A First Course In Linear Algebra

by Shou-te Chang, Minking Eie

Overview

A First Course in Linear Algebra is written by two experts from algebra who have more than 20 years of experience in algebra, linear algebra and number theory. It prepares students with no background in Linear Algebra. Students, after mastering the materials in this textbook, can already understand any Linear Algebra used in more advanced books and research papers in Mathematics or in other scientific disciplines.This book provides a solid foundation for the theory dealing with finite dimensional vector spaces. It explains in details the relation between linear transformations and matrices. One may thus use different viewpoints to manipulate a matrix instead of a one-sided approach. Although most of the examples are for real and complex matrices, a vector space over a general field is briefly discussed. Several optional sections are devoted to applications to demonstrate the power of Linear Algebra.

Product Details

ISBN-13:	9789813143104
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	09/06/2016
Pages:	388
Product dimensions:	6.10(w) x 9.10(h) x 1.00(d)

Table of Contents

Preface v

1 Vector Spaces 1

1.1 A few words on sets and logics 2

1.2 What constitutes a vector space? 8

1.3 Subspaces 18

1.4 How to solve a system of linear equations 24

2 Bases and Dimension 37

2.1 Observations in R² and in R³ 38

2.2 Linear- combinations 42

2.3 Linear dependence and independence 51

2.4 Bases and dimension 61

2.5 Applications 73

2.6 Further examples 83

3 Linear Transformations and Matrices 93

3.1 Linear transformations 94

3.2 Isomorphisms 101

3.3 Range and null space 105

3.4 Matrices and linear transformations 115

3.5 Composites and inverses 126

4 Elementary Matrix Operations 135

4.1 Elementary matrix operations 136

4.2 The rank and the inverse of a matrix 142

4.3 A description of determinants 157

4.4 Applications in linear programming 168

5 Diagonalization 181

5.1 Base change 182

5.2 Eigenvalues and eigenvectors 191

5.3 Diagonalizability 200

5.4 Powers of a square matrix 212

5.5 Differential equations 231

6 Canonical Forms 247

6.1 Cayley-Hamilton theorem 248

6.2 Jordan canonical forms 267

6.3 How to find Jordan canonical forms 279

7 Inner Product Spaces 303

7.1 Inner product spaces 304

7.2 Norm and angle 315

7.3 The spectral theorem 329

7.4 Applications of the spectral theorem 345

7.5 The method of least squares 357

7.6 The L²-approximation of a continuous function 366

Index 373