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Linear Algebra, 5th edition

Published by Pearson (September 7, 2018) © 2019

  • Stephen H. Friedberg Illinois State University
  • Arnold J. Insel Illinois State University
  • Lawrence E. Spence Illinois State University

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For courses in Advanced Linear Algebra.

Illustrates the power of linear algebra through practical applications

Linear Algebra, 5th Edition presents a careful treatment of principal topics. This acclaimed theorem-proof text emphasizes the symbiotic relationship between linear transformations and matrices, but states theorems in the more general infinite-dimensional case where appropriate. Applications to such areas as differential equations, economics, geometry and physics appear throughout. It is suited to a second course in linear algebra that emphasizes abstract vector spaces, and can be used in a first course with a strong theoretical emphasis. Errata list for the 5th Edition

Hallmark features of this title

  • A rigorous treatment of linear algebra that is flexible and organized for a number of course options:
    • Frequently used for a second course that emphasizes abstract vector spaces, but can be used for accelerated students in a first course with a strong theoretical emphasis.
    • Applications to such areas as differential equations, economics, geometry and physics appear throughout, and can be included at the instructor's discretion.
  • Emphasizes the symbiotic relationship between linear transformations and matrices, but states theorems in the more general infinite-dimensional case where appropriate.  
  • Numerous accessible exercises that enrich and extend the text material, offering students a chance to test their understanding by working interesting problems at a reasonable level of difficulty.

New and updated features of this title

  • A streamlined presentation, with clarified exposition informed by extensive reviews from instructors.
  • Revised proofs of some theorems.
  • Additional examples and new exercises throughout.
  • Online solutions to selected theoretical exercises in each section of the book:
    • These exercises each have their exercise number printed within a gray box, and the last sentence of each of these exercises gives a short URL for its online solution.
  • 4 new applications available online of the content in Sections 2.3, 5.3, 6.5, and 6.6. Short URLs at point of use provide easy access to this material. 
  • * Sections denoted by an asterisk are optional.
  1. Vector Spaces
    • 1.1 Introduction
    • 1.2 Vector Spaces
    • 1.3 Subspaces
    • 1.4 Linear Combinations and Systems of Linear Equations
    • 1.5 Linear Dependence and Linear Independence
    • 1.6 Bases and Dimension
    • 1.7* Maximal Linearly Independent Subsets
    • Index of Definitions
  2. Linear Transformations and Matrices
    • 2.1 Linear Transformations, Null Spaces, and Ranges
    • 2.2 The Matrix Representation of a Linear Transformation
    • 2.3 Composition of Linear Transformations and Matrix Multiplication
    • 2.4 Invertibility and Isomorphisms
    • 2.5 The Change of Coordinate Matrix
    • 2.6* Dual Spaces
    • 2.7* Homogeneous Linear Differential Equations with Constant Coefficients
    • Index of Definitions
  3. Elementary Matrix Operations and Systems of Linear Equations
    • 3.1 Elementary Matrix Operations and Elementary Matrices
    • 3.2 The Rank of a Matrix and Matrix Inverses
    • 3.3 Systems of Linear Equations – Theoretical Aspects
    • 3.4 Systems of Linear Equations – Computational Aspects
    • Index of Definitions
  4. Determinants
    • 4.1 Determinants of Order 2
    • 4.2 Determinants of Order n
    • 4.3 Properties of Determinants
    • 4.4 Summary|Important Facts about Determinants
    • 4.5* A Characterization of the Determinant
    • Index of Definitions
  5. Diagonalization
    • 5.1 Eigenvalues and Eigenvectors
    • 5.2 Diagonalizability
    • 5.3* Matrix Limits and Markov Chains
    • 5.4 Invariant Subspaces and the Cayley–Hamilton Theorem
    • Index of Definitions
  6. Inner Product Spaces
    • 6.1 Inner Products and Norms
    • 6.2 The Gram–Schmidt Orthogonalization Process and Orthogonal Complements
    • 6.3 The Adjoint of a Linear Operator
    • 6.4 Normal and Self-Adjoint Operators
    • 6.5 Unitary and Orthogonal Operators and Their Matrices
    • 6.6 Orthogonal Projections and the Spectral Theorem
    • 6.7* The Singular Value Decomposition and the Pseudoinverse
    • 6.8* Bilinear and Quadratic Forms
    • 6.9* Einstein's Special Theory of Relativity
    • 6.10* Conditioning and the Rayleigh Quotient
    • 6.11* The Geometry of Orthogonal Operators
    • Index of Definitions
  7. Canonical Forms
    • 7.1 The Jordan Canonical Form I
    • 7.2 The Jordan Canonical Form II
    • 7.3 The Minimal Polynomial
    • 7.4* The Rational Canonical Form
    • Index of Definitions

Appendices

  • A. Sets
  • B. Functions
  • C. Fields
  • D. Complex Numbers
  • E. Polynomials

Answers to Selected Exercises

Index

About our authors

Stephen H. Friedberg holds a BA in mathematics from Boston University and MS and PhD degrees in mathematics from Northwestern University, and was awarded a Moore Postdoctoral Instructorship at MIT. He served as a director for CUPM, the Mathematical Association of America's Committee on the Undergraduate Program in Mathematics. He was a faculty member at Illinois State University for 32 years, where he was recognized as the outstanding teacher in the College of Arts and Sciences in 1990. He has also taught at the University of London, the University of Missouri and at Illinois Wesleyan University. He has authored or coauthored articles and books in analysis and linear algebra.

Arnold J. Insel received BA and MA degrees in mathematics from the University of Florida and a PhD from the University of California at Berkeley. He served as a faculty member at Illinois State University for 31 years and at Illinois Wesleyan University for 2 years. In addition to authoring and co-authoring articles and books in linear algebra, he has written articles in lattice theory, topology and topological groups.

Lawrence E. Spence holds a BA from Towson State College and MS and PhD degrees in mathematics from Michigan State University. He served as a faculty member at Illinois State University for 34 years, where he was recognized as the outstanding teacher in the College of Arts and Sciences in 1987. He is an author or co-author of 9 college mathematics textbooks, as well as articles in mathematics journals in the areas of discrete mathematics and linear algebra.

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