Math 4571 (Advanced Linear Algebra), Spring 2026
| Course Information | |||||
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| Instructor | Class Times | Office Hours | |||
| Evan Dummit edummit at northeastern dot edu 571 Lake Hall |
MWR 4:35pm-5:40pm 330 Dodge Hall |
M noon-1pm MR 3pm-4pm 571 Lake Hall |
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| For detailed information about the course, please consult the 4571 Course Syllabus. (Note: any information given in class or on this webpage supersedes the written syllabus.) | |||||
| We will use Piazza for any course-related discussion: here is the 4571 Piazza page, which can also be accessed through the course Canvas page. Access is restricted to students in Math 4571. | |||||
| Teaching Assistant | Problem Session | ||||
| Aaron Agulnick | Tuesdays, 1pm-2:30pm, Ryder 143 | ||||
| Neil Gabrielson | Fridays, 3:30pm-5pm, Ryder 147 | ||||
| All homework assignments will be posted on this webpage (see below). Homework assignments will be submitted via Gradescope, which is accessible through Canvas. Please submit scans of your homework pages by 11:59pm Eastern on the due date. Late assignments may be penalized at the grader's discretion. |
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| Use of large-language models ("generative AI", such as ChatGPT or Claude) or equivalent technology in any manner is expressly prohibited in this course. This includes, but is not limited to, summarizing course information, asking for hints or solutions to course assignments, and general information retrival on course topics. Ask questions during class, in office hours, on Piazza, during problem sessions, or via email instead. | |||||
| The instructor will write lecture notes for the course (see below) in lieu of an official textbook as the semester progresses. Reference texts are available upon request. | |||||
| Homework Assignments + Exams | |||||||
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| Some Tips on Problem Solving are available as suggestions for the written assignments. | |||||||
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Homework #1, due Tue Jan 13th.
Problems labeled "Challenge" are optional. However, I highly recommend thinking about them, especially if you are considering applying to graduate school in mathematics, and especially especially if you might want me to write you a recommendation letter. | |||||||
| Homework assignments are to be submitted via the course's Canvas page. To submit an assignment, navigate to "Assignments" and select the appropriate homework assignment. Then attach scans of each page of your assignment (or a pdf) and click Submit. Please submit the pages in order and verify that all pages are included and uploaded correctly. You may resubmit as many times as you like. Assignments are due at 11:59pm eastern time. Late submissions, messy submissions, submissions failing to mark pages on which problems appear, or otherwise unreadable submissions will be penalized at the grader's discretion. | |||||||
| Handouts / Lecture Notes | |||
|---|---|---|---|
| Handout | Topics | ||
| Chapter 0: Preliminaries (23pp, v4.00, posted 1/4) | 0.1 ~ Sets, Numbers, and Functions 0.2 ~ Vectors in Rn 0.3 ~ Complex Numbers, Fields 0.4 ~ Matrices, Systems of Linear Equations, and Determinants 0.5 ~ Induction 0.6 ~ Polynomials |
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| Exam Information | |||
|---|---|---|---|
| Exam | Date, Time, Location | Topics | Review Material |
| Midterm 1 | Wed, Feb 18th In Class |
Homeworks 1-5 Notes 0.1-2.2 |
Review problems to be posted Review session TBA |
| Midterm 2 | Wed, Apr 1st In Class |
Homeworks 6-9 Notes 3.1-4.3 |
Review problems to be posted Review session TBA |
| Final | Date TBA Time TBA Location TBA |
The final is COMPREHENSIVE! Homeworks 1-11 Notes Chapters 0-5 |
Review problems to be posted Review sessions TBA |
| On all exams, calculators are permitted though generally they are not needed, and you are allowed a 1-page note sheet (8.5in by 11in, both sides) on which you may write or type anything. | |||
| Tips For Success In This Course | |||
|---|---|---|---|
| Attend Lecture | Missing lecture is a bad idea! If for any reason you cannot make it to a class, you should review notes from someone who did attend. You are responsible for all material covered in lecture. | ||
| Read the Lecture Notes | The lecture notes are a comprehensive source of material for the course. The notes are intended as review material, although many students like to read them as preparation before attending the lecture on the corresponding topics. Please note that the electronic notes are not identical to the material covered in class: this is by design, so as to provide you a slightly different perspective on the material. | ||
| Solve Homework Problems | Much of the learning in this course will take place as you solve the homework problems. Like many other activities, problem-solving and proof-writing are things that are learned by doing them, not by hearing someone else tell you about them or reading about them in a book. As such, the homework assignments are an integral part of the course, and are fundamental to learning the material. It is highly recommended that you look over the homework assignments as soon as they are available, and work on them well in advance of the deadline: many problems will take substantial time and effort to solve, and you should expect to spend as much time as you need to finish the assignments. | ||
| Attend Problem Sessions | There are weekly problem sessions run by the course TAs. The goal of the problem session is to provide you a location where you can work collaboratively with other students on assignments, and also get TA assistance. Students are highly encouraged to attend the problem sessions and work on the homework assignments there. | ||
| Attend Office Hours | Office hours are specifically reserved for you to receive individual, one-on-one help from the instructor. Office hours will be the most effective when you have already put in effort to learn the material on your own (including trying to solve the homework problems), and when you come in with a list of specific questions or topics you are struggling with. | ||
| Course Schedule | |||
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| The schedule is subject to change! All sections refer to the course lecture notes. | |||
| Week | Schedule | ||
| Week of Jan 5 (class starts 1/7) |
§0.3: Complex Numbers, Fields §0.4.1: Matrix Arithmetic §0.4.2: Systems of Linear Equations §0.4.3: Elementary Matrices and Echelon Forms §0.4.4: Inverses of Matrices No homework this week. |
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| Week of Jan 12 | §0.4.5: Determinants of Matrices §0.4.6: Properties of Determinants §0.5: Induction §0.6: Polynomials Homework #1 due Tuesday, Jan 13th. |
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| Week of Jan 19 (no class 1/19) |
§1.1: The Formal Definition of a Vector Space §1.2: Subspaces §1.3: Linear Combinations and Span §1.4: Linear Independence and Linear Dependence Homework #2 due Tuesday, Jan 20th. |
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| Week of Jan 27 |
§1.5.1: Bases and Their Properties §1.5.2: Existence and Construction of Bases §1.5.3: Dimension §1.5.4: Computing Bases §2.1.1: Linear Transformations Homework #3 due Tuesday, Jan 27th. |
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| Week of Feb 2 |
§2.1.2: Kernel and Image §2.1.3: Algebraic Operations on Linear Transformations §2.1.4: One-to-One Linear Transformations §2.1.5: Isomorphisms of Vector Spaces §2.2.1: The Matrix Associated to a Linear Transformation Homework #4 due Tuesday, Feb 3rd. |
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| Week of Feb 9 | §2.2.2: Algebraic Properties of Associated Matrices §2.2.3: Rank §2.2.4: Inverse Matrices and Inverse Transformations §2.2.5: Change of Basis, Similarity §3.1.1: Inner Products Homework #5 due Tuesday, Feb 10th. |
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| Week of Feb 16 (no class 2/16) |
Review for Midterm 1. MIDTERM 1 in class Wednesday, Feb 18th. §3.1.2: Properties of Inner Products §3.2.1: Orthogonality, Orthonormal Bases, and the Gram-Schmidt Procedure No homework this week. |
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| Week of Feb 23 | §3.2.2: Orthogonal Complements and Orthogonal Projection §3.3.1: Characterizations of Inner Products §3.3.2: The Adjoint of a Linear Transformation §3.4.1: Least-Squares Estimates Homework #6 due Friday, Feb 27th. |
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| Spring Break (no classes) from Mar 2 to Mar 6 | |||
| Week of Mar 9 | §3.4.2: Fourier Series §4.1.1: Eigenvalues and Eigenvectors §4.1.2: Eigenvalues and Eigenvectors of Matrices §4.1.3: Eigenspaces §4.2: Diagonalization Homework #7 due Friday, Mar 13th. |
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| Week of Mar 16 | §4.3.1: Generalized Eigenvectors §4.3.2: The Jordan Canonical Form §4.4.1: Spectral Mapping and the Cayley-Hamilton Theorem Homework #8 due Friday, Mar 20th. |
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| Week of Mar 23 | §4.4.2: The Spectral Theorem for Hermitian Operators §4.4.3: Transition Matrices and Markov Chains §4.4.4: Systems of Linear Differential Equations Homework #9 due Friday, Mar 27th. |
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| Week of Mar 30 | §4.4.5: Matrix Exponentials and the Jordan Form Review for Midterm 2. MIDTERM 2 in class Wed, Apr 1st §5.1: Bilinear Forms No homework this week. |
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| Week of Apr 6 | §5.2.1: Quadratic Forms §5.2.2: Diagonalization of Quadratic Varieties §5.2.3: Definiteness of Real Quadratic Forms §5.2.4: The Second Derivatives Test §5.2.5: Sylvester's Law of Inertia Homework #10 due Friday, Apr 10th. |
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| Week of Apr 13 (classes end 4/17) |
§5.3.1: Singular Values and Singular Value Decomposition §5.3.2: Applications of Singular Value Decomposition §5.3.3: The Moore-Penrose Pseudoinverse Homework #11 due Friday, Apr 17th. |
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| FINAL EXAM is on date TBA at time TBA in location TBA during finals week (Mon Apr 20 to Sun Apr 26) | |||