Haohao Wang – författare

The purpose of this book is twofold: to present some basic ideas in commutative algebra and algebraic geometry and to introduce topics of current research, centered around the themes of Gröbner bases, resultants and syzygies. The presentation of the material combines definitions and proofs with an emphasis on concrete examples. The authors illustrate the use of software such as Mathematica and Singular. The design of the text in each chapter consists of two parts: the fundamentals and the applications, which make it suitable for courses of various lengths, levels, and topics based on the mathematical background of the students. The fundamentals portion of the chapter is intended to be read with minimal outside assistance, and to learn some of the most useful tools in commutative algebra. The applications of the chapter are to provide a glimpse of the advanced mathematical research where the topics and results are related to the material presented earlier. In the applications portion, the authors present a number of results from a wide range of sources without detailed proofs. The applications portion of the chapter is suitable for a reader who knows a little commutative algebra and algebraic geometry already, and serves as a guide to some interesting research topics.This book should be thought of as an introduction to more advanced texts and research topics. Its novelty is that the material presented is a unique combination of the essential methods and the current research results. The goal is to equip readers with the fundamental classical algebra and geometry tools, ignite their research interests, and initiate some potential research projects in the related areas.

Explore vector spaces, transformations, and matrix decompositions with practical applications in coding and image processing.

Key Features

In-depth theoretical conceptsPractical applications and exercisesComprehensive coverage of linear algebra topics

Book Description

This book introduces the fundamental concepts of linear algebra and applies the theorems in computation-oriented applications. It is suitable for a one-semester course and combines definitions and proofs with a focus on computational applications. Examples illustrate the use of software packages such as Mathematica, Maple, and Sage.The journey begins with vector spaces and progresses through linear transformations and operators. It then covers orthogonal bases and matrix decomposition. The material includes a brief introduction to aspects of abstract algebra related to linear algebra, such as groups, rings, modules, fields, and polynomials over fields.Understanding these concepts is crucial for solving complex problems in various fields. This book transitions readers from basic definitions to advanced computational applications, blending theoretical knowledge with practical skills. It is an essential resource for mastering linear algebra and its applications.

What you will learn

Understand vector spacesPerform linear transformationsSolve linear equationsDecompose matricesApply orthogonal basesUse linear algebra in coding

Who this book is for

Ideal for students, educators, and professionals in mathematics, computer science, and engineering. Readers should have a basic understanding of algebra. No advanced prerequisites are necessary.