Advanced Linear Algebra: With an Introduction to Module Theory (1版)

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【簡介】 Certain essential concepts in linear algebra cannot be fully explained in a first course. This is due to a lack of algebraic background for most beginning students. On the other hand, these concepts are taken for granted in most of the mathematical courses at graduate school level. This book will provide a gentle guidance for motivated students to fill the gap. It is not easy to find other books fulfilling this purpose. This book is a suitable textbook for a higher undergraduate course, as well as for a graduate student's self-study. The introduction of set theory and modules would be of particular interest to students who aspire to becoming algebraists. There are three parts to this book. One is to complete the discussion of bases and dimension in linear algebra. In a first course, only the finite dimensional vector spaces are treated, and in most textbooks, it will assume the scalar field is the real number field. In this book, the general case of arbitrary dimension and arbitrary scalar fields is examined. To do so, an introduction to cardinality and Zorn's lemma in set theory is presented in detail. The second part is to complete the proof of canonical forms for linear endomorphisms and matrices. For this, a generalization of vector spaces, and the most fundamental results regarding modules are introduced to readers. This will provide the natural entrance into a full understanding of matrices. Finally, tensor products of vector spaces and modules are briefly discussed. 【目錄】 1.Modules and Vector Spaces 2.Linear Maps 3.Determinant 4.Canonical Forms 5.A Brief Introduction to the Tensor Product

LINEAR ALGEBRA II: ADVANCED TOPICS FOR APPLICATIONS

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This is the second volume of the two-volume book on linear algebra in the University of Tokyo (UTokyo) Engineering Course. The objective of this second volume is to branch out from the standard mathematical results presented in the first volume to illustrate useful specific topics pertaining to engineering applications. While linear algebra is primarily concerned with systems of equations and eigenvalue problems for matrices and vectors with real or complex entries, this volumes covers other topics such as matrices and graphs, nonnegative matrices, systems of linear inequalities, integer matrices, polynomial matrices, generalized inverses, and group representation theory. The chapters are, for the most part, independent of each other, and can be read in any order according to the reader's interest. The main objective of this book is to present the mathematical aspects of linear algebraic methods for engineering that will potentially be effective in various application areas. Request Inspection Copy Sample Chapter(s) Preface Chapter 1: Matrices and Graphs Contents: Preface Matrices and Graphs Nonnegative Matrices Systems of Linear Inequalities Integer Matrices Polynomial Matrices Generalized Inverses Group Representation Theory Bibliography Index Readership: Graduate students majoring in engineering and other mathematical sciences.