## Linear Algebra

#### Dr. Beezer

##### Texts

We will be using A First Course in Linear Algebra, by Robert A. Beezer as our textbook. The most recent public version is Version 3.50 and may be found as a PDF on the book's site at linear.pugetsound.edu, where it is made freely available with an open license. If you prefer, you can use the hardcover version, which is Version 3.00, but this will have major differences in Chapters D and E. See the book's site for information on ordering a physical copy.

The online version of Version 3.50 is at linear.pugetsound.edu/fcla. This has the subtitle “(Beta Version)” presently, and is not the older green-themed version. You may find this version the most useful, and it will also perform well on a mobile device.

However, for this course, this term, we will be using a special “enhanced” and modified version online, which will be discussed in class.

The course web page has some recommendations for open textbooks about proof techniques, which will augment the shorter discussions in the textbook.

##### Course Web Page

Off of buzzard.ups.edu/courses.html you can find the link to the course web page.

##### Office Hours

Office Hours will be online, via Zoom, by appointment. Monday, Thursday and Friday mornings I am in class, and the remainder of weekdays I should be available. Making appointments or simple, non-mathematical questions can be handled via email—my address is beezer@ups.edu. I rarely do not receive your email, and I read all of my email all of the time, usually very shortly after receiving it. Urgency of replying varies by the hour, day and nature of the message. Please do not use email for any question that requires a mathematical reply. Email is not a good medium for discussing mathematics.

##### Computation

Linear algebra is at the heart of many large computations in computer science, physics, chemistry, economics, statistics, and other disciplines. So it is useful to become familiar with relevant software. Futhermore, freed from doing error-prone numerical computations you can concentrate on new ideas and concepts.

For both reasons, we will make extensive use of Sage. Since Sage is open source software, it is available freely in many places. We will be relying this semester on hosted versions at CoCalc, cocalc.com. You already have a course project associated to your UPS email address and by the “last day to drop” you will need to pay \$14 (for the entire semester) so you can move your course project to a members-only server.

There are thorough discussions about Sage integrated into the web version of your textbook. We will discuss in class the use of Sage during examinations. In particular, if you do not own a laptop, investigate procedures now for borrowing one from the library.

##### Homework

There is a nearly complete collection of exercises in the text. Any (or all) of the problems will be good practice as you learn this material. Many of these problems have complete solutions in the text to further aid your understanding. Of course, you are not limited to working just these problems.

None of these problems will be collected, but instead they will form the basis for our fourteen “Problem Sessions” and for discussions at Office Hours. It is your responsibility to be certain that you are learning from these exercises. The best ways to do this are to work the problems diligently as we work through the sections (see attached schedule) and to participate in the classroom discussions. If you are unsure about a problem, then a visit to my office is in order. Making a consistent effort outside of the classroom is the easiest way to do well in this course.

Mathematics not only demands straight thinking, it grants the student the satisfaction of knowing when he [or she] is thinking straight.

―D. Jackson

Mathematics is not a spectator sport.

―Anonymous

I hear, I forget. I see, I remember. I do, I understand.

―Chinese Proverb

An education is not received. It is achieved.

―Anonymous
##### Examinations

There will be seven 50-minute timed exams—they are all listed on the tentative schedule. The lowest of your seven exam scores will be dropped. There will be a comprehensive final exam, but a time has not been set by the Registrar. The final exam cannot be given at any other time and also be aware that I may allow you to work longer on the final exam than just the two-hour scheduled block of time. In other words, plan your travel arrangements accordingly.

As a study aid, I have posted copies of old exams on the course web site. These are offered with no guarantees, since techniques, approaches, emphases, and even notation will change slightly or radically from semester to semester. Some of the solutions contain mistakes, and some of the problem statements have typos. In other words, they are not officially part of this semester's course and I do not maintain them. In particular, I do not advocate working old exams as a primary, or exclusive, technique for learning the material in this course. Use at your own risk: they have not been reviewed for minor mistakes or inconsistencies with this semester's course. I will not entertain questions about the correctness of these materials via email.

Come to examinations prepared to remain in the room for the entire length of the exam. Power off any electronic devices (this includes phones).

##### Writing

This course has been designated as part of the University's Writing in the Major requirement. Thus, there will be two proofs assigned for each chapter. You will be expected to formulate a proof, and write it up clearly. These will be graded on a pass/fail basis. Each chapter's questions will be returned to you with careful comments, and if you do not earn a Pass, then you can resubmit them at the close of the next chapter. You may resubmit a problem for several consecutive chapters in a row, so long as you make a serious effort on each outstanding problem at each opportunity. Once you miss an opportunity to resubmit, or a retry does not contain any new work, or significant comments and hints are ignored, then it will be scored as a No Pass. Failure to follow the directions for submitting these can result in an automatic retry with no feedback. Please read the instructions and details provided with these problems very carefully.

Due dates will be announced in class, generally at the end of a chapter and prior to that chapter's exam. Under no circumstances will they be accepted late. During the first part of the course, we will learn the mathematical typesetting software, , and you will be required to use this tool appropriately when writing your proofs, and you may be required to do a retry soley on the basis of incomplete use of . I might request your source as part of grading your exercises, so make sure you retain these.

These problems ARE YOUR OWN WORK. In other words, no collaboration on formulating the proof, no collaboration on writing the proof, no copying content from the book's source, and no discussion whatsoever with classmates or others familiar with the subject or the exercises. In particular, I do not provide consultation in advance of submission, but rather will provide careful comments on your written submitted work. Late submissions will not be accepted, and forfeit your opportunity to submit retries.

Each section of the textbook contains three reading questions at the end. Once you have read the section prior to our in-class discussion, it will be time to consider these questions. Responses will be due by 6 AM of the day we discuss the section in class, and will not be accepted late. If a question asks for a computation, then provide a complete answer, perhaps with some relevant intermediate computations. If the question requests a yes/no answer, or asks “Why?” then give a thorough explanation using correct English grammar, syntax, and punctuation along with appropriate use of . Cutting and pasting from the textbook without a citation is always plagiarism. And even providing a verbatim quote along with a citation will always get you zero credit.

Runestone information:

• Log in at runestone.academy
• Your username is the first part of your UPS email address—everything up to, but not including, the @ character.
• Your initial password is your UPS ID number (7 digits, starts with a 5).
• Reading questions are towards the bottom, in a subsection of their own.
• The text box will interpret notation, so as we learn how to use you will be expected to format your answers properly.
• My responses will appear here in the problem itself as well, shortly after they are due.

Grades will be based on the following breakdown:

• Writing: 15%

• Examinations: 55%

• Final Examination: 20%

Attendance and improvement will be considered for borderline grades. Scores will be posted anonymously on the web at a link off the course page.

Here are three reminders about important academic policies which are described thoroughly in the “Academic Policies” section of the University Bulletin. The online version is off of

www.pugetsound.edu/academics/academic-resources/university-bulletins/

or a printed copy may be requested from the Registrar's Office (basement of Jones Hall).

• ###### Registration for Courses of Instruction, Non-Attendance.

“Regular class attendance is expected of all students. Absence from class for any reason does not excuse the student from completing all course assignments and requirements.”

Withdrawal grades are often misunderstood. A Withdrawal grade (W) can only be given prior to the university deadline listed on our course schedule, and after that time (barring unusual circumstances), the appropriate grade is a Withdrawal Failing (WF), even if your work has been of passing quality. See the attached schedule for the last day to drop with an automatic ‘W’.

All of your graded work is expected to be entirely your own work, this includes Reading Questions and Writing Exercises. Anything to the contrary is a violation of the university's comprehensive policy on Academic Integrity (cheating and plagiarism). Discovered incidents will be handled strictly, in accordance with this policy. Penalties can include failing the course and range up to being expelled from the university.

##### Purpose

This course is much different from most any mathematics course you have had recently, in particular it is much different than calculus courses. We will begin with a simple idea—a linear function—and build up an impressive, beautiful, abstract theory. We will begin computationally, but soon shift to concentrating on theorems and their proofs. By the end of the course you will be at ease reading and understanding complicated proofs. You will also be very good at writing routine proofs and will have begun the process of learning how to create complicated proofs yourself.

You will see this material applied in subsequent courses in mathematics, computer science, chemistry, physics, economics and other disciplines (though we will not have much time for applications this semester). You will gain a “mathematical maturity” that will be helpful as you pursue upper-division coursework and in any logical, rational, or argumentative activity you might engage in throughout your lifetime. It is not easy material, but your attention and hard work will be amply repaid with an in-depth knowledge of some very interesting and fundamental ideas, in addition to beginning to learn to think like a mathematician.

##### Conduct

Virtual: Please find a quiet place with a good connection to attend online sessions, so that we might better have questions and discussions. I appreciate it if you can leave your camera on—please contact me privately if this is a problem. Please do not use chat to ask questions, or to talk to one another. You can use chat to be recognized with a question, or simply hold up your physical hand close to your camera.

##### University Notices

These are multiple notices the university administration requests we duplicate for you.

• ###### Classroom Emergency Response Guidance.

Please review university emergency preparedness, response procedures and a training video posted at www.pugetsound.edu/emergency/. There is a link on the university home page. Familiarize yourself with hall exit doors and the designated gathering area for your class and laboratory buildings.

If building evacuation becomes necessary (e.g. earthquake), meet your instructor at the designated gathering area so she/he can account for your presence. Then wait for further instructions. Do not return to the building or classroom until advised by a university emergency response representative.

If confronted by an act of violence, be prepared to make quick decisions to protect your safety. Flee the area by running away from the source of danger if you can safely do so. If this is not possible, shelter in place by securing classroom or lab doors and windows, closing blinds, and turning off room lights. Lie on the floor out of sight and away from windows and doors. Place cell phones or pagers on vibrate so that you can receive messages quietly. Wait for further instructions.

• ###### Student Religious Accommodation.

The university provides reasonable religious accommodations for academic courses and programs, and the university policy is found at this location 1 . If you need a reasonable accommodation due to a religious holiday or organized religious activity, you must give me written notice within the first two weeks of class so that we can coordinate the accommodation. If you have questions about the policy, you may contact the University Chaplain. If you have a grievance about the application of the policy or the handling of your request for an accommodation, you may contact the Dean of the Faculty.

www.pugetsound.edu/about/offices-services/human-resources/policies/campus-policies/ student-religious-accommodations-in-academic-courses-or-programs
##### Learning Outcomes

The University Curriculum Committee and accrediting agencies expect to see a list of learning outcomes.

• Understand vector spaces.
• Understand linear transformations between vector spaces.
• Understand that every vector space is isomorphic to the coordinate vector space over its base field.
• Understand that every linear transformation can be represented by a matrix.
• Understand that matrix multiplication is composition of linear transformations.
• Understand that matrix diagonalization is achieved through a similarity transformation and is a matrix representation relative to a basis of eigenvectors.
• Understand that eigenvalues are a property of a linear transformation and are independent of its representation as a matrix.
• Better understand the general concept of an algebraic structure.

Please review these at the end of the semester when they will be easier to understand.

Tentative Daily Schedule
Monday Tuesday Thursday Friday
Jan 18
MLK Day
Jan 19
Section WILA
Syllabus
CoCalc
Jan 21
Section SSLE
Jan 22
Section RREF
Jan 25
Section TSS
Jan 26
Problem Session

Elect CR/NC
Jan 28
Section HSE
Jan 29
Section NM
Feb 1
Problem Session
Writing SLE
Drop w/o Record
Feb 2
Exam SLE
Feb 4
Section VO
Feb 5
Section LC
Feb 8
Section SS
Feb 9
Problem Session
Feb 11
Section LI
Feb 12
Section LDS
Feb 15
Section O
Feb 16
Problem Session
Writing V
Feb 18
Spring Break 1
Feb 19
Spring Break 2
Feb 22
Section MO
Feb 23
Exam V
Feb 25
Section MM
Feb 26
Section MISLE
Mar 1
Section MINM
Mar 2
Problem Session
Mar 4
Section CRS
Mar 5
Section FS
Mar 8
Problem Session
Writing M
Mar 9
Exam M
Mar 11
Section VS
Mar 12
Section S
Mid-Term
Tentative Daily Schedule
Monday Tuesday Thursday Friday
Mar 15
Section LISS
Mar 16
Problem Session
Mar 18
Section B
Mar 19
Section D
Mar 22
Section PD
Mar 23
Problem Session
Writing VS
Mar 25
Section EE
Mar 26
Section PEE
Mar 29
Spring Break 3
Mar 30
Spring Break 4
Apr 1
Section IS
Apr 2
Problem Session
Apr 5
Section SD
Apr 6
Exam VS
Apr 8
Section CP
Apr 9
Problem Session
Apr 12
Section LT
Apr 13
Exam E
Apr 15
Section ILT
Apr 16
Section SLT
Last Day to
Drop w/ a W
Apr 19
Section IVLT
Apr 20
Problem Session
Writing LT
Apr 22
Section VR
Apr 23
Section MR
Apr 26
Problem Session
Apr 27
Exam LT
Apr 29
Problem Session
Apr 30
Section CB
May 3
Problem Session
Writing R
May 4
Exam R
May 6