Home » Programs of Study » Mathematics » Course Outlines » Math 261

Math 261

COURSE DESCRIPTION: (for catalog)

A study of matrices, systems of equations and linear transformations in the setting of finite dimensional vector spaces. This course provides the basic linear algebra necessary for the study of multivariable calculus, differential equations and abstract algebra.

PREREQUISITE:

Mth 252 with a grade of C or better.

INSTRUCTIONAL MATERIALS REQUIRED OF STUDENT: (text, supplies, etc.)

LIST THE PERFORMANCE OBJECTIVES OR STUDENT LEARNING COMPETENCIES AS DESCRIBED IN THE INSTRUCTIONAL LEARNING SYSTEM FOR THIS COURSE:

1. COURSE CONTENT
1. Vector Spaces
2. Matrices
3. Determinants
4. Systems of Linear Equations
5. Transformations
2. PERFORMANCE OBJECTIVES
1. Vector Spaces
1. The student will be able to determine if a mathematical system is a vector space.
2. The student will be able to use the algebra of vectors to solve mathematical problems.
3. The student will be able to determine a basis for a vector space including an orthonormal basis by the Gram-Schmidt process.
2. Matrices
1. The student will be able to apply the algebra of matrices to solve matrix equations.
2. The student will be able to represent a mapping from one vector space to another vector space in matrix form.
3. The student will be able to determine if a mapping is a linear map.
3. Determinants
1. The student will be able to evaluate an n x n determinant using row or column expansion by minors.
2. The student will be able to apply the properties of determinants to solve mathematical and applied problems.
4. Systems of Linear Equations
1. The student will be able to solve a system of linear equations using algebraic method of Gaussian elimination.
2. The student will be able to solve a system of linear equations using the augmented matrix method and elementary row operations.
3. The student will be able to solve a system of linear equations using the inverse matrix method, if possible.
4. The student will be able to find the rank of a matrix and determine the existence and uniqueness of the solution for a system of linear equations from the coefficient matrix.
5. The student will be able to solve a system of linear equations, using Cramer's Rule, if possible.
5. Transformations
1. The student will be able to find the eigen values, eigen vectors and eigen spaces associated with a square matrix.
2. The student will be able to use a similarity transformation to diagonalize a square matrix, if possible.
3. The student will be able to transform coordinate systems for Rn using similarity transformations to simplify mathematical equations.

GENERAL INSTRUCTIONAL METHODS:

The concept should be motivated through applications principally from computer sciences but additionally from other areas as well.

The approach should be mathematically precise but not overly rigorous. Definitions and theorems should be presented clearly to model the precision of language that mathematics requires. Some proofs should be presented carefully in class, accompanied by a discussion of the strategy and informal insight. Mathematical Induction is an especially important proof technique in discrete math and computer science. It should be illustrated with many types of examples.

Two important goals for this course are to develop mathematical reasoning and to achieve a perspective of discrete math as a language for describing certain structures. The reasoning might be developed by a process that includes the following components: 1) careful modeling of problem solving in lecture as well as in the text, 2) assignments that make it clear to the student that original reasoning and problem solving are required - more than just cloning examples, 3) feedback on written work and class discussions.

The perspective of mathematics as a language should be discussed in class. This can motivate the acquisition of vocabulary and the proper usage in order to achieve the goal of precise communication.

Some software packages for discrete mathematics are available. Their use may help motivate students, especially the computer science majors. Some of the video series For All Practical Purposes relate to discrete math and may be effectively used in class or as optional assignments or projects. It is recommended that in addition to regular homework type exercises, some longer term problems be given. These might involve synthesizing several concepts, and could be done individually or in groups. Students might be allowed some choice on projects based on individual interest and these could include programming projects.

The material on graph theory varies considerably among discrete math textbooks. To avoid confusion, it is recommended that the instructor follow the terminology and development of the current text. Be especially careful in bringing in supplementary material since even the basic definitions and theorems from another book may produce inconsistency. It is important to make the students aware of this as well.

EVALUATION PROCESS:

Completion of course objectives and assigned papers and projects according to criteria provided by the instructor will be required. Grading will be in accordance with college standards.