Alignment: Schaum’s Outlines • Linear Algebra • Sixth Edition

Table of Contents

  • Chapter 1 Vectors in 𝐑^𝑛 and 𝐂^𝑛, Spatial Vectors
  • Chapter 2 Algebra of Matrices
  • Chapter 3 Systems of Linear Equations
  • 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 Least-Square 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
  • 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

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