Linear algebra

Linear algebra

Seymour Lipschutz

Book - 2013

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Subjects
Algebras, Linear > Problems, exercises, etc.
Algebras, Linear > Outlines, syllabi, etc.
Published
New York ; London : McGraw-Hill c2013.
Language
English
Main Author
Seymour Lipschutz (-)
Other Authors
Marc Lipson (-)
Edition
5th ed
Item Description
Previous ed.: 2011.
Includes index.
Physical Description
vi, 426 p. : ill. ; 28 cm
ISBN
9780071794565
  • List of Symbols
  • 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 R 3 (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
  • 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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