《MATLAB線性代數簡明教程(Linear Algebra Using MATLAB)》共分為8章，主要內容為：第1章介紹了MATLAB軟件的桌面和MATLAB編程基礎，第2章～第7章分別介紹了線性代數的各種運算，包括矩陣運算、求解線性方程、向量空間和子空間、投影、*小二乘逼近、行列式、特征值和特征向量、相似矩陣和奇異值分解等。此外，基於圖像處理與線性代數有著密切的關繫，第8章簡要介紹了線性代數在圖像處理中的應用。本書結合線性代數理論與MATLAB語言介紹線性代數的應用，列舉了豐富的MATLAB代碼實例，不僅可以加深對線性代數理論的理解，而且可以提高應用線性代數知識解決實際問題的能力。

《MATLAB線性代數簡明教程(Linear Algebra Using MATLAB)》可作為高等院校理工科專業基礎課教材，也是運用MATLAB語言與數學知識解決實際問題的工具書，可供從事經濟、物理、繫統控制、信號處理、圖像處理等領域專業技術人員參考。

Chapter 1Fundament for MATLAB1

1.1MATLAB Desktop1

1.2Vectors and Matrices7

1.3MATLAB Programming11



Chapter 2Introduction to Vectors and Matrices20

2.1Vectors and Linear Combination20

2.2Lengths and Dot Products28

2.3Matrices31



Chapter 3Solving Linear Equations40

3.1Vectors and Linear Equations40

3.2The Idea of Elimination45

3.3Elimination Using Matrices52

3.4Inverse Matrices58



Chapter 4Vector Spaces and Subspaces62

4.1Spaces of Vectors62

4.2The Nullspace of matrix64

4.3The Rank and Linear Dependence68



Chapter 5Orthogonality71

5.1 Projections71

5.2Least Squares Approximations75

5.3Orthogonal Bases and Gram-Schmidt Process80



Chapter 6Determinants85

6.1 The Properties of Determinants85

6.2Permutations and Cofactors92

6.3Cramer’s Rule98



Chapter 7Eigenvalues and Eigenvectors102

7.1Introduction to Eigenvalues102

7.2Similar matrices and diagonalization of matrices105

7.3Singular Value Decomposition109



Chapter 8Linear Algebra in Image Processing112

8.1Digital Image Representation112

8.2Geometric Transformation using Matrix Operation114

8.3Image Restoration using Inverse Matrix117

8.4Image Fusion using Principal Component Analysis119

8.5Image Compression using Singular Value Decomposition120



References122

前言

歡迎使用《MATLAB線性代數簡明教程(Linear Algebra Using MATLAB)》這本書。線性代數是高等院校理工科專業的重要基礎課，也是運用數學知識解決實際問題的有力工具，廣泛應用於經濟、物理、繫統控制、信號處理、圖像處理等領域。線性代數對邏輯推理、抽像思維和基本運算的能力要求較高。在學習線性代數之後，讀者應該知道如何用數學方法解決實際問題，並使用計算機進行科學計算。 而本書采用MATLAB語言解決線性代數問題。MATLAB（MATrix LABoratory，矩陣實驗室）軟件源於矩陣運算。作為一種高度集成的計算機語言，擁有大量的工具箱，具有強大的科學運算能力、靈活的編程過程和高質量的圖形可視化能力。另外，MATLAB軟件與其他編程語言的接口也非常方便。

本書首先介紹了MATLAB軟件的桌面和MATLAB編程基礎(第1章)，然後介紹了線性代數的各種運算(第2章～第7章)，包括矩陣運算、求解線性方程、向量空間和子空間、投影、小二乘逼近、行列式、特征值和特征向量、相似矩陣和奇異值分解等。此外，由於圖像可以看作矩陣，對圖像的運算實際上就是對矩陣的運算。故基於圖像處理與線性代數有著密切的關繫，後介紹了線性代數在圖像處理中的應用(第8章)。本書結合線性代數理論與MATLAB語言介紹線性代數的應用，列舉了豐富的MATLAB代碼實例，不僅可以加深對線性代數理論的理解，而且可以提高應用線性代數知識解決實際問題的能力。你會發現學習線性代數有效的方式就是使用這本教材。

在此，我們感謝對本書提出寶貴意見的專家，特別感謝化學工業出版社編輯的大力支持。我們還感謝武漢大學2016級本科生董志斌、黃宇卓和霍啟正，他們參加了部分文字的輸入和程序調試工作。本書的編寫和出版得到了國家自然科學基金（編號：41901357，編號：41701394）的資助。在此，編者向資助的提供者表示誠摯的謝意。

盡管我們為編寫本書付出了很多的努力，但由於學識水平的限制，本書對一些問題的理解和解釋難免存在不妥之處，敬請讀者批評指正。如對本書有任何意見及建議，請與編者聯繫，郵箱：sli@whu.edu.cn。



編者

2019年9月







Preface

Welcome to Linear Algebra Using MATLAB.Linear algebra is an important basic course for science and technology majors in universities and also is a powerful tool to solve practical problems with mathematical knowledge，which is widely used in economy，physics，system control，signal processing，image processing and so on.Linear algebra requires your ability of logical reasoning，abstract thinking and basic operation.After learning linear algebra，you should know how to solve practical problems with mathematical methods，and use computers to carry out certain scientific calculations.This book focuses on solving linear algebra problems with software.MATLAB（the abbreviation for MATrix LABoratory）software originated from matrix operation.As a highly integrated computer language，MATLAB possesses plenty of toolboxes and provides ability of powerful scientific operations，flexible programming process，and high-quality graphics visualization.In addition，MATLAB software is also very convenient to interface with other programming languages.Thus，MATLAB language is adopted to solve the linear algebra problems in this book.

We first present the guide to MATLAB desktop and fundament to MATLAB programming.Then we introduce various operations of linear algebra using MATLAB，including matrix operation，solving linear equations，vector space and subspace，projections，least square approximation，determinants，eigenvalues and eigenvectors，similar matrices，and singular value decomposition.As colorful image can be treated as multi-dimensional matrix，the operation on image is actually the operation on matrix，which is closely related to linear algebra.Finally，we will give plenty of applications of linear algebra in image processing.This book integrates the theory of linear algebra，the application of linear algebra and the software of MATLAB.The abundant examples can not only deepen the understanding of linear algebra theory，but also improve the ability to apply linear algebra knowledge to solve practical problems.You will be learning the language of linear algebra in the best and most efficient way：by using it.

We are grateful to the many reviewers who helped us shape and refine Linear Algebra Using MATLAB.We also give special thanks to the people at Chemical Industry Press who worked with us in the development of Linear Algebra Using MATLAB.We also thank undergraduate student Zhibin Dong，Yuzhuo Huang and Qizheng Huo from grade 2016 in Wuhan University，who assisted with the development of the code.This book is supported by the project National Natural Science Foundation of China（No.41901357，No.41701394）.We are also grateful to our sponsors.

We have worked hard to make Linear Algebra Using MATLAB，a useful and practical textbook from which to teach and learn linear algebra.Despite our best efforts to ensure accuracy of use，many users will undoubtedly have suggestions for improvement.We welcome your suggestions.The contact email address is  sli@whu.edu.cn.



Editor in Chief

2019.9