First Course in Linear Algebra eBook (3e) : 9781486022090

First Course in Linear Algebra eBook (3e)

Easdown, David
 
Edition
 
3
ISBN
 
9781486022090
ISBN 10
 
148602209X
Published
 
17/02/2014
Published by
 
Pearson Custom Books
Pages
 
Format
 
 
Title type
eBook
$60.00
 
 
 
Description

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About the book

A First Course In Linear Algebra, 3e

An engaging introductory text to linear algebra for new students entering university and returning mature-age students. It aims to make critical algebraic concepts easy to understand. Find out more

Table of contents

Contents

Preface to the Third Edition ix
Preface to the Second Edition xi
Preface to the First Edition xiii

0 Introduction 1
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1 Geometric Vectors 13
1.1 Addition of geometric vectors . . . . . . . . . . . . . . . . . . 18
1.2 Multiplication by a scalar . . . . . . . . . . . . . . . . . . . . 20
1.3 Subtraction of vectors . . . . . . . . . . . . . . . . . . . . . . 24
1.4 List of useful properties . . . . . . . . . . . . . . . . . . . . . 25
1.5 The geometry of parallelograms . . . . . . . . . . . . . . . . . 26
Important Ideas and Useful Facts . . . . . . . . . . . . . . . . . . . . 30
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2 Position Vectors and Components 33
2.1 Magnitude, unit vectors and hat notation . . . . . . . . . . . . 34
2.2 Parallel vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3 Position vectors and components . . . . . . . . . . . . . . . . 37
2.4 Length of a vector . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5 Linear independence for two vectors . . . . . . . . . . . . . . . 44
Important Ideas and Useful Facts . . . . . . . . . . . . . . . . . . . . 49
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Dot Products and Projections 53
3.1 Geometric definition of dot product . . . . . . . . . . . . . . . 55
3.2 Algebraic definition of dot product . . . . . . . . . . . . . . . 57
3.3 Angle between two vectors . . . . . . . . . . . . . . . . . . . . 61
3.4 Projections and orthogonal components . . . . . . . . . . . . . 62
3.5 Another application to geometry in the plane . . . . . . . . . 67
Important Ideas and Useful Facts . . . . . . . . . . . . . . . . . . . . 69
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4 Cross Products 73
4.1 Definition of cross product . . . . . . . . . . . . . . . . . . . . 74
4.2 List of useful properties . . . . . . . . . . . . . . . . . . . . . 76
4.3 Method of expanding brackets . . . . . . . . . . . . . . . . . . 78
4.4 Geometric interpretation . . . . . . . . . . . . . . . . . . . . . 78
4.5 Continuity and the right-hand orientation . . . . . . . . . . . 83
Important Ideas and Useful Facts . . . . . . . . . . . . . . . . . . . . 85
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5 Lines in Space 91
5.1 Parametric vector and scalar equations of a line . . . . . . . . 93
5.2 Cartesian equations of a line . . . . . . . . . . . . . . . . . . . 95
5.3 Finding a line using two points . . . . . . . . . . . . . . . . . 97
5.4 Distance from a point to a line . . . . . . . . . . . . . . . . . . 99
Important Ideas and Useful Facts . . . . . . . . . . . . . . . . . . . . 103
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6 Planes in Space 107
6.1 Vector equation of a plane . . . . . . . . . . . . . . . . . . . . 109
6.2 Cartesian equation of a plane . . . . . . . . . . . . . . . . . . 111
6.3 Finding a plane using three points . . . . . . . . . . . . . . . . 113
6.4 Distance from a point to a plane . . . . . . . . . . . . . . . . . 114
Important Ideas and Useful Facts . . . . . . . . . . . . . . . . . . . . 117
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7 Systems of Linear Equations 121
7.1 Consistent and inconsistent systems . . . . . . . . . . . . . . . 124
7.2 Parametric solutions . . . . . . . . . . . . . . . . . . . . . . . 128
7.3 Augmented matrix of a system . . . . . . . . . . . . . . . . . . 130
7.4 Gaussian elimination . . . . . . . . . . . . . . . . . . . . . . . 132
7.5 Reduced row echelon form . . . . . . . . . . . . . . . . . . . . 140
Important Ideas and Useful Facts . . . . . . . . . . . . . . . . . . . . 143
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

8 Matrix Operations 149
8.1 Addition, subtraction and scalar multiplication . . . . . . . . . 152
8.2 Matrix multiplication . . . . . . . . . . . . . . . . . . . . . . . 153
8.3 Connections with systems of equations . . . . . . . . . . . . . 157
Important Ideas and Useful Facts . . . . . . . . . . . . . . . . . . . . 160
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

9 Matrix Inverses 165
9.1 Identity matrices and inverses . . . . . . . . . . . . . . . . . . 168
9.2 Inverses of two-by-two matrices . . . . . . . . . . . . . . . . . 170
9.3 Powers of a matrix . . . . . . . . . . . . . . . . . . . . . . . . 172
9.4 Using row reduction to find the inverse . . . . . . . . . . . . . 173
9.5 Using inverses to solve systems of equations . . . . . . . . . . 175
9.6 Elementary matrices . . . . . . . . . . . . . . . . . . . . . . . 176
Important Ideas and Useful Facts . . . . . . . . . . . . . . . . . . . . 181
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

10 Determinants 185
10.1 Determinant of a 3 × 3 matrix . . . . . . . . . . . . . . . . . 187
10.2 Cross products revisited . . . . . . . . . . . . . . . . . . . . . 191
10.3 Properties of determinants . . . . . . . . . . . . . . . . . . . . 192
10.4 Orientation of a triangle . . . . . . . . . . . . . . . . . . . . . 195
Important Ideas and Useful Facts . . . . . . . . . . . . . . . . . . . . 200
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

11 Eigenvalues and Eigenvectors 207
11.1 Existence of eigenvalues . . . . . . . . . . . . . . . . . . . . . 214
11.2 Finding eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . 221
11.3 Reflections and rotations in the plane . . . . . . . . . . . . . . 226
Important Ideas and Useful Facts . . . . . . . . . . . . . . . . . . . . 234
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

12 Diagonalising a Matrix 241
12.1 An example which cannot be diagonalised . . . . . . . . . . . 247
12.2 An example of a Markov process . . . . . . . . . . . . . . . . . 250
12.3 The Jordan form of a matrix . . . . . . . . . . . . . . . . . . . 257
Important Ideas and Useful Facts . . . . . . . . . . . . . . . . . . . . 262
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

Hints and Solutions 267
Appendix 1 The Theorem of Pythagoras 295
Appendix 2 Mathematical Implication 301
Appendix 3 Complex Numbers 317
Index 346

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