Mathematics

The course treats some basic parts of discrete mathematics of importance in mathematics, mathematical statistics, computer science and many other subject areas in science and technology. Topics include Number theory: divisibility, prime numbers, the Euclidean algorithm, Chinese remainder theorem, modular arithmetic; Sets, functions and relations, equivalence relations; Combinatorics: the four cases of counting with or without repetition and with or without regard to order, binomial coefficients, the principle of inclusion and exclusion, the method of generating functions; Recursion: recursion formulae and difference equations; Rings and fields: definitions and applications to coding theory; Graph theory: terminology and basic concepts, Eulerian and Hamiltonian graphs.

The aim of the course is to give the necessary knowledge of digital image analysis for further research within the area and to be able to use digital image analysis within other research areas such as computer graphics, image coding, video coding, and industrial image processing problems. The course also prepares students for further studies in computer vision, multispectral image analysis, and statistical image analysis.

This course introduces students to basic concepts from abstract algebra, especially the notion of a group. The course helps prepare students for further study in abstract algebra as well as familiarize them with tools essential in many other areas of mathematics. The course is also intended to help students in the transitions from concrete to abstract mathematical thinking and from a purely descriptive view of mathematics to one of definition and deduction.

This course is the second course in a two semester sequence for the sophomore/junior level undergraduate linear algebra. It helps students understand the abstraction of linear algebra. Linear Algebra is a basic language in mathematics and has many applications in every branch of mathematics. The course covers all the topics such as vector spaces and linear transformations, matrix algebra and analysis, inner product and normed spaces in linear algebra commonly used by analysts, combinatorists, computer scientists, geometers, logicians, number theorists, or topologists. The major goals are: to develop a systematic knowledge of the elements of linear algebra, and the ability to apply the concepts covered in classes; Fields and Vector Spaces, Linear Operators, Determinants and Eigenvalues, The Jordan Canonical Form, Orthogonality, Spectral Theory, Singular Value Decomposition, Matrix Factorization, and Infinite Dimensional Vector Spaces; to understand the elements of linear algebra with an emphasis on concepts, methods of proof, and the communication of mathematical ideas; to see how all these play a key role in many practical applications in today's technological society; Various applications of linear algebra show how linear algebra is essential not only in solving problems involving algebra, geometry, differential equations, optimization, approximation, combinatorics, but also in the fields such as biology, economics, computer graphics, electrical engineering, cryptography, political science as well as sciences; to broaden students' horizons by learning connections of one subject to other areas of linear algebra and mathematics and by mentioning results at the forefront of research.

Textbook: Mark S. Gockenbach, "FINITE-DIMENSIONAL LINEAR ALGEBRA"

Assessment: Midterm (30%), Final (50%), Attendance & Presentations (10%), Homework, Assignments, Quizzes & Class Activity (10%)  

Prerequisite:  Calculus, Linear Algebra I