Welcome to the Course Page of

MATH 2022 3.00 M

Linear Algebra II

Winter 2011


Faculty of Liberal Arts and Professional Studies
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.
April 26: 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.
April 06: As it was announced in the class yesterday, during the examination period
  from April 06 to 21 the MathLab will be open from 10:30 to 15:30.
  Please also note that additional office hours for last minute questions will be
  held on Thursday, April 07 from 15:30 to 17:00.
April 04: Please note the Final Examination will take place on Friday, April 08
  from 19:00 to 22:00 in two class rooms, as follows:
  - surname starting with A - L: SLH A,
  - surname starting with M - Z: SLH B.
  On the exam you will be responsible for all material covered in this course
  according to Course Schedule. No calculator 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 the course web page by April 29.
  Also, please note that this week Wednesday's office hour will be held
  on Tuesday, April 05 from 16:00 to 17:00.
March 30: Practice exercises for Chapter 9:
  9.1:  # 1, 4, 5, 10.
March 29: Practice exercises for Chapter 8:
  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
  8.4:  # 1, 2.
March 25: Please note that the graded Test #2 papers will be returned in the class on
  Tuesday, April 05. I am making avaialable  solutions  to the test questions.
March 18: Please note that Course evaluations are available on-line at the URL link
  http://courseevaluations.yorku.ca until Tuesday, April 5. It is the time for you
  to be the judge. Please go on-line and complete the course evaluation.
March 17: As it was announced in the class yesterday, we will write Test #2 during the
  lecture period from 19:00 to 20:20 on Wednesday, March 23 in two class
  rooms, as follows:
  - surname starting with A - K: CLH M,
  - surname starting with L - Z: CLH B.
  On the test you will be responsible for all the material covered in class from
  Sections 6.1, 6.2, 6.3, 6.4, 7.1, 7.2 and 7.3 of the textbook.
March 11: Please note that the next week there will be two Sessions of Open Forum for all
  Science and Engineering students. Please click  here  to obtain information on day,
  time and venue of the Forums.
March 07: 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.
March 01: Please note that Test #1 papers will be returned in the class on Wednesday,
  March 02.
February 18: Please note that the graded Test #1 papers will hopefully be returned in the class
  on Wednesday, March 02. I am making avaialable solutions to the test questions.
February 10: As it was announced in the class yesterday, we will write Test #1 during the
  lecture period from 19:00 to 20:20 on Wednesday, February 16 in two class
  rooms, as follows:
  - surname starting with A - J: CLH M,
  - surname starting with K - Z: CLH B.
  On the test you will be responsible for all the material covered in class from
  Sections 5.1, 5.2, 5.3, 5.4, 5.5 and 5.6 of the textbook.
February 04: Please note that due to the storm, all classes originally scheduled on Wednesday,
  February 02, will be automatically rescheduled to take place on Tuesday, April 05,
  2011 in their existing assigned rooms. Also, please note that Test #2 has been
  rescheduled for Wednesday , March 23.
  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.
February 02: Please note that due to the storm, effective 1:00 PM, Wednesday, February 02
  all evening classes have been cancelled. Consequently, we will be writing Test #1
  on Wednesday, February 16. Hope, to see in the class on Wednesday, February 09.
January 22: Please note that the MathLab, S525 Ross Building, is open throughout the
  semester. Hours of operation: Monday-Friday from 10:30 A.M. to 3:30 P.M.
  For more information on the MathLab please check the web page at:
January 12: Please note that Test #2 will take place on Wednesday, March 16.
January 07: Please note that Tutorials for the course will start on Wednesday, January 12.
  They will take place on Wednesdays from 6:00 to 6:50 P.M. in CLH M.
  Tutorials are optional. There is no need to register.
January 05: 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.

Course Information

MATH 2022: Session: Winter 2011
  Section: M
  Lectures: W 7:00-9:50 P.M., CLH B
Instructor: Name: Iulduz Raguimov
  Office: S512 Ross
  Phone: 416-736-2100, Ext.66092 or 416-736-5250
  Office Hours: W 4:00-5:00 P.M., F 12:00-1:00 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 AS/SC/AK/MATH 2222 3.00.
Important Dates: January 04: Classes commence.
January 17: The last date to enrol in the course without my permission.
January 31: The last date to enrol in the course with my permission.
  February 16: Test #1
March 04: The last date to drop the course without receiving a final grade.
  March 23: Test #2
  April 05: The last day of classes.
April 07-23: Examination period.

Course Schedule

(subject to any changes announced in class)

WEEK of:



January 05   Introduction, 5.1 Classes begin on Wednesday, January 05.
January 12 5.2-5.3  
January 19  5.4-5.5  
January 26 5.5-5.6  
February 02  --- Classes cancelled due to the storm.
February 09 6.1-6.2  
February 16 6.3, Test #1 Test #1 will take place on Wednesday, February 16.
February 23 No classes Reading/Co-Curricular Week
March 02 6.4, 7.1  
March 09 7.2-7.3  
March 16 7.3, 9.1  
March 23 8.1, Test #2 Test #2 will take place on Wednesday, March 23.
March 30 8.1-8.2  
April 05 8.3, Review The classes end on Tuesday, April 05.
April 06-23 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: Wednesdays, February 16 and March 23). Final Examination:   40% of the overall grade, will take place during the examination period between April 07 and 23, 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-ups 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 for Hawaii or Las Vegas on
April 16  is NOT a legitimate excuse for absence from a final exam on April 20.

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 January 19. 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 April 26, 2011