Math 4571 (Advanced Linear Algebra), Spring 2022
| 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 | MWR noon-1:00pm MWR 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.) | |||||
| 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 Thu Jan 27th. (solutions)
Homework #2, due Thu Feb 3rd. (solutions) Homework #3, due Thu Feb 10th. (solutions) Homework #4, due Thu Feb 17th. (solutions) Homework #5, due Thu Feb 24th. (solutions) Homework #6, due Thu Mar 3rd. (solutions) Take-Home Midterm, due Thu Mar 10th. (solutions) Homework #7, due Mon Mar 28th. (solutions) Homework #8, due Mon Apr 4th. (solutions) Homework #9, due Mon Apr 11th. (solutions) Homework #10, due Wed Apr 20th. (solutions) Homework #11, due Wed Apr 27th. (solutions) Take-Home Final, due Thu May 5th. As with the midterm, the final is take-home, infinite-time, and non-collaboration, but you are encouraged to ask any questions to the instructor. 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, or otherwise unreadable submissions will be penalized at the grader's discretion. | |||||||
| Handouts / Lecture Notes | |||
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| Handout | Topics | ||
| Chapter 0: Preliminaries (19pp, v2.10, posted 1/9) | 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 ~ Polynomials 0.6 ~ Induction |
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| Chapter 1: Vector Spaces (20pp, v3.00, 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, v3.00, posted 2/7) | 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, v3.00, posted 2/22) | 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, v3.00, posted 3/8) | 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 (25pp, v2.00, updated 4/20) (updates in v2.00: added non-matrix example to 5.3.1, fixed typos in 5.3.1, added section 5.3.2) |
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 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 17 (class starts 1/19) |
§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. |
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| Week of Jan 24 | §0.5: Polynomials §0.6: Induction §1.1: The Formal Definition of a Vector Space §1.2: Subspaces Homework #1 due Thursday, Jan 27th. |
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| Week of Jan 31 | §1.3: Linear Combinations and Span §1.4: Linear Independence and Linear Dependence §1.5: Bases and Dimension Homework #2 due Thursday, Feb 3rd. |
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| Week of Feb 7 |
§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 Thursday, Feb 10th. |
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| Week of Feb 14 |
§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 Thursday, Feb 17th. |
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| Week of Feb 21 (no class 2/21) |
§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 Thursday, Feb 24th. |
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| Week of Feb 28 | §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 Thursday, Mar 3rd. |
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| Week of Mar 7 | §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 Thursday, Mar 10th. |
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| Spring Break (no classes) from Mar 14 to Mar 18 | |||
| Week of Mar 21 | §4.1.1: Eigenvalues and Eigenvectors §4.1.2: Eigenvalues and Eigenvectors of Matrices §4.1.3: Eigenspaces §4.2: Diagonalization No homework this week. |
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| Week of Mar 28 | §4.3.1: Generalized Eigenvectors §4.3.2: The Jordan Canonical Form §4.4.1: Spectral Mapping and the Cayley-Hamilton Theorem Homework #7 due Monday, Mar 28th. |
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| Week of Apr 4 | §4.4.2: The Spectral Theorem for Hermitian Operators §4.4.3: Transition Matrices and Incidence Matrices §4.4.4: Systems of Linear Differential Equations Homework #8 due Monday, Apr 4th. |
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| Week of Apr 11 | §4.4.5: Matrix Exponentials and the Jordan Form §5.1: Bilinear Forms §5.2.1: Quadratic Forms Homework #9 due Monday, Apr 11th. |
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| Week of Apr 18 (no class 4/18) |
§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 Wednesday, Apr 20th. |
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| Week of Apr 25 (classes end 4/27) |
§5.3: Singular Value Decomposition Review for Final Exam Homework #11 due Wednesday, Apr 27th. |
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| FINAL EXAM is take-home and due on May 5th. | |||