Linear Algebra Done Right (3rd ed)
出版时间
2014-11-06
ISBN
9783319110806
评分
New edition extensively revised and updated
Covers new topics such as product spaces, quotient spaces, and dual spaces
Features new visually appealing format for both print and electronic versions
Includes almost three times the number of exercises as the previous edition
This best-selling textbook for a second course in linear algebra is aimed at undergrad 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. 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.
The third edition contains major improvements and revisions throughout the book. More than 300 new exercises have been added since the previous edition. Many new examples have been added to illustrate the key ideas of linear algebra. New topics covered in the book include product spaces, quotient spaces, and dual spaces. Beautiful new formatting creates pages with an unusually pleasant appearance in both print and electronic versions.
No prerequisites are assumed other than the usual demand for suitable mathematical maturity. 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 introduced, leading to the finite-dimensional spectral theorem and its consequences. Generalized eigenvectors are then used to provide insight into the structure of a linear operator.
From reviews of previous editions:
“… a didactic masterpiece”
—Zentralblatt MATH
“… a tour de force in the service of simplicity and clarity … 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
Sheldon Axler is Dean of the College of Science & Engineering at San Francisco State University. He has authored many well-received books including Precalculus: A Prelude to Calculus, Algebra & Trigonometry, College Algebra, A Glimpse at Hilbert Space Operators, Harmonic Function Theory, and Holomorphic Spaces.
- 本书彻底摒弃传统教材以行列式为核心的教学路径，将行列式推迟至全书最后章节，转而聚焦于线性算子在有限维向量空间上的结构分析，旨在培养读者对线性代数本质逻辑的深刻理解，而非单纯的矩阵计算技巧。
- 内容涵盖向量空间、线性映射、多项式、特征值、不变子空间、内积空间、复数域与实数域上的算子等高级主题，并新增积空间、商空间及对偶空间等现代数学基础概念，构建严谨且完整的抽象代数体系。
- 作者致力于简化证明过程并强化概念动机，提供大量习题以辅助理解，但需注意其习题难度与正文简略程度存在巨大落差，部分证明步骤省略较多，要求读者具备极强的逻辑推导能力和数学成熟度，不适合依赖直观计算的读者。
- 数学系本科生及研究生，特别是希望深入理解线性代数理论结构、准备从事纯数学研究或需要掌握严格证明技巧的学生，本书是学习高等代数、泛函分析等后续课程的重要理论基石。
- 已具备基础线性代数知识，希望摆脱矩阵运算束缚，从抽象代数视角重构知识体系，提升数学思维严谨性与抽象能力的进阶学习者，适合用于系统性重学和理论深化。
- 不适合工科生、初学者或仅需要掌握矩阵计算、解线性方程组等应用技能的读者，此类人群应选择侧重几何直观与计算方法的传统教材，否则极易因概念抽象、缺乏直观引导而产生挫败感。
- 阅读前务必具备扎实的数学基础，包括集合论、逻辑推理能力及基础线性代数知识，切勿将其作为线性代数入门教材；若感到困惑，应结合其他提供几何直观和计算细节的教材互补阅读，避免陷入纯抽象符号的迷宫。
- 书中证明过程简略，习题难度极大且与正文脱节，读者需做好长期攻坚准备，切勿追求快速阅读；建议配合官方提供的习题解答或相关辅导视频，逐步验证自己的推导过程，确保每一步逻辑严密无误。
- 严禁跳过定义直接看结论，必须逐字理解每个定理的前提条件与证明逻辑；对于商空间、对偶空间等抽象概念，需反复思考其数学动机，若无法建立直观联系，应暂停阅读并寻求其他解释资源，切勿强行推进。
- 读者普遍认为本书是数学思维训练的经典之作，逻辑严密、结构优美，能从根本上纠正对线性代数的错误认知，但其极度抽象的风格和缺乏直观引导的缺陷也备受诟病，被戏称为“用泛函分析降维打击线性代数”。
- 绝大多数评论指出该书完全不适合初学者或非数学专业学生，因其摒弃了矩阵计算和几何直观，导致读者难以建立物理或工程意义上的理解，强行阅读只会带来痛苦且无益于实际应用能力的提升。
- 尽管存在争议，但认可其价值的读者强调，经过长期数学训练后重读此书能体会到其理论之美，建议将其作为辅助教材而非唯一教材，必须搭配注重几何直观和计算技巧的书籍共同使用，以实现理论与实践的平衡。
本导读基于书籍简介、目录、原文摘录、短评和书评生成，不等同于全文精读。
- "You cannot expect to read mathematics the way you read a novel. If you zip though a page in less than an hour, you prabably going too fast."
- "作为上面命题应用的一个例子，考虑组((5,7),(4,3))。F2中这个含有两个向量的组显然是线性无关的，因为任意向量都不是另外一个向量的标量倍)。"
- "在上下文中，基通常都是自明的，但是，当采用符号 M(v, (v1, ..., vn)) 而不采用 M(v) 时，就需要把基明确地写出来。"
- "3.4 定理: 如果 V 是有限维向量空间, 并且 T∈ ℒ (V, W), 那么 range T 是 W 的有限维子空间, 并且 dim V = dim null T + dim range T"
- "这个推论的叙述并未涉及迹，但其简短的证明却用到了迹。在数学中一旦有类似的事情发生，我们可以确信背后一定隐藏着一个很好的定义。"
- "那么可以把 M(T) 的第 k 列视为 T 对 k 个基向量的作用。"
- "一組向量，如果只有係數全為0時才能線性組合成零向量，那麼這組向量線性無關。"
- "一組線性無關的向量，其任意線性組合而得的向量，其表述方式必定唯一。"
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