Linear Algebra with Abstract Vector Spaces

Fall 2015

First, you will learn the fundamental algorithm for solving systems of linear equations, which is called “Gauss-Jordan elimination” or “row-reduction”. Most calculations in the subject boil down to this algorithm. Second, you will learn an abstract framework (“vector spaces and linear maps”) that captures the essential features of systems of equations. This framework is sufficiently general to model many other situations in mathematics. Third, you will learn a method (“bases and coordinates”) for translating any question in the abstract setting into a concrete question about systems of linear equations that can be answered with row reduction. Along the way, we will discuss a number of applications of these ideas to problems in science and engineering. At the end of the day, you will master powerful tools with a very broad range of applicability.

This course will also give you an introduction to abstraction and higher mathematics. We will be careful about definitions and will give careful arguments, i.e., proofs, justifying everything we do. You will increase your ability to write clear and precise explanations of mathematical facts.

First, because it is really useful. Mathematicians sometimes say that all of mathematics is calculus and linear algebra. This isn't completely true, but it's close. In science and engineering, linear algebra is a tool used every day, for example: linear approximations of functions (think interpolation and extrapolation), interpreting data (least squares regressions and lines of best fit), signal processing, computer graphics, internet search, random processes, even the foundations of quantum mechanics.

Second, you are a top student in Math, and this course is designed to challenge students like you.

Third and most importantly, because it is really interesting! You will see the power of the axiomatic method in isolating and analyzing what is essential.

Yes! You did well in calculus, proving that you know how to work hard and master complex material. Although the material in this class is at times somewhat abstract, the underlying ideas are pretty simple (a lot simpler than things like continuity and limits).

The TAs and I are commited to helping you learn this material and succeed in the course. Keep at it and you will do well.

See the last section below for more hints on how to succeed in this class and advanced math classes in general.

- Course: Math 1564-K
- Number: 1564-K1 (90785) and 1564-K2 (91119)
- Title: Linear Algebra with Abstract Vector Spaces
- Credit: 4 hours (4-0-4)

- Basic algorithms: You will learn to perform Gauss and Gauss-Jordan elimination (“row reduction”) on a matrix and interpret the results in terms of solutions of systems of linear equations, including existence, uniqueness, and parameteriztion of solutions. You will also learn to compute least-squares approximations and identify when they are appropriate and relevant. You will also learn to compute and interpret determinants, eigenvalues, and eigenvectors.
- Abstract structures: You will understand and apply the basic abstract structures of linear algebra, namely vector spaces, subspaces, spans and spanning sets, linear independence, and bases, as well as linear maps.
- Coordinates: You will earn how a choice of basis transforms any question in the abstract setting into a concrete question about matrices and vectors which can be solved using the basic algorithms. You will be able to interpret the results of these calculations in the abstract setting.
- Applications: You will become familiar with several applications of basic linear algebra and implementation issues: numerical stability, analysis of DC circuits, Markov chains, the method of powers, and the Google Page Rank algorithm.

- Douglas Ulmer, Professor of Mathematics
- Office: Skiles 234
- E-mail: ulmer@math.gatech.edu
- Web site: http://www.math.gatech.edu/~ulmer
- Office hours:
~~Mondays 2:00 - 3:00~~, 1:30 - 2:30 on Monday 9/28 and Mondays 1:00 - 2:00 thereafter, and Thursdays 3:00 - 4:00, or by appointment. Check my web site for changes.

- TA for Section K1: Xin Xing
- Office: Skiles 140
- E-mail: xxing33@math.gatech.edu
- Office hours: Tuesdays 11:00 - 12:00, Wednesdays 11:00 - 12:00, and in the Math lab (Clough 280) Tuesdays 2:00 - 3:00

- TA for Section K2: Juntao Duan
- Office: Skiles 146A
- E-mail: jduan9@math.gatech.edu
- Office hours: Mondays 1:00 - 2:00, Wednesdays 1:00 - 2:00, and in the Math lab (Clough 280) Mondays 5:00 - 6:00

- Instructional designer: Greg Mayer
- Office: Skiles 116
- E-mail: greg.mayer@gatech.edu

- Lectures: TTh 1:35 - 2:55 in Mason 2117
- Recitation K1: MW 12:05 - 12:55 in College of Computing 52
- Recitation K2: MW 12:05 - 12:55 in Instructional Center 209

- Course topics and timing: See the syllabus for a detailed list of topics.
- Required text: “Linear Algebra” by J. Hefferon. Available as a free download at http://joshua.smcvt.edu/linearalgebra/. A printed and bound version is available from Amazon.
- In-class polling system: You will need to set up an account with Learning Catalytics. (You may already have one if you use a Pearson book in another class. If not, the cost is $12 for six months or $20 for a year.) You will use this system and your laptop, tablet, or phone to answer questions in class.
- All homework assignments, lecture notes, sample exams, etc. will be posted on t-square.
- Part of each homework assignment will use the WebWork on-line homework system. This system generates problems with randomized data, allows unlimited attempts, and grades instantly. These problems are mostly routine drill, but are essential for mastering course material. Assignments will be linked from t-square.
**Important:**To use WebWork from off-campus, you will need to use a VPN. See this OIT page for more on how to set it up. On campus or off, you will also need to configure your browser to allow pop-ups. - The t-square page also links to a Piazza discussion forum. This is the place to post questions and answer questions from your fellow students. It will be the
**fastest**way to get your questions answered. Active participation in Piazza will help you learn the material and will be viewed favorably when grades are assigned.

**Homework**will be assigned weekly, collected, and graded. Part of each assignment will be routine problems done on-line using WebWork, with unlimited attempts and immediate grading. Another part will consist of more challenging problems, to be written up in detail with careful arguments and correct standard English.- There will be frequent short
**quizzes**(using Learning Catalytics) during lectures. Quiz points will be part of the homework grade - There will be two
**in-class exams: on Tuesday, September 22 and Tuesday, November 3**. - The
**final exam will Tuesday December 8th from 2:50pm to 5:40pm, in Mason 2117.**It will be a cumulative exam, covering the whole course.

- Grades will be based on the percentage of possible points earned.
- The in-class exams will each count for 20% of the grade, the final will count for 30%, and homework and quizes will count for 30%.
- Cutoff percentages for A, B, C, D are 90%, 80%, 70%, and 60% respectively.

- Lectures on Tuesday and Thursday will present new material. There will often be quizes to test whether you have understood basic aspects of the preceding lectures. Homework will be assigned on Thursday.
- The Wednesday recitation will be used to
*actively*practice working with the new material, for example by solving problems on worksheets and discussing the answers. - The on-line component of the homework (WebWork) will be due on Sunday night at 11pm.
- The Monday recitation will be used to practice new material and to answer questions related to the more challenging written homework.
- The written homework will be due on Tuesday before lecture and will be returned the following Thursday.
- There will be slight modifications to the schedule on weeks with exams or breaks. Note, however, that homework
**will**be due on exam days.

- There will be
**no make-up exams**. If an in-class exam is missed for an*acceptable and documented*reason (severe illness, death in the family, etc.), the score on that exam will be replaced with the score on the corresponding part of the final exam. (What is “corresponding” will be determined by the professor.) A test missed for unacceptable or undocumented reasons will receive a score of 0. - Arrangements for extensive absences (e.g., for extracurricular activities) must be made by the end of the first week of classes. Students requiring arrangements from the ADAPTS-Disability Services Program should consult their information page and make arrangements as soon as possible.
**Collaboration and plagiarism**: Discussing ideas and homework exercises with your peers is not only acceptable, it is a good idea. However, you must write your own solutions in your own words. Copying another's words or otherwise passing off someone else's work as your own is plagiarism and will result in a score of 0 on the entire assignment in question. Egregious cases will be dealt with more harshly.**ASK**if you have**any questions whatsoever about this**- Homework is due at the beginning of lecture, usually on Tuesdays. Late homework is strongly discouraged. It will be accepted up to 24 hours after the due time and will be assessed a 20% penalty.
- The Georgia Tech Honor Code applies 100% without exception. Know it and live it.
- Show some respect for your professor and your classmates---save Facebook, YouTube, and Technique for other times.

**Read**the book. The high school technique of flipping through the section looking for an example like the problem will not work in college. The key is to read**actively**. This means constantly asking yourself questions like “What is the point of this example? What idea does it illustrate?” In a theorem, “why are these assumptions being made? What happens if we relax one of them?”- It is
*very*beneficial to read the relevant parts of the text**before class**. The discussion will be much more meaningful if you already know what the key issues are. - Your goal should be
**active understanding**. This means being able to apply concepts in a new setting. The best way to develop active understanding is to do lots of problems. The texts have far more good problems than can be graded in this class. Doing them is an excellent way to study. **On writing**: The solution to an exercise or the proof of a theorem is a form of communication---it's a conversation between the reader and the author. The goal is to convince the reader that the solution or proof is correct. Doing this well involves all the same skills as writing an essay, in particular, knowing something about the audience (What can be assumed? What is “trivial” and what requires explanation?) and structuring the discussion so that it is easy to follow. Learning to write mathematics well is an important goal for this course.- Good solutions and proofs usually involve more words than formulas. The author should
**explain**what he or she is doing. I'll give some examples in class. - Working in small groups is a great idea. If you can explain the ideas in this course to your friends, then you really know them.
- The TAs and I are
*resources*. We can't learn the material for you, but we can help a lot, by explaining the concepts and clarifying subtle or tricky points. Take advantage, after all, we're why you are going to Tech, not that other, lesser place down the road. **Ask questions**, in class and in office hours. This is how we know what you need help with, and helping you is what we are trying to do.- The pace will be very fast and if you fall behind it will be very difficult to catch up.
**Keep up!**