Linear Algebra Demystified

CONTENTS
Preface ix
CHAPTER 1          Systems of Linear Equations 1
Consistent and Inconsistent Systems 3
Matrix Representation of a System of
Equations 3
Solving a System Using Elementary
Operations 6
Triangular Matrices 7
Elementary Matrices 18
Implementing Row Operations with
Elementary Matrices 22
Homogeneous Systems 26
Gauss-Jordan Elimination 27
Quiz 31
CHAPTER 2          Matrix Algebra 34
Matrix Addition 34
Scalar Multiplication 35
Matrix Multiplication 36
Square Matrices 40
The Identity Matrix 43
The Transpose Operation 45
The Hermitian Conjugate 49
Trace 50
The Inverse Matrix 52
Quiz 56
CHAPTER 3          Determinants 59
The Determinant of a Third-Order Matrix        61
Theorems about Determinants 62
Cramer’s Rule 63
Properties of Determinants 67
Finding the Inverse of a Matrix 70
Quiz 74
CHAPTER 4          Vectors 76
Vectors in R n 79
Vector Addition 79
Scalar Multiplication 81
The Zero Vector 83
The Transpose of a Vector 84
The Dot or Inner Product 86
The Norm of a Vector 88
Unit Vectors 89
The Angle between Two Vectors 90
Two Theorems Involving Vectors 90
Distance between Two Vectors 91
Quiz 91
CHAPTER 5          Vector Spaces 94
Basis Vectors 100
Linear Independence 103
Basis Vectors 106
Completeness 106
Subspaces 108
Row Space of a Matrix 109
Null Space of a Matrix 115
Quiz 117
CHAPTER 6          Inner Product Spaces 120
The Vector Space R n 122
Inner Products on Function Spaces 123
Properties of the Norm 127
An Inner Product for Matrix Spaces 128
The Gram-Schmidt Procedure 129
Quiz 132
CHAPTER 7          Linear Transformations 135
Matrix Representations 137
Linear Transformations in the Same
Vector Space 143
More Properties of Linear Transformations    149
Quiz 151
CHAPTER 8          The Eigenvalue Problem 154
The Characteristic Polynomial 154
The Cayley-Hamilton Theorem 155
Finding Eigenvectors 159
Normalization 162
The Eigenspace of an Operator A 167
Similar Matrices 170
Diagonal Representations of an Operator      171
The Trace and Determinant and Eigenvalues  177
Quiz 178
CHAPTER 9          Special Matrices 180
Symmetric and Skew-Symmetric Matrices      180
Hermitian Matrices 185
Orthogonal Matrices 189
Unitary Matrices 194
Quiz 197
CHAPTER 10        Matrix Decomposition 199
LU Decomposition 199
Solving a Linear System with an
LU Factorization 204
SVD Decomposition 208
QR Decomposition 212
Quiz 214
Final Exam 217
Hints and Solutions 230
References 248
Index 249

Other Core of CS Books
Instructor's Manual for Elementary Linear Algebra - Grossman, Stanley I
Linear Algebra Schaum Jim Hefferon (All Answer)
Linear algebra - Wikipedia, the free encyclopedia
A First Course in Linear Algebra (A Free Textbook)
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