Advanced Linear Algebra: with an Introduction to Module Theory

✍ Scribed by Shou-Te Chang

• Publisher: World Scientific
• Year: 2024
• Tongue: English
• Leaves: 275
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Contents
Preface
About the Author
Special Notes
1 Modules and Vector Spaces
1.1 Definitions and examples of modules and vector spaces
1.2 Linear combinations and linear independence
1.3 Bases
1.4 Dimension for finite dimensional vector spaces
1.5 Dimension for infinite dimensional vector spaces
1.6 More on cardinal arithmetic
2 Linear Maps
2.1 Linear maps
2.2 Quotient modules and quotient spaces
2.3 Matrices
3 Determinant
3.1 Basics of the determinant
3.2 Matrix operations
3.3 Reduced row echelon form and applications
3.4 Determinant and invertible matrices
3.5 Rank
4 Canonical Forms
4.1 Equivalent matrices
4.2 The normal form
4.3 Structure theorem of finitely generated modules over a PID
4.4 Invariance theorem
4.5 Rational canonical form
4.6 Jordan canonical form
5 A Brief Introduction to the Tensor Product
5.1 Bilinear maps and multilinear maps
5.2 Tensor products of vector spaces
5.3 Tensor products of modules
Index