MAT 315/ MAT 526: Advanced Linear Algebra

Fall 2026, MW 9:30-10:50am, Humanities 2047


Instructor: Ljudmila Kamenova

Office: Math Tower 3-115

Office hours: MW 1:30-2:30pm in Math Tower 3-115; or send me an e-mail: kamenova@math.stonybrook.edu.

TA: Bryan Wong

TA's office hours and recitation: Bryan Wong's web card

Course Description:

Finite dimensional vector spaces over a field, linear maps, isomorphisms, dual spaces, quotient vector spaces, bilinear and quadratic functions, inner products, canonical forms of linear operators, multilinear algebra, tensors. This course serves as an alternative to MAT 310. It is an intensive course, primarily intended for math majors in the Advanced Track program.

MAT 315 STARTS TOGETHER WITH MAT 310, AND WE SPLIT AFTER MAT 310's MIDTERM 1. UNTIL THEN, PLEASE REFER TO THE FOLLOWING WEBPAGE. Here is the webpage for MAT 310: MAT 310.

Major Topics Covered: Matrices and Operations on Matrices; Determinants of Matrices; Vector Spaces and Subspaces; Linear Transformations and Linear Operators; Kernels and Images; Basis for Vector Space and the Dimension of a Vector Space; Eigenvalues, Eigenvectors and the Diagonalization of Linear Operators; the Cayley-Hamilton Theorem; Inner Product Spaces; Self-adjoint Operators, Normal Operators, Orthogonal Operators; the Spectral Theorem.

In addition to the MAT 310 topics, we are also going to cover:

1. Vector spaces over other fields.

2. Quotient spaces.

3. Dual spaces.

4. Polylinear maps and tensors.

5. Symmetric and anti-symmetric tensors.

6. Determinant of a linear operator via polylinear maps.

Textbook: Linear Algebra Done Right (4th Ed.), by Sheldon Axler, Springer 2024.

Here is the electronic version of this textbook which is legally available for free as a PDF file: https://link.springer.com/book/10.1007/978-3-031-41026-0

Sheldon Axler's videos accompanying his book: https://linear.axler.net/LADRvideos.html

Grading: Homework and quizzes each account for 10% of the total grade; each Midterm is worth 20% of the total grade; the Final is worth 40% of the total grade.

Tentative Syllabus:


Week. Lecture Dates. Topics covered from the Textbook.
1. Aug 24-26. Vector spaces and subspaces (1.A, 1.B, 1.C).
2. Aug 31-Sep 2. Span and linear independence, bases, dimension, (2.A, 2.B, 2.C).
3. Sep 7-9. Monday: Labor Day - no class. Linear maps (3.A).
4. Sep 14-16. Null space and range. Matrices (3.B, 3.C).
5. Sep 21-23. Invertibility and isomorphisms. Products and quotients. (3.D, 3.E).
6. Sep 28-30. Monday: MIDTERM 1 on Sep 28 in class (Javits Lectr 109). Duality (3.F).
7. Oct 5-7. Polynomials. Invariant subspaces. Minimal polynomial. (4, 5.A, 5.B).
8. Oct 12-14. Monday: Fall break - no class. Upper triangular matrices (5.C).
9. Oct 19-21. Diagonalizable operators, commuting operators, inner products, norms (5.D, 5.E, 6.A).
10. Oct 26-28. Orthonormal bases. Orthogonal complement, minimization (6.B, 6.C).
11. Nov 2-4. Self-adjoint and normal operators. Spectral theorem (7.A, 7.B).
12. Nov 9-11. Review. Wednesday: MIDTERM 2 on November 11th in class (Humanities 2047).
13. Nov 16-18. Positive operators, isometry. Generalized eigenvalues and eigenvectors. (7.C, 7.D, 8.A).
14. Nov 23-25. Generalized eigenspace decomposition (8.B). Wednesday: Thanksgiving break - no class.
15. Nov 30 - Dec 2. Jordan Form, trace. Bilinear and quadratic forms (8.C, 8.D, 9.A).
16. Dec 7. Alternating multilinear forms, determinants, tensor products (9.B, 9.C, 9.D).
17. Dec 16. Wednesday: FINAL EXAM: 8:00am-10:45am, in the classroom (Humanities 2047).

Homework:

Homework is a fundamental part of this course. Late homework will not be accepted.

Homework will account for 10% of the total grade. Your lowest two scores from the homeworks will be dropped; i.e., your best 8/10 homeworks will be accounted for.

The exercises will be taken from the course textbook. Homework is due to be submitted on Gradescope (the same one that was set up for MAT 310) by the given date at 11:59 PM, as indicated below. Here are the HW assignments after the split from MAT 310.


Number. Due Date. Exercises from the textbook.
1. Oct 19. Problems 5A: 6, 8, 9; 5B: 6,7.
2. Oct 26. Problems 5C: 1; 5D: 2, 3; 5E: 3; 6A: 6.
3. Nov 9. Problems 6B: 2, 8; 6C: 2, 3; 7A: 1.
4. Nov 23. Problems 7B: 2, 8; 7C: 1, 6; 7D: 2.
5. Nov 30. Problems 8A: 6, 22; 8B: 8, 18; 8C: 5.

Accessibility Support Center (SASC) Statement: If you have a physical, psychological, medical or learning disability that may impact your course work, please contact the Student Accessibility Support Center (SASC), Stony Brook Union Suite 107, (631) 632-6748. They will determine with you what accommodations, if any, are necessary and appropriate. All information and documentation is confidential.

Students who require assistance during emergency evacuation are encouraged to discuss their needs with their professors and the staff at the Student Accessibility Support Center (SASC). For procedures and information go to the SASC website.

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