Advanced Linear Algebra--高等线性代数

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Advanced Linear Algebra

Designed for advanced undergraduate and beginning graduate students in linear or abstract al-

gebra,Advanced Linear Algebracovers theoretical aspects of the subject, along with examples,

computations, and proofs. It explores a variety of advanced topics in linear algebra that highlight

the rich interconnections of the subject to geometry, algebra, analysis, combinatorics, numerical

computation, and many other areas of mathematics.

The author begins with chapters introducing basic notation for vector spaces, permutations,

polynomials, and other algebraic structures. The following chapters are designed to be mostly

independent of each other so that readers with different interests can jump directly to the topic

they want. This is an unusual organization compared to many abstract algebra textbooks, which

require readers to follow the order of chapters.

Each chapter consists of a mathematical vignette devoted to the development of one specific

topic. Some chapters look at introductory material from a sophisticated or abstract viewpoint,

while others provide elementary expositions of more theoretical concepts. Several chapters offer

unusual perspectives or novel treatments of standard results.

A wide array of topics is included, ranging from concrete matrix theory (basic matrix compu-

tations, determinants, normal matrices, canonical forms, matrix factorizations, and numerical

algorithms) to more abstract linear algebra (modules, Hilbert spaces, dual vector spaces, bilinear

forms, principal ideal domains, universal mapping properties, and multilinear algebra).

The book provides a bridge from elementary computational linear algebra to more advanced,

abstract aspects of linear algebra needed in many areas of pure and applied mathematics.

Nicholas A. Loehr received his Ph.D. in mathematics from the University of California at San

Diego in 2003, studying algebraic combinatorics under the guidance of Professor Jeffrey Rem-

mel. After spending two years at the University of Pennsylvania as an NSF postdoc, Dr. Loehr

taught mathematics at the College of William and Mary, the United States Naval Academy, and

Virginia Tech. Dr. Loehr has authored over sixty refereed journal articles and three textbooks on

combinatorics, advanced linear algebra, and mathematical proofs. He teaches classes in these

subjects and many others, including cryptography, vector calculus, modern algebra, real analy-

sis, complex analysis, and number theory.

Textbooks in Mathematics

Series editors:

Al Boggess, Kenneth H. Rosen

Classical Vector Algebra

Vladimir Lepetic

Introduction to Number Theory

Mark Hunacek

Probability and Statistics for Engineering and the Sciences with Modeling using R

William P. Fox and Rodney X. Sturdivant

Computational Optimization: Success in Practice

Vladislav Bukshtynov

Computational Linear Algebra: with Applications and MATLAB® Computations

Robert E. White

Linear Algebra With Machine Learning and Data

Crista Arangala

Discrete Mathematics with Coding

Hugo D. Junghenn

Applied Mathematics for Scientists and Engineers

Youssef N. Raffoul

Graphs and Digraphs, Seventh Edition

Gary Chartrand, Heather Jordon, Vincent Vatter and Ping Zhang

An Introduction to Optimization with Applications in Data Analytics and Machine Learning

Jeffrey Paul Wheeler

Encounters with Chaos and Fractals, Third Edition

Denny Gulick and Jeff Ford

Differential Calculus in Several Variables

A Learning-by-Doing Approach

Marius Ghergu

Taking the “Oof!” out of Proofs

A Primer on Mathematical Proofs

Alexandr Draganov

Vector Calculus

Steven G. Krantz and Harold Parks

Intuitive Axiomatic Set Theory

José Luis García

Fundamentals of Abstract Algebra

Mark J. DeBonis

A Bridge to Higher Mathematics

James R. Kirkwood and Raina S. Robeva

Advanced Linear Algebra, Second Edition

Nicholas A. Loehr

https://www.routledge.com/Textbooks-in-Mathematics/book-series/CANDHTEXBOOMTH

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摘要：

本规范文档《Advanced Linear Algebra—高等线性代数》旨在为数学、物理、计算机科学及工程领域的高年级本科生与研究生提供系统化的线性代数进阶理论。内容涵盖向量空间、线性映射、矩阵分解、特征值与特征向量、内积空间、谱定理、Jordan标准形、广义逆矩阵及其在实际问题中的应用。文档从公理化定义出发，以严谨的数学推导与清晰的逻辑结构，帮助读者深入理解线性代数的本质与高阶抽象结构，同时配以典型例题与重要定理的证明，为后续学习泛函分析、数值计算、量子力学、数据科学等课程奠定坚实理论基础。

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