Syllabus for Math 204, Linear Algebra

           Department of Mathematics and Computer Science
           Hobart and William Smith Colleges

           Fall 2020.

           Instructor:  David J. Eck  (eck@hws.edu)
           
           Monday, Wednesday, Friday, 9:50–10:50 AM
                Room Coxe 8.

Please note that this is a tentative syllabus, based on the Colleges' plans for the Fall semester as of mid-July. Changes might be necessary as plans become more definite or as the situation changes. In any case, however, the course should cover the material that is listed here.

About The Course

Linear algebra is the study of vectors, vector spaces, and the mappings between vector spaces. You might already be familiar with the concept of a vector is two or three dimensions, where you can think of a vector as an "arrow" with a direction and a length but not a fixed position. Vectors in n-dimensional space are a major example, but vectors are a more abstract concept that can be applied in many other situations as well.

Linear algebra is often associated with matrices. A matrix is just a bunch of numbers arranged in a rectangular grid. A matrix can represent a "linear map" from one vector space to another, but just as vectors in n-space are only one example of vectors, matrices are only one example of linear mapping. We will work with matrices, but are also interested in the more abstract, more generally applicable idea of linear mapping.

Many questions in linear algebra can be reduced to questions about solutions to systems of linear equations. We will start the course by looking at some systems and how to solve them, and they will be an important tool throughout the rest of the course.

The textbook for the course is Linear Algebra, third edition, by Jim Hefferon. It is available as a free PDF download:

http://joshua.smcvt.edu/linearalgebra/book.pdf

See https://hefferon.net/linearalgebra for more information about the book. The web site also has a link to a solution manual. If you download the book and the solution manual into the same directory, you will be able to use links from each exercise in the book to its solution. There is also an inexpensive printed version of the book, but note that you are not required to buy the printed version and I will not ask you to bring the book to class.

We will follow the general outline of the textbook, and there will be readings from the book, but I will be adding my own perspective and sometimes departing from the book to some extent. The textbook is a fairly standard first course in linear algebra. You will notice that many of the sections in the book are starred, marking them as optional. We will end up covering most of the material from non-optional sections and just a few of the optional ones. The book also has "Topics" at the end of each chapter. Many of the topics are important applications of linear algebra. We might cover some of the topics if time permits. (Applications are covered in detail in a followup course to this one, Math 214: Applied Linear Algebra.)

About proof: This course is more abstract than Calculus courses that you have taken. We will spend a significant part of the course proving things. You will be expected to understand proofs, and you will be asked to write some proofs on the homework. Math 135, which is an introduction to proofs, is a suggested prerequisite for this course. However it is not required. This means that I cannot assume that all students in the course already have a lot of experience with proofs. I will try to explain proof techniques as we go, especially at first, and I encourage you to come in for help if you find it difficult to follow a proof. Talking through a proof in detail can make it a lot clearer what is going on. But don't worry too much—the way to learn to do proofs is to struggle through a variety of examples and see how they work.

About the Pandemic

Dealing with the pandemic will be an issue throughout the semester. While I hope that we will all be able to meet in person and that none of us will get sick or need to be quarantined, only time will tell how realistic that hope may be.

We should expect, according to school policy, to be wearing masks and practicing social distancing in class. No one who is showing symptoms should come to class. It is possible that someone will be asked to self-isolate because of possible exposure to the virus; the policy on that will be set by the Colleges. All of this applies to me as well as to students. In the worst case, we might have to deal with a full shutdown of the Colleges and a transition to remote classes. Detailed policies will be set by the Colleges.

I will try to make the class accessible to students who can't always be there in person. I plan to post reading guides and occasional short videos to supplement the readings from the textbook. I will set up appointment times for individual and group meetings on Zoom. And of course I will always be available on email. If I can't be in class myself, I expect the course to continue either remotely or with a guest lecturer filling in for me.

Homework

I will assign and collect homework weekly. Since the solution manual for the textbook has solutions for every problem in the book, I will make my own homework sets, which will posted on the website for the course. Homework will consist of both computational exercises and exercises that require proof. Under no circumstances will you ever receive credit for answers that are not justified in words or by an appropriate calculation or both. This is true whether or not a question explicitly asks you to justify your answer.

I am hoping (but not requiring) that many people in the course will be willing to write their solutions using LaTeX at overleaf.com. LaTeX is a system that can produce high-quality typeset mathematics. If your work is done at overleaf.com, and if you give me access to your project there, I will be able to view and comment on your work online. I expect hold some demonstrations in class or on Zoom in the first week of classes. For more information about submitting homework, see

http://math.hws.edu/eck/math204/f20/submitting-homework.html

Even though I am not assigning homework from the textbook, it would be a good idea work on a variety of problems from the book. This is part of the reading assignments—you don't really understand some mathematical concept until you have practiced using it. Doing only the graded homework will almost certainly not be enough for you to master the material. Remember that solutions to all of the exercises in the textbook are available in the downloadable solutions manual.

You are allowed and encouraged to discuss the homework with other people in the course. However, you should always write up your own solutions in your own words. When you work with others on homework, you should discuss ideas and approaches, but you should not simply produce line-by-line similar work. Note that since almost all exercises will require some words of explanation, it is very unlikely that your homework solutions will look very similar to someone else's, except in outline.

Note that a many homework assignments will involve proving things. In some cases, I will give you feedback on your proofs and a chance to revise them to raise your grade. It is possible that I will ask to meet with you about your revisions.

Of course, you are also encouraged to come to me for help on the homework!

Tests and Final Evaluation

I plan to give two in-class tests that will cover definitions and basic concepts, along with some straightforward computations and proofs. The two in-class tests are planned for Friday, October 2 and Friday, November 13. It is not entirely clear what the protocol will be for doing tests on paper, but I'm hoping that it will be possible.

Current planning calls for the end of the course, including the final exam, to be run remotely. I don't yet know how that will work, but I am leaning towards a final summary problem set as a kind of take-home exam. It might also include meeting with me on Zoom to discuss your work.

Grading

Here is the approximate weighting for the various components of the course:

             First in-class test:      20%
             Second in-class test:     20%
             Final evaluation:         20%
             Homework:                 40%

About Office Hours

The Colleges' opening plan advises against having in-person meetings with students in Faculty offices. Since my office is large, it might be possible for me to meet with one person there, but only by appointment, since it will not be possible to hold open office hours in my office. I will schedule some open office hours on Zoom. I will also set up times for individual or group appointments on Zoom. Appointments will be made using the Calendar feature in Canvas. It might be possible to schedule in-person appointments at other venues, such as outdoors. It might even be possible for me to hold open office hours outdoors. Details will be announced.

Of course, email is always a good way to contact me. My email address is eck@hws.edu. I welcome comments and questions by email, and I will usually respond to them fairly quickly.

Statements from the Center for Teaching and Learning

At Hobart and William Smith Colleges, we encourage you to learn collaboratively and to seek the resources that will enable you to succeed. The Center for Teaching and Learning (CTL) is one of those resources: CTL programs and staff help you engage with your learning, accomplish the tasks before you, enhance your thinking and skills, and empower you to do your best. Resources at CTL are many: Teaching Fellows provide content support in 12 departments, Study Mentors help you manage your time and responsibilities, Writing Fellows help you think well on paper, and professional staff help you assess academic needs.

Disability Accommodations: If you are a student with a disability for which you may need accommodations, you should self-identify, provide appropriate documentation of your disability, and register for services with Disability Services at the Center for Teaching and Learning (CTL). Disability related accommodations and services generally will not be provided until the registration and documentation process is complete. The guidelines for documenting disabilities can be found at the following website: www.hws.edu/academics/ctl/disability_services.aspx. Please direct questions about this process or Disability Services at HWS to Christen Davis, Coordinator of Disability Services, at ctl@hws.edu or x 3351

Tentative Schedule

Here is a very tentative weekly schedule for this course.

Dates Topics and Readings
Aug. 24, 26, 28 Linear systems and Gauss's method; Section 1.I and 1.I.2.
Vectors in Rn; Sections 1.II.1 and 1.II.2.
Aug. 31; Sept. 2, 4 More on vectors and the geometry of linear systems.
Homogeneous and particular solutions; Section 1.I.3.
Sept. 7, 9, 11 Reduced row-echelon form; Section 1.III.
Accuracy of computation; Topic from the end of Chapter 1.
Sept. 14, 16, 18 Vector spaces and subspaces; Sections 2.I.1 and 2.I.2.
Sept. 21, 23, 25 Linear independence; Section 2.II.
Bases, and dimension; Sections 2.II.1, 2.III.1, 2.III.2.
Sept. 28, 30; Oct. 2 Row space, column space, and null space; Section 2.III.3.
First test, Friday, October 2.
Oct. 5, 7, 9 Linear maps: isomorphisms and homomorphisms; Sections 3.I, 3.II.1.
Oct. 12, 14, 16 Range space and null space; Section 3.II.2.
Matrices and linear maps; Section 3.III.
Oct. 19, 21, 23 Matrix operations, inverses, and the elementary row matrices; Section 3.IV.
Oct. 26, 28, 30 Change of basis; Section 3.V.
Nov. 2, 4, 6 Determinants; Selections from Chapter 4.
Nov. 9, 11, 13 Fields, the complex numbers, and vector spaces over fields.
Second test on Friday, November 13.
Nov. 16, 18, 20 Eigenvalues, eigenvectors, and diagonalizability; Selections from Chapter 5.
Nov. 23 Extra topic, such as: Affine maps and the Chaos Game.
Nov. 30; Dec. 2 Extra topic, such as: Geometric transformations and computer graphics.
TBA Final Evaluation of some sort