Linear Algebra

Last update: December 30, 2005

*This page will be updated at least once a week. Check for announcements and postings regularly, mostly at the bottom of the page!*

**Lecturer:** Detlef Gromoll, Math 5-110

Phone: 632-8286.
Email: *detlef@math.sunysb.edu*

URL: *http://www.math.sunysb.edu/~detlef*

Classes: MWF 9:35-10:30, Harriman Hall 108

Office Hours: W 2:30-4:30pm (Math 5-110), Th 2-3pm {UG Office P-143}, and by appointment.

**Teaching Assistant:** Pedro Solórzano, Math 2-119

Email: *pedro@math.sunysb.edu*

URL: *http://www.math.sunysb.edu/~pedro*

Recitations: Th 9:50-10:45 {R01), Physics P-128

W 3:50-4:45 {R02}, Physics P-117

Office Hours: W 10:30-11:30 and 13:00-15:00, W 17-19 Math Learning Center [MLC] S-240A.

**About this Course:** This is an advanced undergraduate linear algebra course, which will be more rigorous and also cover additional material needed in various areas of mathematics and its applications - including vector spaces, linear transformations, eigenvalues, and inner product spaces. Optional projects may address further topics like elementary Jordan forms.

**Prerequisites:** MAT 211 or equivalent, MAT 200, or permission of instructor.

**Text:** Sheldon Axler, *Linear Algebra (done right)*, second edition, Springer (2004).

This is a new text book for the course. We will cover the first six chapters in complete detail, and selected topics from the remainder of the text. The book costs $39.95 new and $30 used, although I doubt there will be used copies available. Our bookstore had 46 copies on the shelves.

**Grading:** There will be 2 midterms, 60 minutes each, on Friday, October 07, and on Wednesday, November 16; both given in class - *no makeups*. If one midterm exam is missed because of a serious (documented) illness or emergency, the semester grade will be determined based on the balance of the work in the course. A final examination will be held on Wednesday, December 21, 8-10:30am (Period 1).
Students are expected to ensure when they register for this course that they will be available for the final examination, and that they do not have too many final exams on that date.
The final course grades in MAT 310 will be determined as follows:

Homework/Class Participation 30% (120pts), Midterm Tests 20% (80pts) each, Final Exam 30% (120pts)

We will give up to 25% (100pts) extra credit for at most 2 projects, to be assigned below, for a maximum total of 500 points.

*Incompletes* will be granted only if documented circumstances beyond your control prevent you from completing the course work, according to strict University rules.

**Homework/Class Participation:** You can not learn mathematics without doing mathematics. It is essential to be an active participant in class and to solve problems: Each week a homework assignment will be posted further down on this page, normally Wednesday evening. It is due the following week by Thursday 5pm to be collected by our TA. Homework will be graded, returned, and discussed in recitation, which will also include regular short quizzes. While you may work together with others in the class (which can be a rewarding experience), write up your own solutions in your own words. Since homework earns credit, it is assumed that everyone submitting particular problems has solved them individually. The goal of the homework is to understand the material, not to merely hand in some paper. This is a more advanced math course where coherent arguments and rigorous proofs are often required. Late homework will *not* be accepted.

**Approximate Course Schedule:**

*Weeks of* *Material from*

8/29-9/30 Chapters 1,2,3

Midterm 1 - F 10/07 [in class; on Chapters 1,2,3 (except for section on invertibility)]

10/10-11/11 Chapters 4,5,6

Midterm 2 - W 11/16 [in class; on Chapters 3,4,5,6 (first two sections)]

11/11-12/09 Topics from Chapters 7,8,10 (Chapter 9: optional reading)

12/14
Special review session (see below)

Final Exam - W 12/21 [in our classroom, cumulative]

**Special Needs:**
If you have a physical, psychological, medical or learning disability that
may impact your course work, please contact Disability Support Services,
ECC (Educational Communications Center) Building, room 128, (631) 632-6748.
They will determine with you what accommodations are necessary and
appropriate. All information and documentation is confidential.

Students requiring emergency evacuation are encouraged to discuss their
needs with their professors and Disability Support Services. For procedures
and information, go to the following web site.

*http://www.stonybrook.edu/facilities/ehs/fire/disabilities.shtml*

**Week by Week Details:** Assignments are listed here.

8/29-9/02 Study in detail:

Chapter 1 - Complex numbers; Definition of vector space; Properties of vector spaces

Homework problems:
p19 1-4
[due Th 9/08]

9/06-09 Review the first three sections of Chapter 1

Study in detail: Chapter 1 - Subspaces; Sums and direct sums

Homework problems:
pp19-20 5-11,14 Bonus: 15
[due Th 9/15]

9/12-16 Review all of Chapter 1

Study in detail: Chapter 2 - Span and linear Independence; Bases (begin reading)

Homework problems:
pp35-36 1,3,4,6,8,11,13 Bonus: 15
[due Th 9/22]

9/19-23 Study in detail:

Chapter 2 - Bases; Dimension

Chapter 3 - Definitions and examples

Homework problems:
p36 17 pp59-60 1,2,3,5,9,10 Bonus: 13
[due Th 9/29]

9/26-30 Review all of Chapter 2

Study in detail: Chapter 3 - Null spaces and ranges; The matrix of a linear map

Homework problems:
pp60-61 12,16,18,19 Bonus: 24
[due Th 10/06]

10/03-07 Review the first three sections of Chapter 3

Carefully review Chapters 1 through 3 for the (in class) Friday midterm.

Homework problems:
pp61-62 22,23,26 Bonus: 17
[due W 10/19 - see below]

10/10-14 Study in detail:

Chapter 3 - The matrix of a linear map; Invertibility

Homework Problems: p73 1,2,3,5 Bonus: 4
[due M 10/24]

10/17-21 Study in detail:

Chapter 3 - Invertibility (conclusion)

Chapter 4 - Degree; Complex coefficients; Real coefficients

Homework Problems:
pp94-95 2,4,6,10,11,14 Bonus: 8
[due M 10/31]

10/24-28 Study in detail:

Chapter 5 - Invariant subspaces; Polynomials applied to operators;

Upper-triangular matrices; Diagonal matrices

Homework Problems:
pp95-96 15,16,18,22,23 Bonus: 21
[due M 11/07]

10/31-11/04 Study in detail:

Chapter 5 - Invariant subspaces on real vector spaces;

Chapter 6 - Inner products; Norms

Homework Problems:
pp122-123 1,3,4,5,6 Bonus: 8
[due M 11/14]

11/14-18 Study in detail:

Chapter 6 - Norms; Orthonormal bases

Homework Problems:
pp123-124 9,10,13,14 Bonus: 11
[due W 11/23]

11/21-23 Study in detail:

Chapter 6 - Orthogonal projections and minimization problems; Linear functionals

Homework Problems:
pp124-125 15,18,20,21,22 Bonus: 25
[due Th 12/01]

11/28-12/02 Study topics discussed in class:

Chapter 6 - Adjoints;

Chapter 7 - Self-adjoint and normal operators; Spectral theorems

Homework Problems:
p125 26,28,31 pp158-159 4,8,15 Bonus: p125 32
[due Th 12/08]

12/05-09 Study topics discussed in class:

Chapter 7 - Positive operators; Isometries;

Chapter 8 - Generalized eigenvectors; The characteristic polynomial

Homework Problems:
pp160-161 19,21,22 1,4,7,14 Bonus: 15
[due Th 12/15]

12/12 Study topics discussed in class:

Chapter 10 - Trace and determinant of an operator

**Note:** Detailed hardcopy solutions of all assigned homework problems will be available in recitation regularly after the due date.

**Course Review:** W 12/14, 2-5pm, Mathematics P-131.

**Optional Projects:** You may submit work on not more than three of the following 6 independent study type problems, of which the two best will be counted up to 50 points or about 10% extra credit each toward the course grade. Problems are in essence taken from our text, or very close to it. It will often be necessary to first read and understand more of the material of the corresponding sections. Consulting additional sources is always helpful. Problems marked with an asterisk are more difficult or elaborate. If you want to earn extra credit toward a high course grade you must work on at least one of those. All presentations should be reasonably detailed and neatly typed. Formulas can be entered by hand, if necessary. On the average, an optimal treatment of a topic would probably take 2-3 pages. Keep it short, but complete. Problems will be added through end of November. Of course, partial solutions will earn partial credit. Good luck!

*Deadline:* Wednesday, December 21. Turn papers in with the Final, i.e. by 10:30am, at the very latest -- preferably earlier. Sorry, late papers *will not* be accepted anymore after that deadline under any circumstances.

1.* *Fundamental Theorem of Algebra* (.pdf file)

2. *Upper-triangular matrices* (.pdf file)

3. *Orthogonal projections* (.pdf file)

4. *Fibonacci numbers* (.pdf file)

5. *Special basic project* (.pdf file)

6.* *Jordan forms and roots* (.pdf file)

**Statistics and Final Comment.** This course was completed by 50 students.

Median score of the Final Exam: 75/120.

Median score for the entire course (without projects): 256/400.

Projects: 41 (by 30 students), generally well done.

Course grades: Mean 2.76 ~ B-, median 2.95 ~ B.

Overall effort and performance was very good in a challenging course.
Congratulations!