## 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)

Web site:  http://math.hws.edu/eck/math204/

Monday, Wednesday, Friday, 9:50–10:50 AM
Gearan Center 102: Froelich Recital Hall.
```

Please note that this is a somewhat tentative syllabus. Some adaptation might be necessary as the situation changes and as we gain experience with holding class during a pandemic. Detailed polices for the course are subject to discussion, and I welcome your comments at any time. In any case, however, the course should cover the material that is listed in the schedule at the end of this syllabus.

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 in 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 such 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, fourth 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.

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. As the semester begins, I will be experimenting with recording or possibly streaming my lectures. I plan to post reading guides and maybe occasional short videos to supplement the readings from the textbook. I will set up appointment times for individual and group meetings on Zoom. Appointments will be available for anyone in the course, but I will definitely expect people who can't be in class to meet with me on Zoom if they are able to do that. 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 and in Canvas. 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 to hold some demonstrations in class or on Zoom in the first week of classes. The only alternative to submitting homework at overleaf.com is to submit your work as a single PDF file through Canvas. For more details 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 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. For someone who cannot be in class for an exam, we will have to work out some sort of individual accommodation; for example, we might do an oral exam on Zoom or some kind of take-home exam. The same will apply if in-person tests become impossible for the whole class.

The Colleges' schedule for the semester calls for the end of the course, including the final exam, to be run remotely. I have not made a definite decision about the format for the final exam, but I am leaning towards a final summary problem set as a take-home exam, plus individual or group meetings with me on Zoom where you will discuss and defend your work on the exam. We can discuss this as the end of the semester nears.

Here is the weighting for the various components of the course. Some adjustment might be necessary, for example if the course is forced to go entirely online.

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

The Colleges' opening plan advises against having in-person meetings with students in Faculty offices. Since my office is large, however, it might be possible for me to meet there with one person at a time. However, that would be by appointment only, since we can't have groups of people waiting in the hall. It might also be possible to meet somewhere other than my office. If you would like to try to schedule an in-person meeting, you should contact me.

I will schedule a few open, drop-in 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. Details will be announced.

Zoom links for office hours will be posted on Canvas.

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

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 an approximate weekly schedule for this course. We should cover all of the topics listed here, but might not follow the schedule exactly. See the course web site for accurate scheduling information, posted weekly.

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. [Continued next week.]
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.
Dec. 9 Final Evaluation of some sort
Scheduled Final Exam Period: Wednesday, Dec. 9, 8:30AM