Instructor: Ljudmila Kamenova
Office: Math Tower 3-115
Office hours: TR 11am-12 noon in Math Tower 3-115; or send me an e-mail: kamenova@math.stonybrook.edu.
TA: Ceyhun Elmacioglu
TA's office hours and recitation: Ceyhun Elmacioglu'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. 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 2023.
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 accounts for 20% of the total grade; each Midterm is worth 20% of the total grade; the Final is worth 40% of the total grade.
| Week. | Lecture Dates. | Topics covered from the Textbook. |
|---|---|---|
| 1. | Jan 28 - 30. | Vector spaces and subspaces (1.A, 1.B, 1.C). |
| 2. | Feb 4 - 6. | Span and linear independence, bases, dimension, (2.A, 2.B, 2.C). |
| 3. | Feb 11 - 13. | Linear maps, null space and range (3.A, 3.B). |
| 4. | Feb 18 - 20. | Matrices, invertibility and isomorphisms (3.C, 3.D). |
| 5. | Feb 25 - 27. | Products and quotients, duality (3.E, 3.F). |
| 6. | March 4 - 6. | MIDTERM 1 on March 4th in class (Humanities 1006; it covers everything up to 3.D?), Polynomials (4). |
| 7. | March 11 - 13. | Invariant subspaces. Minimal polynomial. Upper triangular matrices (5.A, 5.B, 5.C). |
| 8. | March 18 - 20. | Spring break, no classes. |
| 9. | March 25 - 27. | Diagonalizable operators, commuting operators, inner products, norms (5.D, 5.E, 6.A). |
| 10. | April 1 - 3. | Orthonormal bases. Orthogonal complement, minimization (6.B, 6.C). |
| 11. | April 8 - 10. | Self-adjoint and normal operators (7.A), MIDTERM 2 on April 10th in class (Lgt Engr Lab 152; it covers everything up to 6.A). |
| 12. | April 15 - 17. | Spectral theorem. Positive operators, isometry (7.B, 7.C, 7.D). |
| 13. | April 22 - 24. | Generalized eigenvalues, nilpotent operators, generalized eigenspace decomposition (8.A, 8.B). |
| 14. | April 29 - May 1. | Jordan Form, trace, quadratic forms (8.C, 8.D, 9.A). | 15. | May 6 - 8. | Alternating multilinear forms, determinants, tensor products (9.B, 9.C, 9.D). |
| 16. | May 15. | Final exam: 11:15am-1:45pm, in the classroom (Lgt Engr Lab 152). |
Homework is a fundamental part of this course. Late homework will not be accepted. Homework will account for 20% of the total grade. The exercises will be taken from the course textbook. Homework is due in your recitation in the week indicated below and should be handed to your recitation instructor. Here are the HW assignments after the split from MAT 310.
| Number. | Due Week. | Exercises from the textbook. |
|---|---|---|
| 1. | Week of March 26 in recitation. | Problems 5A: 6, 8, 9; 5B: 6,7. |
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