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Lipschutz, Seymour and Lipson, Marc. Schaum's Outline of Linear Algebra, 5th Edition. US: McGraw-Hill, 2012.

Schaum's Outline of Linear Algebra, 5th Edition

Authors: Seymour Lipschutz and Marc Lipson

Published:  November 2012

eISBN: 9780071794572 0071794573 | ISBN: 9780071794565
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  • 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