Syllabus for Math 204, Linear Algebra

           Department of Mathematics and Computer Science
           Hobart and William Smith Colleges

           Spring 2020.

           Instructor:  David J. Eck  (eck@hws.edu)
           
           Monday, Wednesday, Friday, 11:00–12:00 AM
                Room Coxe 8.

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 it is the more abstract, more generally applicable idea of linear mapping that we are really interested in.

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 http://joshua.smcvt.edu/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 every 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. A copy of the printed version will be on reserve at the library during the semester.

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 try to cover all of the non-optional sections and just a few of the optional ones. The book also has "Topics" at the end of each chapter. Except for the very first one, on computer algebra systems, I will not cover them; however, some of them will be covered in student presentations.

I also plan to introduce the algebra system, MatLab, to be used as an aid in some basic matrix manipulations. MatLab is complicated, but we will only cover enough to use it for some basic calculations.


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 be handed out in class and 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.

Even though I am not assigning homework from the textbook, you should work on a wide 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.

You are allowed and encouraged to work on 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.

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


Project and Presentation

Each student in the class will work on a short project and present it to the class. The idea is to bring some material into the course that is not covered in regular lectures. The textbook includes a large number of "Topic" sections, and many of the topics would be appropriate for the type of project that I have in mind. The book also has some more advanced or interesting exercises, generally marked with a "?", and some of those could make good topics. We might even be able to come up with something from outside the book.

You will consult with me to select a topic. I will expect you to give a ten-to-twenty minute presentation to the class—maybe even longer, depending on your topic. Assuming that your topic is based on the textbook, much of the material for the presentation will come from the book; however, I would generally expect you to do some additional research. You will also turn in a write-up (or possibly PowerPoint slides), which ideally could be made available to the class. Details will be subject to negotiation.

The project/presentation will count for 5% of your grade for the course. Grading will be simple: you will get 4 out of 5 points for a reasonable effort, or a full 5 points for an especially nice job. Hopefully, I won't have to go below 4 points for anyone in the class!


Tests and Final Exam

There will be two in-class tests in addition to a final exam. The tests will be given on Friday, February 21 and Friday, April 3.

The final exam will take place during the officially scheduled exam time for the course, which is Saturday, May 9, at 7:00 PM. The final exam will be comprehensive, but more than half will be about material covered in the last segment of the course.


Grading

Your numerical grade for the course will be computed as follows:

            First Test    18%
            Second Test   18%
            Homework      35%
            Project        5%
            Final Exam    24%

I reserve the right to adjust your grade downwards if you miss more than a couple of classes without a good excuse. In my grading scale, an A corresponds to 90–100%, B to 80–89%, C to 65–79%, D to 55–64%, and F to 0–54%. Grades near the endpoints of a range get a plus or minus.


Attendance, Etc.

I assume that you understand the importance of attending class. While I do not take attendance in every class, I expect you to be present unless circumstances make that impossible.

If you miss a test or the final exam without an extremely good excuse, you will receive a grade of zero. If you think you have a sufficient excuse for missing a test, please discuss it with me, in advance if possible; there is the possibility of a make-up test if circumstances warrant.

Although it should not need to be said, I expect you to maintain a reasonable level of decorum in class. This means that there is usually no eating or drinking in class. Cell phones are turned off. There is no walking in late or unnecessary walking in and out of the room during lecture. I strongly discourage any use of technology in the classroom, including tablets and laptops.


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, Q Fellows support you in courses that require math, 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: http://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.


Office Hours, E-mail, WWW

My office is room 313 in Lansing Hall. My office phone extension is 3398. I am on campus most days, and you are welcome to come in anytime you can find me there. Here are my tentative regular office hours but note that I am often available outside these times and that your office visits are certainly not restricted to my regular office hours!

         MWF 9:45–10:45 AM
         Tuesday 11:00‐12:00

My e-mail address is eck@hws.edu.

The home page for this course is http://math.hws.edu/eck/math204/index.html. This page will contain a weekly guide to the course, including assignments and labs. You will want to bookmark this page. This courses does not use Canvas.


Tentative Schedule

Here is a tentative weekly schedule of readings for this course. We will try to follow this schedule, but some adjustments might be necessary. The actual readings for each week will be listed on the course web page.

Dates Topics / Readings
Jan. 22 and 24 Linear Systems and Gauss's Method
Chapter One, Sections I.1 and I.2
Jan. 27, 29 and 31 Homogeneous and Particular Solutions; Vectors in Rn
Chapter One, Sections I.3, II.1, II.2
Feb. 3, 5, and 7 Reduced Echelon Form; Computer Algebra
Chapter One, Sections III.1 and III.2 and second Topic
Feb. 10, 12, and 14 Vector Spaces and Linear Independence
Chapter Two, Sections I.1, I.2, and II.1
Feb. 17, 19, and 21 Bases and Dimension
Chapter Two, Sections III.1 and III.2
Test on Friday, February 21
Feb. 24, 26, and 28 Vectors and Linear Systems; Isomorphisms
Chapter Two, Section III.3; Chapter 3, Sections I.1 and I.2
Mar. 2, 4, and 6 Homomorphisms
Chapter Three, Sections II.1, II.2, and III.1
Mar. 9, 11, and 13 Homomorphisms and Matrices
Chapter Three, Sections III.2 and IV.1
Spring Break, March 14–22
Mar. 23, 25, and 27 Matrix Operations and Inverses
Chapter Three, Sections IV.2, IV.3, and IV.4
Mar. 30; Apr. 1 and 3 Projection
Chapter Three, Sections VI.1 and VI.2
Test on Friday, April 3
Apr. 6, 8, and 10 Change of Basis; Introduction to Determinants
Chapter Three, Sections V.1 and V.2; Chapter Four, Section I.1
Apr. 13, 15, and 17 Determinants
Chapter Four, Sections II.2, II.3, and III.1
Apr. 20, 22, and 23 Complex Vector Spaces
Chapter Five, Sections I.1, I.2, and II.1
Apr. 27 and 29; May 1 Eigenvalues and Eigenvectors
Chapter Five, Sections II.2 and II.3
May 4 Last day of class; wrap up the course!
May 9 Final Exam: Saturday, May 9, 7:00 PM