
Schaum's Outline of Linear Algebra, 5th Edition
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Content
- Cover Page
- SCHAUM'S outlines® Linear Algebra, Fifth Edition
- Copyright Page
- Preface
- List of Symbols
- Contents
- Chapter 1 Vectors in Rn and Cn, Spatial Vectors
- 1.1 Introduction
- 1.2 Vectors in Rn
- 1.3 Vector Addition and Scalar Multiplication
- 1.4 Dot (Inner) Product
- 1.5 Located Vectors, Hyperplanes, Lines, Curves in Rn
- 1.6 Vectors in R3 (Spatial Vectors), ijk Notation
- 1.7 Complex Numbers 1.8 Vectors in Cn
- 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
- 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 Direct-Sum 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 Self-Adjoint 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
- Index
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