Schaum's outline of linear algebra.

Description

Contents List of Symbols Chapter 1 Vectors in R[sup(n)] and C[sup(n)], Spatial Vectors 1.1 Introduction 1.2 Vectors in R[sup(n)] 1.3 Vector Addition and Scalar Multiplication 1.4 Dot (Inner) Product 1.5 Located Vectors, Hyperplanes, Lines, Curves in R[sup(n)] 1.6 Vectors in R[sup(3)] (Spatial Vectors), ijk Notation 1.7 Complex Numbers 1.8 Vectors in C[sup(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 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 A B C D E F G H I J K L M N O P Q R S T U V W Z