Math 4571 (Advanced Linear Algebra), Spring 2023
| 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, Snell Library 123 | R noon-1:00pm WR 3:00pm-4:00pm Online via Zoom |
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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.) | |||||
| Teaching Assistant | Problem Session | ||||
| Connor Anderson anderson.con at northeastern | Friday, 3:00pm-5:00pm, Ryder 143 | ||||
| All homework assignments will be posted on this webpage (see below), and are to be submitted via the 4571 Canvas page. 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. The course will generally follow the presentation in "Linear Algebra" by Friedberg, Insel, and Spence (4th or 5th edition), but it is not necessary to purchase the textbook for this course. | |||||
| 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 Fri Jan 20th. (solutions)
Homework #2, due Fri Jan 27th. (solutions) Homework #3, due Fri Feb 3rd. (solutions) Homework #4, due Fri Feb 10th. (solutions) Homework #5, due Fri Feb 17th. (solutions) Homework #6, due Fri Feb 24th. (solutions) Midterm Exam, due Fri Mar 3rd. (solutions) Homework #7, due Fri Mar 17th. (solutions) Homework #8, due Fri Mar 24th. (solutions) Homework #9, due Fri Mar 31st. (solutions) Homework #10, due Fri Apr 7th. (solutions) Homework #11, due Fri Apr 14th. (solutions) Homework #12, due Fri Apr 21st. (solutions) Takehome Final Exam, due Fri Apr 28th. Students with an A average on the 12 homeworks plus the midterm are exempt from the final exam and will automatically receive an A in the course. All other students are required to take and submit the final exam. You will be notified explicitly whether you are exempt from the final exam, or not exempt, after you submit homework 12 on Gradescope. 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 11pm 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 (21pp, v3.10, posted 1/1) | 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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| Chapter 1: Vector Spaces (21pp, v3.50, posted 1/24) | 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 |
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| Chapter 2: Linear Transformations (22pp, v3.50, posted 2/1) | 2.1 ~ Linear Transformations (Definition and Examples, Kernel and Image, Algebraic Operations, One-to-One Transformations, Isomorphisms) 2.2 ~ Matrices Associated to Linear Transformations (Associated Matrices, Algebraic Properties, Rank, Inverses, Change of Basis and Similarity) |
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| Chapter 3: Inner Product Spaces (23pp, v3.50, posted 2/13) | 3.1 ~ Inner Product Spaces 3.2 ~ Orthogonality 3.3 ~ Linear Transformations and Inner Products 3.4 ~ Applications of Inner Products |
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| Chapter 4: Eigenvalues, Diagonalization, and the Jordan Canonical Form (32pp, v3.50, posted 3/13) | 4.1 ~ Eigenvalues, Eigenvectors, and The Characteristic Polynomial 4.2 ~ Diagonalization 4.3 ~ Generalized Eigenvectors and the Jordan Canonical Form 4.4 ~ Applications of Diagonalization and the Jordan Canonical Form (Cayley-Hamilton, The Spectral Theorem, Stochastic Matrices and Markov Chains, Systems of Linear Differential Equations, Matrix Exponentials) |
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| Chapter 5: Bilinear and Quadratic Forms (26pp, v2.50, posted 3/29) | 5.1 ~ Bilinear Forms 5.2 ~ Quadratic Forms (Definition, Diagonalization over R, Definiteness, Second Derivatives Test, Sylvester's Law of Inertia) 5.3 ~ Singular Values and Singular Value Decomposition (Singular Value Decomposition, Pseudoinverses) |
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| 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 (or Textbook) | The lecture notes and the textbook are comprehensive sources 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 is a 2-hour weekly problem session run by the course TAs. This session runs on Fridays ahead of the time the homework is due. 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. | ||
| 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 9 (class starts 1/9) |
§0.1: Sets, Numbers, and Functions §0.2: Vectors in Rn §0.3: Complex Numbers, Fields §0.4: Matrices and Systems of Linear Equations §0.5: Induction No homework this week. |
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| Week of Jan 16 (no class 1/16) |
§0.6: Polynomials §1.1: The Formal Definition of a Vector Space §1.2: Subspaces Homework #1 due Friday, Jan 20th. |
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| Week of Jan 23 | §1.3: Linear Combinations and Span §1.4: Linear Independence and Linear Dependence §1.5: Bases and Dimension Homework #2 due Friday, Jan 27th. |
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| Week of Jan 30 |
§1.5: Bases and Dimension §2.1.1: Linear Transformations §2.1.2: Kernel and Image §2.1.3: Algebraic Operations on Linear Transformations Homework #3 due Friday, Feb 3rd. |
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| Week of Feb 6 |
§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 §2.2.2: Algebraic Properties of Associated Matrices Homework #4 due Friday, Feb 10th. |
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| Week of Feb 13 | §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 17th. |
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| Week of Feb 20 (no class 2/20) |
§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 24th. |
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| Week of Feb 27 | §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, Mar 3rd. |
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| Spring Break (no classes) from Mar 6 to Mar 10 | |||
| Week of Mar 13 | §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 17th. |
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| Week of Mar 20 | §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 24th. |
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| Week of Mar 27 | §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 31st. |
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| Week of Apr 3 | §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 7th. |
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| Week of Apr 10 | §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 14th. |
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| Week of Apr 17 (no class 4/17, classes end 4/19) |
§5.4: Pseudoinverses Homework #12 due Friday, Apr 21st. |
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| FINAL EXAM is take-home and due on April 28th. | |||