Linear algebra is growing in importance. 3D entertainment, animations in movies and video games are developed using linear algebra. Animated characters are generated using equations straight out of this book. Linear algebra is used to extract knowledge from the massive amounts of data generated from modern technology.

The Fourth Edition of this popular text introduces linear algebra in a comprehensive, geometric, and algorithmic way. The authors start with the fundamentals in 2D and 3D, then move on to higher dimensions, expanding on the fundamentals and introducing new topics, which are necessary for many real-life applications and the development of abstract thought. Applications are introduced to motivate topics.

The subtitle, A Geometry Toolbox, hints at the book's geometric approach, which is supported by many sketches and figures. Furthermore, the book covers applications of triangles, polygons, conics, and curves. Examples demonstrate each topic in action.

This practical approach to a linear algebra course, whether through classroom instruction or self-study, is unique to this book.

New to the Fourth Edition:

Ten new application sections.
A new section on change of basis. This concept now appears in several places.
Chapters 14-16 on higher dimensions are notably revised.
A deeper look at polynomials in the gallery of spaces.
Introduces the QR decomposition and its relevance to least squares.
Similarity and diagonalization are given more attention as are eigenfunctions.
A longer thread on least squares, running from orthogonal projections to a solution via SVD and the pseudoinverse.
More applications for PCA have been added.
More examples, exercises, and more on the kernel and general linear spaces.
A list of applications has been added in Appendix A.

The book gives instructors the option of tailoring the course for the primary interests of their students: mathematics, engineering, science, computer graphics, and geometric modeling.

Table of Contents

Preface

Descartes' Discovery
Here and There: Points and Vectors in 2D
Lining Up: 2D Lines
Changing Shapes: Linear Maps in 2D
2 2 Linear Systems
Moving Things Around: Aﬃne Maps in 2D
Eigen Things
3D Geometry
Linear Maps in 3D
Aﬃne Maps in 3D
Interactions in 3D
Gauss or Linear Systems
Alternative System Solvers
General Linear Spaces
Eigen Things Revisited
The Singular Value Decomposition
Breaking It Up: Triangles
Putting Lines Together: Polylines and Polygons
Conics
Curves

Appendices

Applications

Glossary

Select Exercise Solutions

Bibliography

Biography

Gerald Farin (deceased) was a professor in the School of Computing, Informatics, and Design Systems Engineering (CIDSE) at Arizona State University. He received his doctoral degree in mathematics from the University of Braunschweig, Germany. His extensive experience in geometric design started at Daimler-Benz. He was a founding member of the editorial board for the journal Computer-Aided Geometric Design (Elsevier), and he served as co-editor in chief for more than 20 years. He published more than 100 research papers. Gerald also organized numerous conferences and authored or edited 29 books. This includes his much read and referenced textbook Curves and Surfaces for CAGD and his book on NURBS. In addition to this book, Gerald and Dianne co-authored The Essentials of CAGD, Mathematical Principles for Scientific Computing and Visualization both also published by AK Peters/CRC Press.

Dianne Hansford, received her Ph.D. from Arizona State University. Her research interests are in the field of geometric modeling with a focus on industrial curve and surface applications related to mathematical definitions of shape. Together with Gerald Farin (deceased), she delivered custom software solutions, advisement on best practices, and taught on-site courses as a consultant. She is a co-founder of 3D Compression Technologies. She is now lecturer in the School of Computing, Informatics, and Design Systems Engineering (CIDSE) at Arizona State University, primarily teaching geometric design, computer graphics, and scientific computing and visualization. In addition to this book, Gerald and Dianne co-authored The Essentials of CAGD, Mathematical Principles for Scientific Computing and Visualization both also published by AK Peters/CRC Press.

線性代數的重要性日益增加。3D 娛樂、電影和視頻遊戲中的動畫都是使用線性代數開發的。動畫角色是根據本書中的方程式生成的。線性代數被用來從現代技術產生的大量數據中提取知識。

這本受歡迎的教科書第四版以全面、幾何和算法的方式介紹線性代數。作者從 2D 和 3D 的基本概念開始，然後進入更高的維度，擴展基本概念並引入許多實際應用和抽象思維發展所需的新主題。應用案例的引入旨在激發讀者對主題的興趣。

副標題《幾何工具箱》暗示了本書的幾何方法，並且有許多草圖和圖形作為支持。此外，本書涵蓋了三角形、多邊形、圓錐曲線和曲線的應用。示例展示了每個主題的實際應用。

這本書提供的線性代數課程的實用方法，無論是通過課堂教學還是自學，都是獨一無二的。

第四版的新內容：
- 十個新的應用部分。
- 一個關於基底變換的新部分。這個概念現在出現在幾個地方。
- 第 14 至 16 章關於更高維度的內容有顯著修訂。
- 更深入地探討了空間畫廊中的多項式。
- 引入了 QR 分解及其與最小二乘法的相關性。
- 相似性和對角化得到了更多的關注，特徵函數也是如此。
- 關於最小二乘法的內容更長，從正交投影到通過 SVD 和偽逆的解決方案。
- 增加了更多的 PCA 應用。
- 增加了更多的示例、練習，以及有關核和一般線性空間的內容。
- 附錄 A 中增加了一個應用列表。

本書使教師能夠根據學生的主要興趣（數學、工程、科學、計算機圖形學和幾何建模）來調整課程。

目錄
前言
1. 笛卡爾的發現
2. 這裡和那裡：2D 中的點和向量
3. 排成一行：2D 線
4. 改變形狀：2D 中的線性映射
5. 2x2 線性系統
6. 移動物體：2D 中的仿射映射
7. 特徵事物
8. 3D 幾何
9. 3D 中的線性映射
10. 3D 中的仿射映射
11. 3D 中的互動
12. 高斯或線性系統
13. 替代系統求解器
14. 一般線性空間
15. 再次探討特徵事物
16. 奇異值分解
17. 拆分：三角形
18. 組合線條：折線和多邊形
19. 圓錐曲線
20. 曲線

附錄
1. 應用
2. 詞彙表
3. 選擇練習解答

參考文獻

傳記
Gerald Farin（已故）曾是亞利桑那州立大學計算、資訊與設計系統工程學院的教授。他在德國布倫瑞克大學獲得數學博士學位。他在幾何設計方面的豐富經驗始於戴姆勒-奔馳。他是《計算機輔助幾何設計》（Elsevier）期刊編輯委員會的創始成員，並擔任主編超過 20 年。他發表了 100 多篇研究論文。Gerald 還組織了許多會議，並撰寫或編輯了 29 本書籍。其中包括他廣受閱讀和引用的教科書《CAGD 的曲線與曲面》和他的 NURBS 書籍。除了這本書，Gerald 和 Dianne 共同撰寫了《CAGD 的要素》和《科學計算與可視化的數學原則》，這兩本書也由 AK Peters/CRC Press 出版。

Dianne Hansford，在亞利桑那州立大學獲得博士學位。她的研究興趣集中在幾何建模領域，特別是與形狀的數學定義相關的工業曲線和曲面應用。她與已故的 Gerald Farin 一起提供定制軟件解決方案、最佳實踐建議，並作為顧問教授現場課程。她是 3D 壓縮技術公司的共同創始人。她目前是亞利桑那州立大學計算、資訊與設計系統工程學院的講師，主要教授幾何設計、計算機圖形學以及科學計算與可視化。除了這本書，Gerald 和 Dianne 共同撰寫了《CAGD 的要素》和《科學計算與可視化的數學原則》，這兩本書也由 AK Peters/CRC Press 出版。

Gerald Farin (deceased) was a professor in the School of Computing, Informatics, and Design Systems Engineering (CIDSE) at Arizona State University. He received his doctoral degree in mathematics from the University of Braunschweig, Germany. His extensive experience in geometric design started at Daimler-Benz. He was a founding member of the editorial board for the journal Computer-Aided Geometric Design (Elsevier), and he served as co-editor in chief for more than 20 years. He published more than 100 research papers. Gerald also organized numerous conferences and authored or edited 29 books. This includes his much read and referenced textbook Curves and Surfaces for CAGD and his book on NURBS. In addition to this book, Gerald and Dianne co-authored The Essentials of CAGD, Mathematical Principles for Scientific Computing and Visualization both also published by AK Peters/CRC Press.

Dianne Hansford, received her Ph.D. from Arizona State University. Her research interests are in the field of geometric modeling with a focus on industrial curve and surface applications related to mathematical definitions of shape. Together with Gerald Farin (deceased), she delivered custom software solutions, advisement on best practices, and taught on-site courses as a consultant. She is a co-founder of 3D Compression Technologies. She is now lecturer in the School of Computing, Informatics, and Design Systems Engineering (CIDSE) at Arizona State University, primarily teaching geometric design, computer graphics, and scientific computing and visualization. In addition to this book, Gerald and Dianne co-authored The Essentials of CAGD, Mathematical Principles for Scientific Computing and Visualization both also published by AK Peters/CRC Press.

傑拉德·法林（已故）曾是亞利桑那州立大學計算、資訊與設計系統工程學院（CIDSE）的教授。他在德國布倫瑞克大學獲得數學博士學位。他在幾何設計方面的豐富經驗始於戴姆勒-奔馳（Daimler-Benz）。他是期刊《計算機輔助幾何設計》（Computer-Aided Geometric Design，Elsevier）編輯委員會的創始成員，並擔任主編超過20年。他發表了超過100篇研究論文。傑拉德還組織了許多會議，並撰寫或編輯了29本書籍。其中包括他廣受閱讀和引用的教科書《CAGD的曲線與曲面》（Curves and Surfaces for CAGD）以及他關於NURBS的書籍。除了這本書，傑拉德和黛安共同撰寫了《CAGD的要素》（The Essentials of CAGD）、《科學計算與可視化的數學原則》（Mathematical Principles for Scientific Computing and Visualization），這兩本書也由AK Peters/CRC Press出版。

黛安·漢斯福德在亞利桑那州立大學獲得博士學位。她的研究興趣集中在幾何建模領域，特別是與形狀的數學定義相關的工業曲線和曲面應用。她與傑拉德·法林（已故）合作提供定制軟體解決方案、最佳實踐建議，並作為顧問教授現場課程。她是3D壓縮技術公司的共同創辦人。她目前是亞利桑那州立大學計算、資訊與設計系統工程學院（CIDSE）的講師，主要教授幾何設計、計算機圖形學以及科學計算與可視化。除了這本書，傑拉德和黛安共同撰寫了《CAGD的要素》（The Essentials of CAGD）、《科學計算與可視化的數學原則》（Mathematical Principles for Scientific Computing and Visualization），這兩本書也由AK Peters/CRC Press出版。