Linear algebra is one of the important subjects which are treated in many areas such as the natural sciences, computer sciences, engineering, economy, etc. Up to date linear algebra is also one of the useful theories that supply the fundamental and systematic methods to erect those academic areas developed diversly and deeply in scholarly pursuits. According to the various different types of requirements in those areas, the contents of linear algebra have been greatly influenced. It is really difficult to work that one can obtain a book containing the useful contents with these all requirements for undergraduate students. As a foundation of such a book, this book is edited toward the basic theories required in each academic area, and this book is of the form like a lecture note consisting of mainly theoretical aspects and was mainly written for one semester course in linear algebra at the junior undergraduate level. The one of the most important aims of the book is to induce themselves to find the methods analyzing concretely vector structures of a finite dimensional vector spaces over a given ground field, and was centered on making them understood important properties and concepts appearing in vector spaces having more complicated structures through the use of the methods. For the purpose the book provide sufficient examples to explain the meanings inside given  efinitions, lemmas, propositions, and theorems and help out to solve the exercises given in each section of the book. One may omit the sections having advanced  oncepts in chapters 6, 7, and 8, whenever one teaches junior undergraduate students without obstructing the flaw of the aim of the book. The contents of the book consist of eight parts. First, the book contains basic concepts with respect to the ground fields of vector spaces. In fact many  other linear algebra books avoid the details of the fields, not even its definition and the related basic facts with appropriate examples. However it follows from the definite meaning of the field that one can see more easily the structures of the vector spaces, and it is very important matter how the vector space is defined on the ground fields which consists of so called scalars. For this reason, the book introduces the basic concepts of fields in chapter 1 and matrices defined from the given ground fields together with the related problems. In chapter 2, we investigate the basic concepts and structures of a general vector space over a field and try to expect the explicit geometric structures of vector spaces over the field through the three dimensional real vector space over the real number system. In chapter 3, the methods to find the solution set of system of linear equations are explained more precisely together with many additional examples. From the examples, the reader can easily understand the procedure to characterize the general solution set. Next, in chapter 4, we study methods to analyze vector structures indirectly by using the linear transformations with the corresponding matrices over the fields, which preserve the given operations on the vector spaces. And, in section 4.4, we study a way changing matrices of linear transformations through the use of transitive matrices. In chapter 5, the definition of determinant of a square matrix is defined by using the cofactor expansions of row vectors or column vectors in the matrix instead of using the sign function of permutations. The reason is also precisely explained in section 5.1. Here, we study many basic properties of the deteminant. In chapter 6, to see more explicit properties for the vector structures of finite dimensional vector spaces, the inner product spaces are introduced together with the related properties. From the facts it is shown that every n-dimensional vector space over a field has the same vector structures as the n-dimension real vector space over the real number system. In chapter 7, we investigate certain vector structures of a finite dimensional vector space over a field through the eigenvalues and the eigenvectors of a square matrix, which are produced by an endomorphism on the vector space. As their applications, many important topics like  agonalizing a square matrix by a certain type of invertible matrix, particularly a symmetric matrix, are introduced in detail with many examples. The entire chapter 8 has been written newly into two sections 8.1 and 8.2. The main topic of chapter 8 is to verify if every square matrix with complex entries can be represented as a block diagonal matrix, so called a matrix in Jordan canonical form (or Jordan normal form), formed of Jordan blocks. In the section 8.1 as preliminaries, ascending (Jordan) chains and descending chains produced from eigenspaces of a given square matrix are introduced precisely. They are important to characterize the Jordan canonical form of a square matrix. Through the section 8.2, the whole procedure to get the form is described step-by-step at great length with many interesting examples and plentiful figures. Finally we would like to thank the Kyungmoon Publishers for editing and publishing this book all the while. As the first edition, we think that many errors could be found in various ways. We hope that readers will inform us about the errors if any. We promise to be contented to correct them at the next edition. We also hope that this book will be used usefully to understand the fundamental concepts of linear algebra

Chapter 1 Preliminaries

1.1 Fields _2

1.2 Matrices and Matrix Operations _8

Chapter 2 Vector Spaces

2.1 Vector Spaces _24

2.2 Vectors in Euclidean Spaces _33

2.3 Subspaces _45

2.4 Bases of Vector Spaces _54

Chapter 3 System of Linear Equations

3.1 Gauss-Jordan Elimination Method _78

3.2 Inverse Matrix _110

3.3 Elementary Matrix Multiplications _123

3.4 Row and Column Spaces _134

Chapter 4 Linear Transformations and Matrices

4.1 Linear Transformations _152

4.2 Matrix Representations of Linear Transformations _171

4.3 Compositions of Linear Transformations and Their Matrices _184

4.4 Change of Basis _191

Chapter 5 Determinants

5.1 Definition of Determinant _204

5.2 Properties of Determinant _215

5.3 Cramer’s Rule _227

Chapter 6 Inner Products

6.1 Inner Products _236

6.2 Gram-Schmidt Theorem _249

6.3 QR-Decomposition _271

Chapter 7 Eigenvalues and Their Applications

7.1 Cayley-Hamilton Theorem _278

7.2 Eigenvalues and Their Applications _290

7.3 Diagonalization of Square matrices _305

7.4 Diagonalization of Symmetric Matrices _321

7.5 Quadratic Forms _332

Chapter 8 Jordan Canonical Forms

8.1 Jordan Chains _352

8.2 Jordan Canonical Forms _369

Answers to Exercises _419

References _559

Index _561

Hong Goo Park

Department of Mathematics

Hanyang University

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