• Linear Algebra with Applications, 7 th edition, by Steven J. Leon, Prentice-Hall, 2006
  

• Contemporary Abstract Algebra, 6 th edition, by Joseph A. Gallian, Houghton-Mifflin, 2006

Course Contributions

MAT 573 covers the basic principles of linear and abstract algebra including an in-depth look at early elementary group theory. This course is required of all secondary school mathematics teacher certification candidates, and for many of them, it serves as the first significant introduction to abstraction. Definitions are carefully stated, most theorems carefully proved, and abundant concrete examples given. The student should begin to see the type of questions asked by algebraists and how algebra interacts with other branches of mathematics. The course provides ample opportunity to construct contextual proofs and have them criticized.

Topical Outline

1. Systems of Linear Equations, Vectors, Matrices
2. Determinants and their Properties
3. Vector Spaces, Subspaces, Linear Independence and Spanning Sets, Bases, Dimension
4. Row Space and Column Space of a Matrix
5. Linear Transformations and Similarity
6. Brief Review: Sets, Mappings, Equivalence Relations, Complex Numbers
7. Integers: Divisibility, Fundamental Theorem of Arithmetic, Integers Modulo n
8. Groups: Binary Operations, Groups, Subgroups, Cyclic Groups, Permutations, Symmetric Group, Isomorphisms

Grade

Quizzes

There will be quizzes (days to be announced in class) which will emphasize definitions and results seen in class. A missed quiz is assigned a grade of zero. THE LOWEST QUIZ GRADE WILL BE DROPPED.

Tests/Final

THERE WILL BE NO MAKE-UP TESTS. If an exam is missed because of an emergency, written documentation is to be provided to justify the absence, in which case the weight of that exam will be incorporated with the other grades. THE FINAL IS A COMPREHENSIVE EXAM.

Homework

Homework problems are listed on this site and in the course syllabus. They will be collected periodically. You will have e-mail assignments (posted on this site) that will be due the next class day by 3:00 pm. These assignments will consist of two or three questions based on the reading due for that day.

Student Outcomes

Upon completion of MAT 573, students should be able to demonstrate satisfactory knowledge of the major concepts of linear and abstract algebra, and they should be able to construct symbolically accurate and mathematically correct proofs of basic facts in elementary linear algebra and group theory . In particular, students should:

1. Possess a basic but solid knowledge of the mathematical objects encountered during their math career (such as numbers, matrices, polynomials, functions, sets) and recognize the algebraic similarities they share (NCTM 1.4, 1.5.14; INTASC 1, 7; CCT I(3,4,5), CCT II(1) )

2. Understand the concept of binary operation on a set and the idea of an algebraic structure (a set endowed with operations that satisfy certain axioms) (NCTM 1.2, 1.4, 1.5.14; INTASC 1; CCT I(3,4,5) )

3. Be able to prove elementary facts about vector spaces, groups and rings by logically combining definitions and theorems (NCTM 1.1, 1.2, 1.3, 1.4, 1.5.14; INTASC 1, 4, 8; CCT I(3,4,5), CCT II(1,2) )

4. Make connections between groups and familiar geometric objects such as a regular n-gon, tetrahedron, cube and so on (via the dihedral and symmetric groups) (NCTM 1.1, 1.4; INTASC 1; CCT I(3,4,5) )

5. Acquire knowledge about the ways to obtain new algebraic structures out of old ones (e.g. sub-structures, quotient structures, direct products) (NCTM 1.1, 1.2, 1.4, 1.5.14; INTASC 1; CCT I(3,4,5) )

6. Learn new ways of denoting familiar objects (e.g. cyclic notation for permutations) (NCTM 1.3; INTASC 1; CCT I(3,4,5) )

7. Most importantly, recognize when two algebraic structures are abstractly “the same” (in other words, internalize the concept of isomorphism) (NCTM 1.1, 1.2, 1.4, 1.5.14; INTASC 1, 7; CCT I(3,4,5), CCT II(1) )

Disabilities

Before you receive course accommodations in this class, you will need to make an appointment with the Disability Resource Office. However, if you have other information you would like to speak with me about, or if you have emergency medical information to share with me, or if you need special arrangements in case the building must be evacuated, please let me know as soon as possible.