^1. (No need to find P −^1 .)
Blank page.
Problem 8. A discrete dynamical system is described by:
xn+1 = 17xn + 9yn, yn+1 = − 30 xn − 16 yn.
Given that x 0 = 1, y 0 = −1, find x 30 , y 30.
Continue on the next page.
Problem 9. Find the least squares fit of a parabola y = a + bx + cx^2 to the data (xi, yi): (0, 1), (1, 0), (2, 5), (3, 6).
Problem 10. Find the numbers a, b, c which make the matrices below diagonalizable. (No need to diagonalize them.)
a.
0 3 a 0 0 0
(^) , b.
2 1 b 3 0 3 − 1 c 0 0 2 2 0 0 0 3
Problem 12. Mark each statement either True or False. You do not have to justify your answer.
a. A system of 4 linear equations in 3 variables is always inconsistent. b. If A is a 4 × 3 matrix, then there exists a vector ~b such that A~x = ~b has no solutions. c. If the matrix A has 6 rows and 9 columns, then dim(N ul(A)) ≥ 3. d. If a 5 × 6 matrix A has rank 4, then dim(N ul(A)) = 1. e. If T : Rn^ → Rn^ is the orthogonal projection onto a subspace W , then the standard matrix of T is diagonalizable. f. The rank of any upper-triangular n × n matrix is the number of nonzero entries on its diagonal. g. If ~v 1 ,... , ~vn span R^5 , then n must be equal to 5. h. If ~v 1 ,... , ~vm are linearly independent vectors in Rn, then they form a basis of a subspace W of Rn. i. If the system A^2 ~x = ~b is consistent, then A~x = ~b must also be consistent. j. The matrix A =
[
]
is the matrix of rotation by some angle θ in R^2.
Blank page.
