Preface
CHAPTER 1 SYSTEMS OF LINEAR EQUATIONS
1.1 Solving Systems of Linear Equations
Linear Equations
Systems of Linear Equations
General Systems of Linear Equations
Gaussian Elimination
Elementary Replacement and Scale Operations
Row-Equivalent Pairs of Matrices
Elementary Row Operations
Reduced Row Echelon Form
Row Echelon Form
Intuitive Interpretation
Application: Feeding Bacteria
Mathematical Software
Algorithm for the Reduced Row Echelon Form
Summary 1.1
Key Concepts 1.1
General Exercises 1.1
Computer Exercises 1.1
1.2 Vectors and Matrices
Vectors
Linear Combinations of Vectors
Matrix–Vector Products
The Span of a Set of Vectors
Interpreting Linear Systems
Row-Equivalent Systems
Consistent and Inconsistent Systems
Caution
Application: Linear Ordinary Differential Equations
Application: Bending of a Beam
Summary 1.2
Key Concepts 1.2
General Exercises 1.2
Computer Exercises 1.2
1.3 Kernels, Rank, Homogeneous Equations
Kernel or Null Space of a Matrix
Homogeneous Equations
Uniqueness of the Reduced Row Echelon Form
Rank of a Matrix
General Solution of a System
Matrix–Matrix Product
Indexed Sets of Vectors: Linear Dependence and Independence
Using the Row-Reduction Process
Determining Linear Dependence or Independence
Application: Chemistry
Summary 1.3
Key Concepts 1.3
General Exercises 1.3
Computer Exercises 1.3
CHAPTER 2 VECTOR SPACES
2.1 Euclidean Vector Spaces
n-Tuples and Vectors
Vector Addition and Multiplication by Scalars
Properties of as a Vector Space
Linear Combinations
Span of a Set of Vectors
Geometric Interpretation of Vectors
Application: Elementary Mechanics
Application: Network Problems, Traffic Flow
Application: Electrical Circuits
Summary 2.1
Key Concepts 2.1
General Exercises 2.1
Computer Exercises 2.1
2.2 Lines, Planes, and Hyperplanes
Line Passing Through Origin
Lines in
Planes in
Lines and Planes in
General Solution of a System of Equations
Application: The Predator–Prey Simulation
Application: Partial-Fraction Decomposition
Application: Method of Least Squares
Summary 2.2
Key Concepts 2.2
General Exercises 2.2
Computer Exercises 2.2
2.3 Linear Transformations
Functions, Mappings, and Transformations
Domain, Co-domain, and Range
Various Examples
Injective and Surjective Mappings
Linear Transformations
Using Matrices to Define Linear Maps
Injective and Surjective Linear Transformations
Effects of Linear Transformations
Effects of Transformations on Geometrical Figures
Composition of Two Linear Mappings
Application: Data Smoothing
Summary 2.3
Key Concepts 2.3
General Exercises 2.3
Computer Exercises 2.3
2.4 General Vector Spaces
Vector Spaces
Theorems on Vector Spaces
Linearly Dependent Sets
Linear Mapping
Application: Models in Economic Theory
Summary 2.4
Key Concepts 2.4
General Exercises 2.4
Computer Exercises 2.4
CHAPTER 3 MATRIX OPERATIONS
3.1 Matrices
Matrix Addition and Scalar Multiplication
Matrix–Matrix Multiplication
Pre-multiplication and Post-multiplication
Dot Product
Special Matrices
Matrix Transpose
Symmetric Matrices
Skew–Symmetric Matrices
Non-commutativity of Matrix Multiplication
Associativity Law for Matrix Multiplication
Elementary Matrices
More on the Matrix–Matrix Product
Vector–Matrix Product
Application: Diet Problems
Dangerous Pitfalls
Summary 3.1
Key Concepts 3.1
General Exercises 3.1
Computer Exercises 3.1
3.2 Matrix Inverses
Solving Systems with a Left Inverse
Solving Systems with a Right Inverse
Analysis
Square Matrices
Invertible Matrices
Elementary Matrices and LU Factorization
Computing an Inverse
More on Left and Right Inverses of Non-square Matrices
Invertible Matrix Theorem
Application: Interpolation
Summary 3.2
Key Concepts 3.2
General Exercises 3.2
Computer Exercises 3.2
CHAPTER 4 DETERMINANTS
4.1 Determinants: Introduction
Properties of Determinants
An Algorithm for Computing Determinants
Algorithm without Scaling
Zero Determinant
Calculating Areas and Volumes
Summary 4.1
Key Concepts 4.1
General Exercises 4.1
Computer Exercises 4.1
4.2 Determinants: Properties
Minors and Cofactors
Work Estimate
Direct Methods for Computing Determinants
Cramer’s Rule
Computing Inverses Using Determinants
Vandermonde Matrix
Application: Coded Messages
Review of Determinant Notation and Properties
Summary 4.2
Key Concepts 4.2
General Exercises 4.2
Computer Exercises 4.2
CHAPTER 5 VECTOR SUBSPACES
5.1 Column, Row, and Null Spaces
Introduction
Revisiting Kernels and Null Spaces
The Row Space and Column Space of a Matrix
Summary 5.1
Key Concepts 5.1
General Exercises 5.1
Computer Exercises 5.1
5.2 Bases and Dimension
Basis for a Vector Space
Coordinate Vector
Isomorphism and Equivalence Relations
Finite-Dimensional and Infinite-Dimensional Vector Spaces
Linear Transformation of a Set
Dimensions of Various Subspaces
Summary 5.2
Key Concepts 5.2
General Exercises 5.2
Computer Exercises 5.2
5.3 Coordinate Systems
Coordinate Vectors
Changing Coordinates
Mapping a Vector Space into Itself
Similar Matrices
More on Equivalence Relations
Further Examples
Summary 5.3
Key Concepts 5.3
General Exercises 5.3
Computer Exercises 5.3
CHAPTER 6 EIGENSYSTEMS
6.1 Eigenvalues and Eigenvectors
Eigenvectors and Eigenvalues
Using Determinants in Finding Eigenvalues
Distinct Eigenvalues
Bases of Eigenvectors
Application: Powers of a Matrix
Characteristic Equation and Characteristic Polynomial
Diagonalization Involving Complex Numbers
Application: Dynamical Systems
Further Dynamical Systems in
Analysis of a Dynamical System
Application: Economic Models
Application: Systems of Linear Differential Equations
Epilogue: Eigensystems without Determinants
Summary 6.1
Key Concepts 6.1
General Exercises 6.1
Computer Exercises 6.1
CHAPTER 7 INNER-PRODUCT VECTOR SPACES
7.1 Inner-Product Spaces
Inner-Product Spaces and Their Properties
The Norm in an Inner-Product Space
Distance Function
Mutually Orthogonal Vectors
Orthogonal Projection
Angle between Vectors
Orthogonal Complements
Orthonormal Bases
Subspaces in Inner-Product Spaces
Application: Work and Forces
Application: Collision
Summary 7.1
Key Concepts 7.1
General Exercises 7.1
Computer Exercises 7.1
7.2 Orthogonality
The Gram–Schmidt Process
Unnormalized Gram–Schmidt Algorithm
Modified Gram–Schmidt Process
Linear Least-Squares Solution
Gram Matrix
Distance from a Point to a Hyperplane
Summary 7.2
Key Concepts 7.2
General Exercises 7.2
Computer Exercises 7.2
CHAPTER 8 ADDITIONAL TOPICS
8.1 Hermitian Matrices and the Spectral Theorem
Hermitian Matrices and Self-Adjoint Mappings
Self-Adjoint Mapping
The Spectral Theorem
Unitary and Orthogonal Matrices
The Cayley–Hamilton Theorem
Quadratic Forms
Application: World Wide Web Searching
Summary 8.1
Key Concepts 8.1
General Exercises 8.1
Computer Exercises 8.1
8.2 Matrix Factorizations and Block Matrices
Permutation Matrix
LU-Factorization
LLT-Factorization: Cholesky Factorization
LDLT-Factorization
QR-Factorization
Singular-Value Decomposition (SVD)
Schur Decomposition
Partitioned Matrices
Solving a System Having a 2 × 2 Block Matrix
Inverting a 2 × 2 Block Matrix
Application: Linear Least-Squares Problem
Summary 8.2
Key Concepts 8.2
General Exercises 8.2
Computer Exercises 8.2
8.3 Iterative Methods for Linear Equations
Richardson Iterative Method
Jacobi Iterative Method
Gauss–Seidel Method
Successive Overrelaxation (SOR) Method
Conjugate Gradient Method
Diagonally Dominant Matrices
Gerschgorin’s Theorem
Infinity Norm
Convergence Properties
Power Method for Computing Eigenvalues
Application: Demographic Problems, Population Migration
Application: Leontief Open Model
Summary 8.3
Key Concepts 8.3
General Exercises 8.3
Computer Exercises 8.3
APPENDIX A DEDUCTIVE REASONING AND PROOFS
A.1 Introduction
A.2 Deductive Reasoning and Direct Verification
A.3 Implications
A.4 Method of Contradiction
A.5 Mathematical Induction
A.6 Truth Tables
A.7 Subsets and de Morgan Laws
A.8 Quantifiers
A.9 Denial of a Quantified Assertion
A.10 Some More Questionable ‘‘Proofs’’
Summary Appendix A
Key Concepts Appendix A
General Exercises Appendix A
APPENDIX B COMPLEX ARITHMETIC
B.1 Complex Numbers and Arithmetic
B.2 Fundamental Theorem of Algebra
B.3 Abel–Ruffini Theorem
ANSWERS/HINTS FOR GENERAL EXERCISES
REFERENCES
INDEX
as a Vector Space 
