Fundamentals of Linear Algebra
Marcel B. Finan
Arkansas Tech University
c All Rights Reserved
,2
PREFACE
Linear algebra has evolved as a branch of mathematics with wide range of
applications to the natural sciences, to engineering, to computer sciences, to
management and social sciences, and more.
This book is addressed primarely to second and third your college students
who have already had a course in calculus and analytic geometry. It is the
result of lecture notes given by the author at The University of North Texas
and the University of Texas at Austin. It has been designed for use either as a
supplement of standard textbooks or as a textbook for a formal course in linear
algebra.
This book is not a ”traditional” book in the sense that it does not include
any applications to the material discussed. Its aim is solely to learn the basic
theory of linear algebra within a semester period. Instructors may wish to in-
corporate material from various fields of applications into a course.
I have included as many problems as possible of varying degrees of difficulty.
Most of the exercises are computational, others are routine and seek to fix
some ideas in the reader’s mind; yet others are of theoretical nature and have
the intention to enhance the reader’s mathematical reasoning. After all doing
mathematics is the way to learn mathematics.
A solution manual to the text is available by request from the author. Email:
Marcecl B. Finan
Austin, Texas
March, 2001.
,Contents
1 Linear Systems 5
1.1 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . 5
1.2 Geometric Meaning of Linear Systems . . . . . . . . . . . . . . . 8
1.3 Matrix Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Elementary Row Operations . . . . . . . . . . . . . . . . . . . . 13
1.5 Solving Linear Systems Using Augmented Matrices . . . . . . . . 17
1.6 Echelon Form and Reduced Echelon Form . . . . . . . . . . . . . 21
1.7 Echelon Forms and Solutions to Linear Systems . . . . . . . . . . 28
1.8 Homogeneous Systems of Linear Equations . . . . . . . . . . . . 32
1.9 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2 Matrices 43
2.1 Matrices and Matrix Operations . . . . . . . . . . . . . . . . . . 43
2.2 Properties of Matrix Multiplication . . . . . . . . . . . . . . . . . 50
2.3 The Inverse of a Square Matrix . . . . . . . . . . . . . . . . . . . 55
2.4 Elementary Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.5 An Algorithm for Finding A−1 . . . . . . . . . . . . . . . . . . . 63
2.6 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3 Determinants 75
3.1 Definition of the Determinant . . . . . . . . . . . . . . . . . . . . 75
3.2 Evaluating Determinants by Row Reduction . . . . . . . . . . . . 79
3.3 Properties of the Determinant . . . . . . . . . . . . . . . . . . . . 83
3.4 Finding A−1 Using Cofactor Expansions . . . . . . . . . . . . . . 86
3.5 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.6 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4 The Theory of Vector Spaces 101
4.1 Vectors in Two and Three Dimensional Spaces . . . . . . . . . . 101
4.2 Vector Spaces, Subspaces, and Inner Product Spaces . . . . . . . 108
4.3 Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.4 Basis and Dimension . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.5 Transition Matrices and Change of Basis . . . . . . . . . . . . . . 128
4.6 The Rank of a matrix . . . . . . . . . . . . . . . . . . . . . . . . 133
3
, 4 CONTENTS
4.7 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5 Eigenvalues and Diagonalization 151
5.1 Eigenvalues and Eigenvectors of a Matrix . . . . . . . . . . . . . 151
5.2 Diagonalization of a Matrix . . . . . . . . . . . . . . . . . . . . . 164
5.3 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6 Linear Transformations 175
6.1 Definition and Elementary Properties . . . . . . . . . . . . . . . 175
6.2 Kernel and Range of a Linear Transformation . . . . . . . . . . . 180
6.3 The Matrix Representation of a Linear Transformation . . . . . . 189
6.4 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
Marcel B. Finan
Arkansas Tech University
c All Rights Reserved
,2
PREFACE
Linear algebra has evolved as a branch of mathematics with wide range of
applications to the natural sciences, to engineering, to computer sciences, to
management and social sciences, and more.
This book is addressed primarely to second and third your college students
who have already had a course in calculus and analytic geometry. It is the
result of lecture notes given by the author at The University of North Texas
and the University of Texas at Austin. It has been designed for use either as a
supplement of standard textbooks or as a textbook for a formal course in linear
algebra.
This book is not a ”traditional” book in the sense that it does not include
any applications to the material discussed. Its aim is solely to learn the basic
theory of linear algebra within a semester period. Instructors may wish to in-
corporate material from various fields of applications into a course.
I have included as many problems as possible of varying degrees of difficulty.
Most of the exercises are computational, others are routine and seek to fix
some ideas in the reader’s mind; yet others are of theoretical nature and have
the intention to enhance the reader’s mathematical reasoning. After all doing
mathematics is the way to learn mathematics.
A solution manual to the text is available by request from the author. Email:
Marcecl B. Finan
Austin, Texas
March, 2001.
,Contents
1 Linear Systems 5
1.1 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . 5
1.2 Geometric Meaning of Linear Systems . . . . . . . . . . . . . . . 8
1.3 Matrix Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Elementary Row Operations . . . . . . . . . . . . . . . . . . . . 13
1.5 Solving Linear Systems Using Augmented Matrices . . . . . . . . 17
1.6 Echelon Form and Reduced Echelon Form . . . . . . . . . . . . . 21
1.7 Echelon Forms and Solutions to Linear Systems . . . . . . . . . . 28
1.8 Homogeneous Systems of Linear Equations . . . . . . . . . . . . 32
1.9 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2 Matrices 43
2.1 Matrices and Matrix Operations . . . . . . . . . . . . . . . . . . 43
2.2 Properties of Matrix Multiplication . . . . . . . . . . . . . . . . . 50
2.3 The Inverse of a Square Matrix . . . . . . . . . . . . . . . . . . . 55
2.4 Elementary Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.5 An Algorithm for Finding A−1 . . . . . . . . . . . . . . . . . . . 63
2.6 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3 Determinants 75
3.1 Definition of the Determinant . . . . . . . . . . . . . . . . . . . . 75
3.2 Evaluating Determinants by Row Reduction . . . . . . . . . . . . 79
3.3 Properties of the Determinant . . . . . . . . . . . . . . . . . . . . 83
3.4 Finding A−1 Using Cofactor Expansions . . . . . . . . . . . . . . 86
3.5 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.6 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4 The Theory of Vector Spaces 101
4.1 Vectors in Two and Three Dimensional Spaces . . . . . . . . . . 101
4.2 Vector Spaces, Subspaces, and Inner Product Spaces . . . . . . . 108
4.3 Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.4 Basis and Dimension . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.5 Transition Matrices and Change of Basis . . . . . . . . . . . . . . 128
4.6 The Rank of a matrix . . . . . . . . . . . . . . . . . . . . . . . . 133
3
, 4 CONTENTS
4.7 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5 Eigenvalues and Diagonalization 151
5.1 Eigenvalues and Eigenvectors of a Matrix . . . . . . . . . . . . . 151
5.2 Diagonalization of a Matrix . . . . . . . . . . . . . . . . . . . . . 164
5.3 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6 Linear Transformations 175
6.1 Definition and Elementary Properties . . . . . . . . . . . . . . . 175
6.2 Kernel and Range of a Linear Transformation . . . . . . . . . . . 180
6.3 The Matrix Representation of a Linear Transformation . . . . . . 189
6.4 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
