Math 4571 (Advanced Linear Algebra, Course #37878), Spring 2020
| Course Information | |||||
|---|---|---|---|---|---|
| Instructor | Class Times | Office Hours | |||
| Evan Dummit edummit at northeastern dot edu |
Effective Thursday, March 12th, all Northeastern in-person meetings are cancelled until further notice. | ||||
| In lieu of office hours, 4571 now has a Piazza page. Links to all of the live lecture material are hosted there. | |||||
| 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). Homework assignments are now to be submitted via the 4571 Blackboard page . | |||||
| 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 | |||||||
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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 16th. (solutions)
Homework #2, due Thu Jan 23rd. (solutions) Homework #3, due Thu Jan 30th. (solutions) Homework #4, due Thu Feb 6th. (solutions) Homework #5, due Thu Feb 13th. (solutions) Homework #6, due Thu Feb 27th. (solutions) Homework #7, due Thu Mar 19th. (solutions) Homework #8, due Thu Mar 26th. (solutions) Homework #9, due Thu Apr 2nd. (solutions) Homework #10, due Thu Apr 9th. (solutions) Homework #11, due Thu Apr 16th. (solutions) | |||||||
| Lecture Slides | |||
|---|---|---|---|
| These are slides corresponding to the material that would have been covered in the lectures. | |||
| Date | Material | ||
| Thu, Mar 12th | Class "cancelled" (Notes 4.1 covered in Mar 11th lecture) | ||
| Wed, Mar 18th | Lecture 24: Diagonalization (Notes 4.2) | ||
| Thu, Mar 19th | Lecture 25: Generalized Eigenvectors (Notes 4.3.1) | ||
| Mon/Wed, Mar 23rd/25th | Lecture 26: Jordan Canonical Form (Notes 4.3.2) | ||
| Wed/Thu, Mar 25th/26th | Lecture 27: Applications of Jordan Canonical Form 1 (Notes 4.4.1 + 4.4.4 + 4.4.5) | ||
| Mon, Mar 30th | Lecture 28: Applications of Jordan Canonical Form 2 (Notes 4.4.2 + 4.4.3) | ||
| Mon/Wed, Apr 6th/8th | Lecture 29: Bilinear Forms (Notes 5.1.1 + 5.1.2) | ||
| Wed/Thu, Apr 8th/9th | Lecture 30: Quadratic Forms 1 (Notes 5.2.1 + 5.2.2) | ||
| Mon, Apr 13th | Lecture 31: Quadratic Forms 2 (Notes 5.2.3 + 5.2.4) | ||
| Handouts / Lecture Notes | |||
|---|---|---|---|
| Handout | Topics | ||
| Chapter 0: Preliminaries (19pp, v2.10, updated 1/12) (changes in 2.10: added section 0.4.3 on echelon forms) |
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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| Supplement: Determinants, Formally (8pp, v1.00, posted 1/12) | 0.A ~ Determinants, Formally | ||
| Chapter 1: Vector Spaces (20pp, v2.00, posted 1/12) | 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, v2.00, posted 1/26) | 2.1 ~ Linear Transformations 2.2 ~ Kernel and Image 2.3 ~ Algebraic Properties of Linear Transformations 2.4 ~ Matrices Associated to Linear Transformations |
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| Chapter 3: Inner Product Spaces (21pp, v2.00, posted 2/16) | 3.1 ~ Inner Product Spaces 3.2 ~ Orthogonality 3.3 ~ Applications of Inner Products 3.4 ~ Linear Transformations and Inner Products |
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| Chapter 4: Eigenvalues, Diagonalization, and the Jordan Canonical Form (32pp, v2.00, 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 |
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| Chapter 5: Bilinear and Quadratic Forms (13pp, v1.00, posted 4/6) | 5.1 ~ Bilinear Forms 5.2 ~ Quadratic Forms |
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| Exam Information | |||
|---|---|---|---|
| Exam | Date, Time, Location | Topics | Review Material |
| Midterm 1 (Exam), (Solutions) |
Thu, February 13th In Class |
Homeworks 1-5 Notes Chapters 1-2 Topics List |
Review Problems Review Answers In-class review Wed Feb 12th |
| Midterm 2 (Exam), (Solutions) |
Sat, April 4th or alternate date by arrangement |
Homeworks 6-9 Notes Chapters 3-4 Topics List |
Review Problems Review Answers Review lecture Thu Apr 2nd |
| Final | Sat, April 18th or alternate date by arrangement |
The final is COMPREHENSIVE! Homeworks 1-11 Notes Chapters 1-5 |
Topics List Review session Fri Apr 17th |
| Bring your University ID to all exams. Cell phones, electronic or calculating devices, books, and notes of any kind will NOT be permitted in exams. | |||
| 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. | ||
| Attend Problem Sessions | There is an optional weekly problem session run by the course's TA. This session is held in 544 Nightingale from noon-1:30pm on Fridays. 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 assistance from the TA. | ||
| Use Tutoring Services | The university offers a wide variety of (free) tutoring services. Please consult the Northeastern Tutoring page for additional information on peer tutoring. The Math Tutoring Center, located in 540B NI, is specifically set up for tutoring in mathematics courses. Tutoring appointments can be made via MyNEU (which will provide lists of available appointments for the tutors for your specific classes); walk-in tutoring is very limited and tends to be unavailable near exam dates. | ||
| 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, Systems of Linear Equations, and Determinants §0.5: Polynomials No homework this week. |
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| Week of Jan 13 | §0.6: Induction §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 #1 due Thursday, Jan 16th. |
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| Week of Jan 20 (no class 1/20) |
§1.5: Bases and Dimension Homework #2 due Thursday, Jan 23rd. |
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| Week of Jan 27 |
§2.1: Linear Transformations §2.2: Kernel and Image §2.3.1: Algebraic Operations on Linear Transformations §2.3.2: One-to-One Linear Transformations Homework #3 due Thursday, Jan 30th. |
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| Week of Feb 3 |
§2.3.3: Isomorphisms of Vector Spaces §2.4.1: The Matrix Associated to a Linear Transformation §2.4.2: Algebraic Properties of Matrices Associated to Linear Transformations §2.4.3: Inverse Matrices, Change of Basis, Similarity Homework #4 due Thursday, Feb 6th. |
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| Week of Feb 10 | Review for Midterm 1. Homework #5 due Thursday, Feb 13th. |
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| MIDTERM 1 will be held in class on Thursday, February 13th | |||
| Week of Feb 17 (no class 2/17) |
§3.1.1: Inner Products §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 24 | §3.2.2: Orthogonal Complements and Orthogonal Projection §3.4.1: Least-Squares Estimates §3.4.2: Fourier Series Homework #6 due Thursday, Feb 27th. |
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| Spring Break (no classes) from Feb 29 to Mar 8 | |||
| Week of Mar 9 | §3.4: Linear Transformations and Inner Products §4.1.1: Eigenvalues and Eigenvectors §4.1.2: Eigenvalues and Eigenvectors of Matrices §4.1.3: Eigenspaces No homework this week. |
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| Effective Thursday, March 12th, all in-person meetings are cancelled. | |||
| Week of Mar 16 | §4.2: Diagonalization §4.3.1: Generalized Eigenvectors Homework #7 due Thursday, Mar 19th. |
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| Week of Mar 23 | §4.3.2: The Jordan Canonical Form §4.4.1: Spectral Mapping and the Cayley-Hamilton Theorem §4.4.2: Transition Matrices and Incidence Matrices Homework #8 due Thursday, Mar 26th. |
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| Week of Mar 30 | §4.4.3: Systems of Linear Differential Equations §4.4.4: Matrix Exponentials and the Jordan Form §4.4.5: The Spectral Theorem for Hermitian Operators Review for Midterm 2 Homework #9 due Thursday, Apr 2nd. |
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| MIDTERM 2 will be held on Saturday, April 4th (alternate dates of Fri Apr 3rd and Sun Apr 5th are available) | |||
| Week of Apr 6 | §5.1: Bilinear Forms §5.2.1: Quadratic Forms §5.2.2: Diagonalization of Quadratic Varieties Homework #10 due Thursday, Apr 9th. |
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| Week of Apr 13 (classes end 4/14) |
§5.2.3: The Second Derivatives Test §5.2.4: Sylvester's Law of Inertia Review for Final Exam Homework #11 due Thursday, Apr 16th. |
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| FINAL EXAM will be held on Saturday, April 18th (alternate date of Sun Apr 19th is available) | |||