Abstract Vector Spaces

Fall 2010

The theory of abstract vector spaces and linear transformations is called “linear algebra”. Like calculus, linear algebra is built from a few simple but powerful ideas and, again like calculus, it is a cornerstone of modern mathematics. A rigorous course in the subject, such as this one, will cover the key ideas of the subject, train students to reason carefully in abstract situations, and enable them to construct and write proofs. This is the gateway to higher mathematics.

- Douglas Ulmer, Professor and Chair, School of Mathematics
- Office: Skiles 118B
- E-mail: ulmer@math.gatech.edu
- Web site: http://www.math.gatech.edu/~ulmer
- Office hours: Tuesdays 3:00 - 4:00, Wednesdays 1:00 - 3:00, or by appointment. Check my web site for changes.

- Title: Abstract Vector Spaces
- Number: 2406-M (82680)
- Course Meetings: MW 3:05 - 4:25 in Skiles 271
- Objectives: (1) Learn the basic theory of abstract vector spaces and linear transformations, how these notions relate to systems of linear equations, and selected other applications, such as to differential equations or geometry. (2) Learn to construct and write mathematical proofs.
- Audience: Students planning to major in Applied Mathematics or Discrete Mathematics, students minoring in Mathematics, and anyone planning to take proof-oriented, upper-division mathematics courses such as Abstract Algebra, Number Theory, Analysis, etc.
- Prerequisites: Math 1502, 1512, 1522, 15X2, or the equivalent
- Topics:
- Basic proof techniques
- Vector spaces, subspaces, span, linear independence, bases, dimension
- Linear transformations, nullspace, range, invertibility
- Inner products, orthonormal bases, Gramm-Schmidt process, direct sums and orthogonal complements
- Coordinates, change of coordinate formulas
- Definition and properties of determinants
- Eigenvalues, eigenvectors, and diagonalization
- Operators on inner product spaces: adjoint, orthogonal, symmetric, and normal operators; spectral theorem
- Time permitting, additional topics such as the Jordan and rational canonical forms

- Text: “Linear Algebra” (fourth edition) by Friedberg, Insel, and Spence
- Alternates: “Linear Algebra Done Right” by Axler has a very slimmed-down, elegant treatment. It might be useful if you find the main text too verbose. “Linear Algebra” by Hoffman and Kunze is the gold standard, high-end treatment. Check it out if you are hungry for more.
- Notes on Introduction to Mathematical Arguments by UC Berkeley Professor Michael Hutchings
- See the t-square page for other notes on proofs by my colleague Professor Chris Heil and by UC Berkeley Professor George Bergman.
- Check the course t-square page for discussion, assignments, etc.

- Homework will be assigned regularly, collected, and graded.
- There will be frequent short quizes using
**clickers**at the beginning of class. - There will be three
**in-class exams: on Monday, September 20; Monday, October 25; and Monday, November 22**. - The
**final exam will Monday, December 13, 2:50 - 5:40, in Skiles 271.** - Grades will be based on the percentage of possible points earned. The-in class exams will each count for 15% of the grade, the final will count for 35%, and homework and quizes will count for 20%.
- Cutoff percentages for A, B, C, D are 90%, 80%, 70%, and 60% respectively.

- 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 famiy, etc.), the score on that exam will be replaced with the score on the corresponding part of the final exam. 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. **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 and proofs 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 assigmemnt 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 class, usually on Mondays. Late homework is strongly discouraged. It will be accepted up to 48 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.
- Experience shows that attendance in class is very much to the student's benefit. It is strongly encouraged and offers the opportunity to earn quiz points.
- Class time will be used to discuss difficult ideas, points of confusion, etc., not to go over basics in the text. Read assignments before class and come prepared with questions.
- Show some respect for your professor and your classmates---save Facebook, YouTube, and Technique for other times.