Welcome to the Course Page of

MATH 2022 3.00 M

Linear Algebra II

Summer 2011


Department of Mathematics and Statistics
Faculty of Science and Engineering
York University
4700 Keele Street
Toronto, Ontario  M3J 1P3

Course Organization:

Course Information
Course Schedule
Exams  Information
Links & Other Resources


This area will contain announcements and solutions to tests. Announcements made in class will be posted here in reverse chronological order and will not be repeated in class. Please take a minute to read the announcements carefully, as they often get updated. You may need to Reload/Refresh this page.
August 14: Please click  here  to see  your final exam mark and unofficial letter grade
  for the course. Also, please note that your official grade will be sent to you
  by the Registrar's Office upon the approval by the Senate.
July 29: As it was announced in the class yesterday, the Final Examination will
  take place on Wednesday, August 03 from 19:00 to 22:00 in SLH B.
  On the exam you will be responsible for all material covered in this course
  according to the Course Schedule. No calculator or other aid is permitted
  on the exam. There will be several questions, some with a few parts. Marks
  for each question will be indicated. You have to show all your work, and
  you should explain your solutions where appropriate. Unofficial final grades
  for the course will (hopefully) be posted to this web page by August 20.
July 27: Please note that the graded Test #2 papers will be returned in the class on
  Thursday, July 28. I am making avaialable  solutions  to the test questions.
July 26: Practice exercises for Chapters 8 and 9:
  8.1:  # 1, 2, 4, 13, 14, 16
  8.2:  # 1, 5, 7, 9, 13, 14, 16, 23, 25
  8.3:  # 1, 3, 7, 9
  9.1:  # 1, 4, 5, 10.
July 19: As it was announced in the class today, we will write Test #2 during the
  lecture period from 18:00 to 19:20 on Thursday, July 21. On the test you
  will be responsible for all the material covered in class from Sections 5.6,
  6.1 - 6.4 and 7.1 - 7.2 of the textbook.
July 18: Practice exercises for Chapter 7:
  7.1:  # 1, 2, 3, 4, 11, 14, 15
  7.2:  # 1, 2, 6, 8, 9, 10, 24, 28, 30
  7.3:  # 1, 4, 6, 10, 23.
July 14: Please note that the graded Test #1 papers will be returned in the class on
  Thursday, July 14. I am making avaialable  solutions  to the test questions.
July 12: Please note that Course Evaluations are available on-line at the URL link
  http://courseevaluations.yorku.ca  until Friday, July 29. It is the time for you
  to be the judge. Please go on-line and complete the course evaluation.
July 10: Practice exercises for Chapter 6:
  6.1:  # 1, 2, 4, 6, 12, 14
  6.2:  # 1, 2, 3, 5, 6, 8, 9, 13, 21, 27
  6.3:  # 1, 2, 3, 4, 10, 12, 15, 23, 27, 32, 33
  6.4:  # 1, 3, 7, 18, 23, 25, 26.
July 01: As it was announced in the class yesterday, we will write Test #1 during the
  lecture period from 18:00 to 19:20 on Tuesday, July 05. On the test you will
  be responsible for all the material covered in class from Sections 5.1 - 5.5 of
  the textbook.
June 25: I am making available Errata Corrections to the 6th Edition of the textbook.
June 20: Welcome back!
  To view and/or print  PDF  files you need to download the Acrobat Reader 
  Practice exercises for Chapter 5:
  5.1:  # 1, 2, 3, 5, 6, 12, 15, 16, 18,  23, 24
  5.2:  # 1, 4, 6, 7, 9, 18, 19, 20
  5.3:  # 4, 5, 6, 7, 13, 15, 18
  5.4:  # 1, 2, 3, 7, 11, 12, 14, 15
  5.5:  # 1, 2, 4, 5, 9, 16
  5.6:  # 1, 2, 4, 14.
  Please note that Tutorials for the course will be done in MathLab, S525 Ross
  Building, which is open throughout the term: Monday-Thursday from 4:20 to
  5:50 P.M., also Mondays 10.30-12:00 P.M. and Thursdays 1:30-3:00 P.M.
  For more information on the MathLab please check the web page at:

Course Information

MATH 2022: Session: Summer 2011
  Section: M
  Lectures: MW 6:00-9:00 P.M., CLH J
Instructor: Name: Iulduz Raguimov
  Office: S512 Ross
  Phone: 416-736-2100, EXT 66092 or 416-736-5250
  Office Hours: W 4:30-5:30 P.M., R 4:00-5:00 P.M., also by appointment
  E-mail: raguimov@mathstat.yorku.ca
Grading: Two Tests:  30% each
  Final Examination: 40%

Course Description:  This course is required in Specialized Honours Statistics and all Applied Mathematics, Mathematics and Mathematics for Commerce programs except the BA program in Mathematics for Commerce. Topics include inner product spaces, linear transformations, orthogonality, eigenvalues, diagonalization, least squares and quadratic forms. Similar to AS/SC/AK/MATH 2222 3.00 but at a more advanced level. Additional topics: General linear transformations, matrix of a linear transformation from V to W relative to bases for V and W, effect of a change of bases. For a more detailed list of topics with references to the textbook, please see Course Schedule.


Linear Algebra with Applications, Sixth Edition

by W. Keith Nicholson
McGraw-Hill Ryerson, ISBN 0-07-098510-3
Course Prerequisites: AK/AS/SC/MATH 1021 3.00 or AS/SC/MATH 2021 3.00.
  If you have not taken MATH 1021 3.00, it is essential that you take it first and then take
MATH 2022.
Course Credit  
Exclusion:  AS/SC/AK/MATH 2222 3.00.
Important Dates: June 20: Classes commence.
June 24: The last date to enrol in the course without my permission.
July 02: The last date to enrol in the course with my permission.
  July 05: Test #1
July 15: The last date to drop the course without receiving a final grade.
  July 21: Test #2
  July 29: The last day of classes.
August 02-12: Examination period.

Course Schedule

(subject to any changes announced in class)

WEEK of:



June 20   Introduction, 5.1-5.3 Classes begin on Tuesday, July 21.
June 27  5.4-5.6  
July 04 6.1-6.2, Test #1 Test #1 will take place on Tuesday, July 05.
July 11  6.3-6.4, 7.1  
July 18 7.2-7.3, Test #2 Test #2 will take place on Thursday, July 21.
July 25 9.1, 8.1-8.3, Review The classes end on Thursday, July 28.
August 02-12 Final Examination TBA

Note: The course will not cover all the sections of each chapter from the textbook.


Throughout the term, practice exercises will be posted to the Announcements section. In order to develop your understanding of the course material you must do all assigned problems. These will not be collected for grading and hence will not directly affect your grade. However, it is extremely important that you complete all assigned exercises. Do not memorize them. Rather, learn how to solve problems like them. Learning Mathematics is not a spectacular sport, it is a participatory event. There is nothing like a bit of computation to strengthen understanding in mathematics! Use the Solutions Manual only after you have tried the problem. Successful students must keep up with homework and seek help for points they do not understand as soon as possible. Do not fall behind! It is recommended to read the relevant sections of the textbook before every class. Next to the classes, working out the answers to the problems is the most important preparation for the tests and exam that will contain for the most part but not exclusively, questions very similar to those from the textbook. It is quite practical to work on your own or together in small groups. Each student should do at least two hours of independent study for every lecture hour. The amount you learn in this course and the grade you receive will be proportional to the amount of time you spend working on problems.

Exams Information

Lecture/Exam Rules:

Please turn off all cell phones and pagers before entering the lecture hall. For tests and exam cell phones, calculators, digital dictionaries, palms, pagers or other electronic devices are NOT allowed. All such devices as well as all books, papers, knapsacks, and briefcases must be left at the front of the lecture hall. Anyone caught with electronic devices will be charged with Academic Dishonesty (see the next page). The only items you may have at your seat are pens, pencils, student ID and purses. You are responsible for all material covered in lectures.
Note: Photo identification and signing-in are required at all
tests and exam.

Final Grade:

The official grades at York University are LETTER grades. Qualitative descriptions of these grades can be found in the York Calendar at this link. Your test marks will be on a numerical scale because these have to be weighted and averaged in various ways, but THERE IS NO AUTOMATIC PREDETERMINED CORRELATION BETWEEN NUMERICAL GRADES AND LETTER GRADES in this course. In particular, the "numerical" grades you receive are NOT to be interpreted as percentages; they are merely 'raw scores'. When the tests are returned in class, the distribution of 'raw scores' will be announced along with an approximate conversion to letter grades. The median letter grade for the course will likely fall between C and C+.
The composition of the final grade is as follows:
Two Tests (80 minutes each written tests held in lecture periods):   30% each (Tentative dates for the tests: Tuesday, July 05 and Thursday, July 21). Final Examination:   40% of the overall grade, will take place during the examination period between August 02 and 12, 2011.
Students have five days from the date of the return of test papers to appeal their marks. Some test papers (randomly selected) are photocopied before they are handed back, so we can detect subsequent alterations. Once you have written both tests and your final exam grade is higher than your average test grade, your final exam will be counted as 60% and each test as 20% of your overall grade for the course.

Make-up Policy:

No permission will be given to a student to write tests in advance of their scheduled dates. No make up will be done for the tests. A missed test will be counted as zero, except under extreme circumstances in which case the corresponding percentage of the overall grade will be "forwarded" to the final exam. If you miss a term-test  for medical reasons you must turn in  within one week following the test date the Attending Physician's Statement form provided  here after getting it filled in by your doctor. No other type of medical note will be accepted.  However, missing tests is extremely dangerous and not recommended. As experience has shown, students who miss class tests because of some 'mysterious illnesses' will usually average 30% on the final exam. A student who misses the final examination will be allowed to write a make-up exam only if both of the following conditions are met:
1) the student notifies me (raguimov@mathstat.yorku.ca) or the Department of Mathematics and Statistics (Undergraduate Office, N502/503 Ross Building, 416-736-2100, Ext. 55902 or 33969) in advance that the exam will be missed,
2) the student submits within one week following the exam date the Attending Physician's Statement form provided  here after getting it filled in by his/her doctor .
Students who miss the
final examination and do not meet both conditions will receive a grade of F. It is student's responsibility to fill out and submit the Deferred Standing Agreement Form.
Note: Do not make vacation/job plan until the final exam date is known: having a plane ticket to Banff or Hawaii for
August 06  is NOT a legitimate excuse for absence from a final exam on August 09.

Religious Observance:

 York University is committed to respecting the religious beliefs and practices of all members of the community and making accommodations for observances of special significance to adherents. If any of the dates specified in the course schedule for in-class tests pose such a conflict, students should contact me (raguimov@mathstat.yorku.ca) by June 28. Please note that if the final exam date poses a conflict, students must complete the Final Exam Accommodation Form, which can be obtained from the Registrarís Office

Academic Honesty:

Students are responsible for familiarizing themselves with University Policies and Regulations. Conduct that violates the ethical or legal standards of the university community or of oneís programme or specialization may result in serious consequences. Refer to the Senate Policy on  Academic Honesty.  Photo identification and signing-in are required at all tests and exam. Also, some test papers (randomly selected) will be photocopied before they are returned so that subsequent alterations can be detected.


Individual questions can be discussed by e-mail, or in person after class, or during office hours. Please send all e-mail notes as plain text within the body of the message. Do not send attachments nor HTML-formatted mail. Also, if the name of your account is an alias, I will not know who the mail is from unless you sign it; it also risks being accidentally discarded as junk mail. The course web page will be up-dated regularly to include important announcements made in class, such as the material to be covered on the tests. E-mail notes requesting such information contained on the web page will be answered the last.

Links and Other Resources

1-  Study Groups
2-  York Undergraduate Mathematics Program
3Links to Linear Algebra sites
4-  How To Write Proofs


Questions and comments regarding this Web page please send to raguimov@mathstat.yorku.ca

© 2004-2011, All Rights Reserved, York University & Iouldouz S. Raguimov

Last updated August 14, 2011