Instructor: Filip Zivanovic, office hours: My SB card
TA: Ceyhun Elmacioglu, office hours and recitation times: Ceyhun Elmacioglu's SB card
NB. Most information and material related to the course will be posted on this page.
The exceptions are:
Gradescope will be used to submit homework assignments and to see your graded homework.
Brightspace will be used to make announcements, and solutions to problem sets will also be posted there.
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, and 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.
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.
Grading:
Homework (first five homeworks from MAT310 and the six ones below) 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.
Letter grades are calculated as follows: A 90-100% B 80-89% C 70-79% D 60-69% F <60%
Midterm 1 will be held on Wednesday, October 1st, in class (Javits Lectr 111)
Midterm 2 will be held on Thursday, April 10th, in class (Earth&Space177).
The final exam will be held on Wednesday. December 10, 11:15-1:45 pm, location will be shown here in due course
Syllabus:
MAT310 part / MAT315 part (tentative)
Week. Lecture Dates. Topics covered from the Textbook.
Aug 25-27 Vector spaces. Subspaces (1A, 1B, 1C).
Sep 1-3 Labor day - no class. Span and linear independence (2A).
Sep 8-10 Bases and dimension. Linear maps (2B, 2C, 3A).
Sep 15-17 Null space and range. Matrices (3B, 3C).
Sep 22-24 Invertibility and isomorphisms. Products and quotients (3D, 3E).
Sep 29-Oct 1 Duality (3F). Midterm 1 (in class)
Oct 6-8 Monday MAT310. Wednesday: Polynomials. Invariant subspaces. (4, 5A).
Oct 13-15 Fall break - no class on Monday. Wednesday: Minimal polynomial. Upper-triangular matrices. (5B, 5C)
Oct 20-22 Diagonalization. Commuting operators. Inner products and norms. (5D, 5E, 6A).
Oct 27-29 Orthonormal bases. Orthogonal complements and projections. Self-adjoint and normal operators (6B, 6C, 7A).
Nov 3-5 MIDTERM 2 on Wed Nov 5 in class (Earth&Space177; covers everything up to 6A).
Nov 10-12 Spectral theorem. Positive operators. Isometries (7B, 7C, 7D).
Nov 17-19 Generalized eigenvectors. Generalized eigenspaces. Jordan Form. (8A, 8B, 8C).
Nov 24-26 Trace. Bilinear and quadratic forms (8D, 9A). Thanksgiving - no class on Wednesday
Dec 1-3 Alternating multilinear forms. Determinants. Tensor products (9B, 9C, 9D).
Dec 8-10 FINAL EXAM on Wednesday 11:15 am-1:45 pm; Location will be shown here in due course
Homework:
Homework is a fundamental part of this course. Late homework will not be accepted. Homework (first five homeworks from MAT310 and the six ones below)
will account for 20% of the total grade. Your lowest three scores from the homeworks will be dropped; i.e., your best 8/11 homeworks will be accounted for.
The exercises will be taken from the course textbook. Homework is due to be submitted on Gradescope by Wednesdays, 11:59 PM, as indicated below:
Number Due Exercises from the textbook.
Oct 22 Problems 4: 4, 6; 5A: 6, 9; 5B: 6, 7.
Oct 29 Problems 5C: 1; 5D: 2, 3; 5E: 3; 6A: 6, 15.
Nov 12 Problems 6B: 2, 8; 6C: 2, 3; 7A: 1, 6
Nov 19 Problems 7B: 2, 8; 7C: 1, 6; 7D: 2, 4
Nov 26 Problems 8A: 6, 20, 22; 8B: 8, 18; 8C: 5.
Dec 3 Problems 8D: 2, 5, 13; 9A: 2, 5, 10.
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), ECC (Educational Communications Center) Building, room 128, (631) 632-6748. They will determine with you what accommodations, if any, are necessary and appropriate. All information and documentation is confidential.
Students who might require special evacuation procedures in the event of an emergency are urged to discuss their needs with both the instructor and DSS. For important related information, click here.
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