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MAT 224 A: Linear Algebra II

Winter Semester: Jan. 2013

CourseMAT 224 A “Linear Algebra II“.

Prereq. MAT 120, MAT 200 with a minimum grade of C, and one of MAT 31, 113, or 114.

Credit may be obtained for only one of MAT 121 or 224
Course Description: CUCA Calendar, or


Course Instructor:

Name: Dr. A. N. (Tom) Tavouktsoglou, PhD
Professor of Mathematics & Chemistry

Office: Faculty House F.H.-204 (main level).

Tel: 780-479-9360.

e-mail: tomtavou@math.concordia.ab.ca

web: http://www.math.concordia.ab.ca/tomtavou/

Note: Students must phone (780-479-9360) and leave a message (or send an e-mail) ahead of
time, if they are unable to present themselves for a test/exam..

Office Hours: M./T./W./R./F.: 12:05 – 12:55

Books & Supplies (Available at Concordia University College of Alberta Bookstore):

  1. H. Anton, “Elementary Linear Algebra“, 10th Edition, John Wiley & Sons Inc.

  2. A copy of “Mathematica® – Student Version” appropriate for the student’s own computer is recommended (but not required).

  3. A copy of the Students’ Solutions Manual is on Reserve at the Library.

Course ScheduleM/W/F: 11:10 – 12:00 in T-103 (Math Lab); please be punctual.

Additional Bibliography(optional – Do NOT purchase)

  1. S. Axler, “Linear Algebra Done Right”, 2nd Edition, Springer-Verlag New York Inc., 1997.

  2. D. C. Lay, “Linear Algebra and its Applications”, Addison – Wesley.

  3. W. K. Nicholson, “Linear Algebra with Applications”, PWS Publishing Co.

  4. G. Schay, “Introduction to Linear Algebra”, Jones and Bartlett Publishers.

  5. T. Lawson, “Linear Algebra”,, John Wiley & Sons Inc.

  6. Many books on the subject are available at the Library. Students may wish to consult one or more of these as alternative sources of information, presentation, style and for additional problems.


  • Chapter Numbers refer to the textbook.

  • The syllabus and schedule are tentative and subject to modifications.

  • Approximately one week’s notice will be given before each test.

  1. Review (Directed Study; first two weeks of classes)

Chapter 4 General Vector Spaces
4.1 Real Vector Spaces
4.2 Subspaces
4.3 Linear Independence
4.4 Coordinates and Basis
4.5 Dimension
4.6 Change of Basis
4.7 Row Space, Column Space and null Space
4.8 Rank, Naullity and the Fundamental Matrix Spaces
4.9 Matrix Transformations from 
Rn to Rm
4.10 Properties of Matrix Transformations
Test #1, Weight: 20%, 50 min, Date: T.B.A.

  1. Chapter 5
    5.1 Eigenvalues and Eigenvectors
    5.2 Diagonalization
    Appendix B: Complex Numbers
    5.3 Complex Vector Spaces
    Application: Differential Equations (*)

  2. Chapter 6 Inner Product Spaces
    6.1 Inner Products
    6.2 Angle and Orthogonality in Inner Product Spaces
    6.3 Orthonormal Bases; Gram-Schmidt Process; QR-Decomposition
    6.4 Best Approximation; Least Squares
    Application: Least Squares Fitting to Data (*)
    Application: Function Approximation; Fourier Series (*)
    Test #2, Weight: 20%, 50 min, Date: T.B.A.

  3. Chapter 7 Diagonalization and QuadraticForms
    7.1 Orthogonal Matrices
    7.2 Orthogonal Diagonalization
    7.3 Quadratic Forms
    7.4 Optimization Using Quadratic Forms
    7.5 Hermitian, Unitary, and Normal Matrices

  4. Chapter 8 Linear Transformations
    8.1 General Linear Transformations
    8.2 Isomorphism
    8.3 Compositions and Inverse Transformations
    8.4 Matrices of General Linear Transformations
    8.5 Similarity

    Test #3, Weight: 20%, 50 min, Date: T.B.A.

  5. Chapter 9 Additional Topics (as time permits)
    (*) If time permits

Final Examination (cumulative),. Weight: 40%,. Length: 2 hours,
Date: as per Exam Schedule, issued by the Registrar’s Office.

Test/Exam Regulations:

  1. Tests/Exams may not be missed. There are NO MAKE-UP TESTS.

  2. Students must inform the instructor ahead of time by telephone (leave a message at: 780-479-9360) or by e-mail (tomtavou@math.concordia.ab.ca), if unable to present themselves for a test due to illness or other serious extenuating circumstances.

  3. Students must contact the instructor in person as soon as possible after a missed test. At that time, they should present a physician’s note or other appropriate proof of the reason the test was missed. In this case, the weight of the missed test will be added to the final exam.

  4. A missed test is considered non-excused, if either rule 2 or 3 above has not been adhered to. The missed test in this case is assigned a grade of zero.

  5. All tests/exams are closed book.

  6. Use of calculators is NOT permitted. Use of Mathematica® is permitted and/or required.

  7. All Concordia University College of Alberta rules and regulations apply; in particular the ones regarding Academic Honesty. See CUCACalendar.


Students should use the math lab outside classroom time as much as possible. The assigned homework includes problems that need to be done on the computer. Tests and exams will also include questions that need to be answered using the computer and Mathematica®.

Instructor-Student Communication:

  1. Students should know how to access their e-mail ”yourname@student.concordia.ab.ca” and to check it regularly. Istructors use it to send course announcements to students.

  2. The instructor regrets that he is unable to answer homework questions over e-mail.

Lecture Format:

Much of this course will be directed/supervised study. Students are expected to do a considerable amount of self-study. Classroom time will be used for discussion and clarification of concepts, and for examples. Working in groups is strongly encouraged.

Cellular Telephone Policy:

All cellular phones should be turned off upon enrty into any mathematics classroom or laboratory. A cellular phone may be left on, only with the express permission of the instructor. Permission MUST be obtained before the class commences. In tests/exams, all cellular phones must be de-activated and placed in the student’s bag. The bag must be left at the front of the class throughout the duration of the test/exam.

Evaluation Scheme:

  1. The weights of tests/exams are indicated under “Syllabus/Tests/Exams” above.

  2. Tests/exams are marked out of 100.

  3. The final weighted average is calculated as a percentage and reported on the alpha grading scale as follows:

% Range

Alpha Scale

98 – 100


95 – 97


90 – 94


85 – 89


80 – 84


75 – 79


70 – 74


65 – 69


60 – 64


55 – 59


50 -54


00 – 49


  1. The Bell curve will not be applied. Students know their standing in the course at any point in time as a result of their marks in the tests. Marks are not scaled.

  2. Assignments are given on a regular basis. However, they are not collected or graded. It is the responsibility of the individual student to do his/her homework.

  3. All final marks will be reported on the Four-Point Grading Scale as follows (also see Calendar ):

Four-Point Grading Scale


Alpha Grade

Grade Point Value









Very Good


















Minimal Pass






How to Do Well:

  1. Do not skip classes. Attend regularly. If you miss a class due to illness, obtain the notes and do the homework as soon as possible.

  2. Try to read ahead by one lecture.

  3. In class, pay attention; scrutinize every sentence or statement; never be embarrassed to ask.

  4. Look up your notes as soon as possible after a lecture.

  5. Study systematically, using pencil and paper; do not simply read.

  6. Study notes and textbook carefully. Make sure that you have understood a concept well, before proceeding to the next one. Test your understanding of the concept by mentally defining it or pretending to explain it to someone else; be clear, concise and precise.

  7. After studying solved examples, whether from your textbook or notes, make sure that you can solve the same problems independently.

  8. Try the homework problems after you have done steps 6 and 7 above.

  9. If a particular problem gives you trouble, leave it for a while; come back to it at a later time (or next day). If still unsuccessful, ask in class. If questions still persist, discuss them with your colleagues, and seek help from your professor.

  10. Use the math lab on a regular basis for your homework and for computer practice.

  11. Working in small groups often helps, provided the group participants are all willing to actively contribute to the work and discussions, and they are not simply passive recipients of work done by others.

  12. Remember… Question time is all the time! Ask!

  13. Study regularly. Do not “fall behind”. Every topic is built on the previous one.

  14. Do not cram before exams. Exams are designed to test your understanding of the material and your competence in the course according to established criteria and standards; they are also meant to rank students according to their performance in the course. They are not meant to trick or fool them.

  15. Review often. Reviewing enhances the understanding, aids learning and builds up confidence.

  16. Enjoy the course! I wish you well!

When difficulties arise:
If you feel you need help, do not hesitate to seek it. Here is a list of people who can assist you in various circumstances…

  1. About the course… Contact your professor, Dr. A. N. (Tom) Tavouktsoglou, F.H.-204,
    Tel. 780-479-9360, 

  2. General academic or personal… Contact the Associate Vice President of Student and Enrollment Services, Dr. Jonathan Strand, jonathan.strand@concordia.ab.ca

  3. Psychological, Personal… Contact the Campus Counselling Psychologist, Barbara Van Ingen, barbara.vaningen@concordia.ab.ca

  4. Spiritual, Pastoral… Contact the Campus Chaplain, Rev. Dr. Garry Dombrosky garry.dombrosky@concordia.ab.ca

  5. Career Counseling… Contact our campus career practitioner, Ms. Doreen Kooy, 

  6. Assistance in writing… Make an appointment by contacting Student and Enrollment Services.

Academic Hierarchy:

Dr. Vladimir Pitchko, Associate Professor of Chemistry, Chair of the Science Division,
F.H. – 305, Tel.: 780-379-9376,