THE UNIVERSITY OF WESTERN ONTARIO

LONDON CANADA

DEPARTMENT OF MATHEMATICS

__Mathematics 3121A/9050A: Advanced Linear Algebra __

**Fall 2014**

**
**

**INSTRUCTOR:** Lex Renner

**TEXTBOOK:** Linear Algebra, 4th Edition, by S. Friedberg, A. Insel and L. Spence,
published by Prentice Hall, ISBN: 0-13-008451-4.

**PREREQUISITES:** Mathematics 2120 A/B.

**COURSE OUTLINE:** A continuation of the material of Mathematics 2120A/B including
properties of complex numbers and the principal axis theorem; singular value decomposition; linear
groups; similarity; Jordan canonical form; Cayley-Hamilton theorem; bilinear forms; Sylvester's theorem.
This is an in-depth development of the theory of linear operators and bilinear forms.

**EVALUATION OF STUDENT PERFORMANCE:**

Assignments---worth 40%

Midterm #1---October 17, 2014----worth 10%

Midterm #2---November 21, 2014----worth 10%

Final Exam---December 10, 2014----worth 40%

Assignment #1, DUE Sept 26: p.322--327: #3,4,11,12,18,19.

Assignment #2, DUE Oct 17: p.336--338: #4b),9,10,14,17. p.352--356: #2b),2c),6,7,11.

Assignment #3, DUE Oct 24: p.366: #6,12,18. p.375: #2b),4,6,7.9.

Assignment #4, DUE Nov 21.

Assignment #5. DUE Dec **.

5.4 Invariant Subspaces and Cayley-Hamilton Theorem

6.1 Inner product spaces, norms, Cauchy-Schwarz inequality, triangle inequality

6.2 Gram-Schmidt process

6.3 Adjoints

6.4 Normal and self-adjoint operators

Five questions: 1. Five True or False questions. 2. General adjoint and ``perp'' question. 3. Find the matrix of T^* question. 4. Characteristic polynomial/Cayley-Hamilton question. 5. Normal and self-adjoint question (four short parts).

7.1 Jordan Canonical Form

Chapter 5: Sections 5.2, 5.4.

Chapter 6: Sections 6.1, 6.2, 6.3, 6.4, 6.5, 6.6.

Chapter 7: Sections 7.1, 7.2, 7.3, 7.4.

Here is a rough guide to the 3hr exam.

1.[20] True or False.

2.[10] Basic T:V-->V question.

3.[10] Cayley-Hamilton question.

4.[10] Find A^n question.

5.[10] Gram-Schmidt question.

6.[10] Adjoint question.

7.[10] Normal or self-adjoint question.

8.[10] Jordan Canonical form question.

9. [5] Jordan Canonical form question.

10.[5] Rational Canonical form question.

**IMPORTANT SENATE POLICY:**

Students are responsible for ensuring that their selection of courses is appropriate and accurately recorded and that all course prerequisites have been successfully completed. If the student does not have the requisites for a course, and does not have written special permission from his or her Dean to enroll in the course, the student may be removed from the course and it will be deleted from the student's record. This decision may not be appealed. A student will receive no adjustment to his or her fees in the event that he or she is dropped from a course for failing to have the necessary prerequisites.

**STATEMENT ON ACADEMIC OFFENCES:**

** **"Scholastic offences are taken seriously and students are directed to
read the appropriate policy, specifically, the definition of what
constitutes a Scholastic Offence, at the following Web site:
http://www.uwo.ca/univsec/handbook/appeals/scholoff.pdf** "**

**NEW MEDICAL EXCUSE REGULATIONS:**

If you are unable to meet a course requirement due to illness or other

serious circumstances, you must provide valid medical or other

supporting documentation to the Dean's Office as soon as possible and

contact your instructor immediately. It is the student's responsibility

to make alternative arrangements with their instructor once the

accommodation has been approved and the instructor has been informed. In the event of a missed final exam, a "Recommendation of Special Examination" form must be obtained from the Dean's Office immediately.

For further information please see:

http://www.uwo.ca/univsec/handbook/appeals/medical.pdf

A student requiring academic accommodation due to illness, should use the Student Medical Certificate when visiting an off-campus medical facility or request a Record's Release Form (located in the Dean's Office) for visits to Student Health Services.

The form can be found here:

https://studentservices.uwo.ca/secure/medical_document.pdf