Table of Contents

Chapter 1 Vectors in 𝐑^𝑛 and 𝐂^𝑛, Spatial Vectors
 1.1 Introduction
 1.2 Vectors in 𝐑^𝑛
 1.3 Vector Addition and Scalar Multiplication
 1.4 Dot (Inner) Product
 1.5 Located Vectors, Hyperplanes, Lines, Curves in 𝐑^𝑛
 1.6 Vectors in 𝐑³ (Spatial Vectors), 𝐢𝐣𝐤 Notation
 1.7 Complex Numbers
 1.8 Vectors in 𝐂^𝑛

Chapter 2 Algebra of Matrices
 2.1 Introduction
 2.2 Matrices
 2.3 Matrix Addition and Scalar Multiplication
 2.4 Summation Symbol
 2.5 Matrix Multiplication
 2.6 Transpose of a Matrix
 2.7 Square Matrices
 2.8 Powers of Matrices, Polynomials in Matrices
 2.9 Invertible (Nonsingular) Matrices
 2.10 Special Types of Square Matrices
 2.11 Complex Matrices
 2.12 Block Matrices

Chapter 3 Systems of Linear Equations
 3.1 Introduction
 3.2 Basic Definitions, Solutions
 3.3 Equivalent Systems, Elementary Operations
 3.4 Small Square Systems of Linear Equations
 3.5 Systems in Triangular and Echelon Forms
 3.6 Gaussian Elimination
 3.7 Echelon Matrices, Row Canonical Form, Row Equivalence
 3.8 Gaussian Elimination, Matrix Formulation
 3.9 Matrix Equation of a System of Linear Equations
 3.10 Systems of Linear Equations and Linear Combinations of Vectors
 3.11 Homogeneous Systems of Linear Equations
 3.12 Elementary Matrices
 3.13 𝐿𝑈 Decomposition

Chapter 4 Vector Spaces
 4.1 Introduction
 4.2 Vector Spaces
 4.3 Examples of Vector Spaces
 4.4 Linear Combinations, Spanning Sets
 4.5 Subspaces
 4.6 Linear Spans, Row Space of a Matrix
 4.7 Linear Dependence and Independence
 4.8 Basis and Dimension
 4.9 Application to Matrices, Rank of a Matrix
 4.10 Sums and Direct Sums
 4.11 Coordinates
 4.12 Isomorphism of 𝑉 and 𝐾^𝑛
 4.13 Full Rank Factorization
 4.14 Generalized (Moore–Penrose) Inverse
 4.15 LeastSquare Solution

Chapter 5 Linear Mappings
 5.1 Introduction
 5.2 Mappings, Functions
 5.3 Linear Mappings (Linear Transformations)
 5.4 Kernel and Image of a Linear Mapping
 5.5 Singular and Nonsingular Linear Mappings, Isomorphisms
 5.6 Operations with Linear Mappings
 5.7 Algebra 𝐴(𝑉) of Linear Operators

Chapter 6 Linear Mappings and Matrices
 6.1 Introduction
 6.2 Matrix Representation of a Linear Operator
 6.3 Change of Basis
 6.4 Similarity
 6.5 Matrices and General Linear Mappings

Chapter 7 Inner Product Spaces, Orthogonality
 7.1 Introduction
 7.2 Inner Product Spaces
 7.3 Examples of Inner Product Spaces
 7.4 Cauchy–Schwarz Inequality, Applications
 7.5 Orthogonality
 7.6 Orthogonal Sets and Bases
 7.7 Gram–Schmidt Orthogonalization Process
 7.8 Orthogonal and Positive Definite Matrices
 7.9 Complex Inner Product Spaces
 7.10 Normed Vector Spaces (Optional)

Chapter 8 Determinants
 8.1 Introduction
 8.2 Determinants of Orders 1 and 2
 8.3 Determinants of Order 3
 8.4 Permutations
 8.5 Determinants of Arbitrary Order
 8.6 Properties of Determinants
 8.7 Minors and Cofactors
 8.8 Evaluation of Determinants
 8.9 Classical Adjoint
 8.10 Applications to Linear Equations, Cramer’s Rule
 8.11 Submatrices, Minors, Principal Minors
 8.12 Block Matrices and Determinants
 8.13 Determinants and Volume
 8.14 Determinant of a Linear Operator
 8.15 Multilinearity and Determinants

Chapter 9 Diagonalization: Eigenvalues and Eigenvectors
 9.1 Introduction
 9.2 Polynomials of Matrices
 9.3 Characteristic Polynomial, Cayley–Hamilton Theorem
 9.4 Diagonalization, Eigenvalues and Eigenvectors
 9.5 Computing Eigenvalues and Eigenvectors, Diagonalizing Matrices
 9.6 Diagonalizing Real Symmetric Matrices and Quadratic Forms
 9.7 Minimal Polynomial
 9.8 Characteristic and Minimal Polynomials of Block Matrices

Chapter 10 Canonical Forms
 10.1 Introduction
 10.2 Triangular Form
 10.3 Invariance
 10.4 Invariant DirectSum Decompositions
 10.5 Primary Decomposition
 10.6 Nilpotent Operators
 10.7 Jordan Canonical Form
 10.8 Cyclic Subspaces
 10.9 Rational Canonical Form
 10.10 Quotient Spaces

Chapter 11 Linear Functionals and the Dual Space
 11.1 Introduction
 11.2 Linear Functionals and the Dual Space
 11.3 Dual Basis
 11.4 Second Dual Space
 11.5 Annihilators
 11.6 Transpose of a Linear Mapping

Chapter 12 Bilinear, Quadratic, and Hermitian Forms
 12.1 Introduction
 12.2 Bilinear Forms
 12.3 Bilinear Forms and Matrices
 12.4 Alternating Bilinear Forms
 12.5 Symmetric Bilinear Forms, Quadratic Forms
 12.6 Real Symmetric Bilinear Forms, Law of Inertia
 12.7 Hermitian Forms

Chapter 13 Linear Operators on Inner Product Spaces
 13.1 Introduction
 13.2 Adjoint Operators
 13.3 Analogy between 𝐴(𝑉) and 𝐂, Special Linear Operators
 13.4 SelfAdjoint Operators
 13.5 Orthogonal and Unitary Operators
 13.6 Orthogonal and Unitary Matrices
 13.7 Change of Orthonormal Basis
 13.8 Positive Definite and Positive Operators
 13.9 Diagonalization and Canonical Forms in Inner Product Spaces
 13.10 Spectral Theorem
 Appendix A Multilinear Products
 Appendix B Algebraic Structures
 Appendix C Polynomials over a Field
 Appendix D Odds and Ends