University of British Columbia - Mathematics 221 Final Examination, April 2009, Exams of Algebra

The final examination for mathematics 221 at the university of british columbia, held on april 28, 2009. The examination consists of 12 problems covering various topics in linear algebra, including systems of equations, determinants, eigenvectors, and orthogonal projections. Students were not allowed to use notes or calculators during the exam, which lasted 2.5 hours.

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The University of British Columbia
Final Examination - April 28, 2009
Mathematics 221
All Sections
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.
1 10
2 10
3 10
4 10
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6 10
7 10
8 10
9 10
10 10
11 10
12 10
Total 120
Page 1 of 20 pages
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Download University of British Columbia - Mathematics 221 Final Examination, April 2009 and more Exams Algebra in PDF only on Docsity!

The University of British Columbia Final Examination - April 28, 2009 Mathematics 221 All Sections

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 20 pages

Problem 1. Find all values of c such that the system of equations below is consistent. For these values of c write the general solution of the system in the parametric vector form.

x 1 + 4 x 3 − 2 x 4 = 1 −x 1 + x 2 − 7 x 3 + 7 x 4 = 2 2 x 1 + 3 x 2 − x 3 + cx 4 = 11

Problem 3. The population P (t) (in hundreds) of a colony of rabbits in year t is given in the table: t 0 2 4 6 P 5 6 8 9

Find the equation P (t) = a + bt of the least squares line that best fits the data and use it to estimate the population at time t = 7.

Problem 4. Let W = Span{ w~ 1 , ~w 2 }, where

w ~ 1 =

 (^) , w~ 2 =

If T : R^3 → R^3 is the orthogonal projection onto W , find the standard matrix of T.

Problem 6. If

xn+1 = 0. 7 xn + 0. 6 yn yn+1 = 0. 3 xn + 0. 4 yn

and x 0 = 0, y 0 = 3, find the limiting values of xk, yk as k → ∞.

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Problem 8. Find a formula for Ak, where

A =

[

]

You may leave your final answer as a product of three matrices.

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Problem 9. Consider the matrix

A =

a. Find a basis for N ul(A). b. Find a basis for Col(A).

c. Find the coordinate vector of

 (^) relative to the basis of Col(A) which you found in

part b. d. Find the dimension of N ul(AT^ ).

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Problem 11. Let

A =

a. Find a nonzero vector ~v such that A~v = 2~v. b. Find all eigenvalues of A. c. Find a matrix P such that P −^1 AP is diagonal, if it exists. If such a P does not exist, explain why. (No need to find P −^1 .)

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