Linear algebra done right

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Linear Algebra Done Right (线性代数应该这样学)

The fourth and most recent edition of Linear Algebra Done Right is an Open Access book available in English, Chinese, Farsi, Greek, and Portuguese. Other languages are coming soon. Electronic versions of this fourth edition, which has a Creative Commons BY-NC license, are legally available without cost at the links below.

《线性代数应该这样学》（Linear Algebra Done Right）的第四版（最新版）是一本开放获取（Open Access）教材，目前提供英语、中文、波斯语、希腊语和葡萄牙语版本，其他语言版本也即将推出。该第四版采用知识共享（Creative Commons）BY-NC 许可协议，可通过下方链接合法免费获取电子版。

The fourth edition of Linear Algebra Done Right contains over 250 new exercises and over 70 new examples, along with several new topics and multiple improvements throughout the book. See page xvi in the English file linked above for a list of major improvements and additions in the fourth edition.

《线性代数应该这样学》第四版包含了超过 250 道新习题和 70 多个新例题，同时增加了若干新主题，并对全书进行了多处改进。有关第四版主要改进和新增内容的列表，请参阅上述英语版 PDF 文件的第 xvi 页。

The hardcover print version of the fourth edition of Linear Algebra Done Right is available from Amazon at the link below.

《线性代数应该这样学》第四版的精装印刷版可通过以下亚马逊链接购买。

Print versions of translations of the fourth edition into Greek and into Portuguese are available at booksellers in appropriate countries. Print versions of translations of the third edition into Basque and Chinese are available from Amazon at the links below.

第四版的希腊语和葡萄牙语译本印刷版可在相关国家的书店购买。第三版的巴斯克语和中文译本印刷版可通过以下亚马逊链接购买。

This best-selling textbook for a second course in linear algebra is aimed at undergraduate math majors and graduate students. The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces.

这本畅销教材专为线性代数进阶课程设计，面向数学专业本科生及研究生。书中采用了一种新颖的方法，将行列式的内容推迟到书末介绍。全书聚焦于线性代数的核心目标：理解有限维向量空间上线性算子的结构。

The author has taken unusual care to motivate concepts and to simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. No prerequisites are assumed other than the usual demand for suitable mathematical maturity.

作者在激发概念动机和简化证明方面倾注了非凡的心血。每一章中丰富有趣的习题有助于学生理解并掌握线性代数中的对象。除了要求具备相应的数学素养外，本书不预设其他先修知识。

Thus the text starts by discussing vector spaces, linear independence, span, basis, and dimension. The book then deals with linear maps, eigenvalues, and eigenvectors. Inner product spaces are then introduced, leading to the finite-dimensional spectral theorem and its consequences such as the singular value decomposition. Generalized eigenvectors are then used to provide insight into the structure of a linear operator. Determinants are cleanly introduced via alternating multilinear forms.

因此，本书从讨论向量空间、线性无关、张成、基和维数开始。随后探讨线性映射、特征值和特征向量。接着引入内积空间，进而推导出有限维谱定理及其推论，如奇异值分解。随后利用广义特征向量深入剖析线性算子的结构。最后，通过交替多线性形式简洁地引入了行列式。

Excerpts from Reviews (书评摘录)

  • “Altogether, the text is a didactic masterpiece.” — zbMATH

  • “Axler demotes determinants (usually quite a central technique in the finite dimensional setting, though marginal in infinite dimensions) to a minor role. To so consistently do without determinants constitutes a tour de force in the service of simplicity and clarity; these are also well served by the general precision of Axler’s prose…”

  • “The most original linear algebra book to appear in years, it certainly belongs in every undergraduate library.” — Choice

  • “The determinant-free proofs are elegant and intuitive.” — American Mathematical Monthly

  • “Clarity through examples is emphasized… the text is ideal for class exercises… I congratulate the author and the publisher for a well-produced textbook on linear algebra.” — Mathematical Reviews

  • “If you liked the previous editions, you will like this new edition even better!” — Monatshefte für Mathematik

  • “总而言之，这是一部教学杰作。” —— zbMATH

  • “Axler 将行列式（在有限维设置中通常是核心技术，但在无限维中作用边缘化）降级为次要角色。如此始终如一地摒弃行列式，是为追求简洁与清晰而进行的壮举；Axler 精准的文笔也为这些目标提供了有力支撑……”

  • “这是近年来出现的最具原创性的线性代数书籍，绝对应该被每一所大学的图书馆收藏。” —— Choice

  • “不使用行列式的证明既优雅又直观。” —— American Mathematical Monthly

  • “强调通过例题来体现清晰度……该书非常适合课堂练习……我向作者和出版商表示祝贺，这是一本制作精良的线性代数教材。” —— Mathematical Reviews

  • “如果你喜欢之前的版本，你会更喜欢这个新版本！” —— Monatshefte für Mathematik

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