Math 4571: Advanced Linear Algebra ~ Homepage

Math 4571 (Advanced Linear Algebra), Spring 2025

Course Information
Instructor		Class Times		Office Hours
Evan Dummit edummit at northeastern dot edu 571 Lake Hall		MWR 4:35pm-5:40pm 115 Snell Library		M 12:05pm-1:05pm R 3:00pm-4:00pm 571 Lake Hall
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			Thursdays, 2:45pm-4:15pm, 003 Snell Library
Toby Busick-Warner			Fridays, 3:30pm-5:00pm, 003 Snell Library
All homework assignments will be posted on this webpage (see below). Homework assignments will be collected via Gradescope on 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. Exams are take-home and non-collaborative, and are submitted the same way as homework assignments.
The instructor will write lecture notes for the course (see below) in lieu of an official textbook as the semester progresses.

Homework Assignments + Exams
Some Tips on Problem Solving are available as suggestions for the written assignments.
Homework #1, due Fri Jan 17th. Homework #2, due Fri Jan 24th. 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
Chapter 1: Vector Spaces (21pp, v4.00, posted 1/17)	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 1.5 ~ Bases and Dimension

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
The schedule is subject to change! All sections refer to the course lecture notes.
Week	Schedule
Week of Jan 6 (class starts 1/6)	§0.1: Sets, Numbers, and Functions §0.2: Vectors in Rⁿ §0.3: Complex Numbers, Fields §0.4: Matrices and Systems of Linear Equations No homework this week.
Week of Jan 13	§0.4: Matrices and Systems of Linear Equations §0.5: Induction §0.6: Polynomials §1.1: The Formal Definition of a Vector Space Homework #1 due Friday, Jan 17th.
Week of Jan 20 (no class 1/20)	§1.2: Subspaces §1.3: Linear Combinations and Span §1.4: Linear Independence and Linear Dependence Homework #2 due Friday, Jan 24th.
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 Friday, Jan 31st.
Week of Feb 3	§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 Friday, Feb 7th3rd.
Week of Feb 10	§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 Friday, Feb 14th.
Week of Feb 17 (no class 2/17)	§3.1.2: Properties of Inner Products §3.2.1: Orthogonality, Orthonormal Bases, and the Gram-Schmidt Procedure §3.2.2: Orthogonal Complements and Orthogonal Projection Homework #6 due Friday, Feb 21st.
Week of Feb 24	§3.3.1: Characterizations of Inner Products §3.3.2: The Adjoint of a Linear Transformation §3.4.1: Least-Squares Estimates §3.4.2: Fourier Series Midterm exam due Friday, Feb 28th.
*Spring Break (no classes) from Mar 3 to Mar 7*
Week of Mar 10	§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 14th.
Week of Mar 17	§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 21st.
Week of Mar 24	§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 28th.
Week of Mar 31	§4.4.5: Matrix Exponentials and the Jordan Form §5.1: Bilinear Forms §5.2.1: Quadratic Forms §5.2.2: Diagonalization of Quadratic Varieties Homework #10 due Friday, Apr 4th.
Week of Apr 7	§5.2.3: Definiteness of Real Quadratic Forms §5.2.4: The Second Derivatives Test §5.2.5: Sylvester's Law of Inertia §5.3: Singular Value Decomposition Homework #11 due Friday, Apr 11th.
Week of Apr 14 (classes end 4/15)	§5.4: Pseudoinverses Homework #12 due Friday, Apr 18th.
FINAL EXAM is take-home and due on Fri April 25th.