Instructor: Ljudmila Kamenova
Office: Math Tower 3-115
Office hours: Th 11:30am-12:30pm, and Th 1:30-2:30pm in Math Tower 3-115; Tu 1:30-2:30pm - administrative undergraduate advising hours; or send me an e-mail: kamenova@math.stonybrook.edu.
TA: Alessandro Pilastro
TA's office hours and recitation: Alessandro Pilastro'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 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; Selfadjoint Operators, Normal Operators, and 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 30% of the total grade; each Midterm is worth 20% of the total grade; the Final is worth 30% of the total grade.
| Week. | Lecture Dates. | Topics covered from the Textbook. |
|---|---|---|
| 1. | Jan 22 - 26. | Intro to course, R^n and C^n, Vector spaces and subspaces (1.A, 1.B, 1.C) |
| 2. | Jan 29 - Feb 2. | Span, Linear independence, Bases, dimension. (2.A, 2.B, 2.C) |
| 3. | Feb 5 - 9. | Vector space of linear maps, null space, range (3.A, 3.B), MIDTERM 1 on Feb 8 in class (it covers everything up to 3.B). |
| 4. | Feb 12 - 16. | Matrices, inverses and isometries. Products, quotient vector spaces. (3.C, 3.D, 3.E) |
| 5. | Feb 19 - 23. | Duality. Zeros of polynomials, division algorithm for polynomials. (3.F, 4) |
| 6. | Feb 26 - March 1. | Factorization of polynomials. Invariant subspaces. (4, 5.A) |
| 7. | March 4 - 8. | Minimal polynomial. Upper triangular matrices. (5.B, 5.C) |
| 8. | March 11 - 15. | Spring break. |
| 9. | March 18 - 22. | Diagonalizable operators, commuting operators. Inner products, norms. (5.D, 5.E, 6.A) |
| 10. | March 25 - 29. | Orthonormal bases. Orthogonal complement, minimization. (6.B, 6.C) |
| 11. | April 1 - 5. | Self-adjoint and normal operators (7.A), MIDTERM 2 on April 4 in class (it covers everything up to 6.A). |
| 12. | April 8 - 12. | Spectral theorem. Positive operators, isometry. (7.B, 7.C, 7.D) |
| 13. | April 15 - 19. | Generalized eigenvalues, nilpotent operators. Generalized eigenspace decomposition. (8.A, 8.B) |
| 14. | April 22 - 26. | Jordan Form. Trace, quadratic forms. (8.C, 8.D, 9.A) | 15. | April 29 - May 3. | Alternating multilinear forms, determinants. Tensor products. (9.B, 9.C, 9.D) |
| 16. | May 14. | Final exam: 2:15-5pm, in the classroom. |
Homework is a fundamental part of this course. Late homework will not be accepted. Homework will account for 30% 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.
| Number. | Due Week (in recitation). | Exercises from the textbook. |
|---|---|---|
| 1. | Week of Jan 31 in recitation. | Problems 1.A.9, 1.A.10, 1.A.15, 1.B.3, 1.B.7, 1.C.3, 1.C.4, 1.C.22. |
| 2. | Week of Feb 7 in recitation. | Problems 2.A.3, 2.A.9, 2.A.19, 2.B.7, 2.B.9, 2.B.10, 2.C.3, 2.C.4, 2.C.20. |
| 3. | Week of Feb 14 in recitation. | Problems 3.A.3, 3.A.7, 3.A.12, 3.A.15, 3.B.3, 3.B.10, 3.B.22, 3.B.25. |
| 4. | Week of Feb 21 in recitation. | Problems 3.C.3, 3.C.5, 3.C.17, 3.D.2, 3.D.8. |
| 5. | Week of Feb 28 in recitation. | Problems 3.E.4, 3.E.13, 3.E.17, 3.F.5, 3.F.7. | 6. | Week of March 6 in recitation. | Problems 4.6, 4.7, 4.13, 4.14, 5.A.13. | 7. | Week of March 20 in recitation. | Problems 5.B.9, 5.B.15, 5.C.7, 5.C.12, 5.C.14. | 8. | Week of March 27 in recitation. | Problems 5.D.8, 5.E.6, 5.E.10, 6.A.9, 6.A.14. | 9. | Week of April 17 in recitation. | Problems 6.B.9, 6.C.5, 6.C.12, 7.A.7, 7.A.27. | 10. | Week of April 24 (you may turn it in late). | Problems 7.B.11, 7.B.20, 8.A.22, 8.B.8, 8.B.18. |
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