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Elementary Linear Algebra

Stephen Andrilli and David Hecker
Publisher: 
Academic Press
Publication Date: 
2016
Number of Pages: 
806
Format: 
Hardcover
Edition: 
5
Price: 
145.00
ISBN: 
9780128008539
Category: 
Textbook
[Reviewed by
Michele Intermont
, on
04/6/2017
]

Joint Review of:

Linear Algebra is a subject taught in the service of several masters. A craving for the abstractness of vector spaces and a desire to showcase the art of proof tugs against a desire for computational tools. In classical terms, this dichotomy is often framed as offering a course for students of mathematics versus a course for students of engineering. These two texts display the broad range of approaches one might take to a linear algebra course.

Andrilli and Hecker’s Elementary Linear Algebra looks intimidating and at 750 pages it is certainly hefty. This stems from their focus on providing numerous exercises and plenty of examples. The authors also state that easing the transition from computation to abstract thinking is one of their goals. It is not clear to me that they succeed in this, although they do provide plenty of proofs in the text, and a few opportunities for writing proofs in the exercises. Instead, they succeed at building an algorithmic feel for the subject, with many summaries of how to do things like diagonalizing a matrix in six easy steps or testing a set of vectors for linear independence in three steps.

Each section in Andrilli and Hecker’s text ends with a list of new vocabulary and highlights. Of course this is a sound idea, but it is a bit overwhelming to find 10 pages of text followed by as many highlighted statements, and twice as many new vocabulary words. Most sections have similar proportions. Andrilli and Hecker apply this same diligence to the exercises, and there the effect is lovely. It allows the reader to choose more practice when desired, without it being necessary to complete every exercise.

Happily, the text includes a chapter devoted to applications and another on numerical methods. Coding Theory, Ohm’s Law, Computer Graphics as well as several applications to other mathematical topics are presented and mention is given throughout the book when the necessary background for an application has been completed. The last chapter, on numerical methods, collects the topics of LDU, QR, SVD decompositions and also deals with partial pivoting and finding dominant eigenvalues, all relevant topics for today’s average student.

At the other end of the spectrum is the book A First Course in Linear Algebra by Eie and Chang. Where Andrilli and Hecker tip the scales towards computation and leaving nothing unwritten, Eie and Chang take a more general approach. Chapter 1 defines a vector space over a field, and it defines the term field too. This tendency towards abstraction continues throughout the text. There are not enough exercises. For the sophisticated reader, the text provides some concreteness along with the general theory. For example, the first chapter not only defines abstract vector space; it also describes solving systems of linear equations and introduces echelon form. And the examples provided really are for those just learning the subject.

Eie and Chang don’t trouble themselves to think about applications of linear algebra, but they do discuss the Jordan Canonical Form and the Spectral Theorem. The text itself is well-written. There are only a few places where the phrasing seems a bit awkward, and this is no way diminishes the readability. Each chapter begins with a brief synopsis which nicely captures the heart of the chapter, and there are enough boldface headings and vocabulary to keep the reader oriented to the task at hand.

Each of these texts will appeal to a subset of students and faculty. While the subject matter is the same, the approaches differ tremendously. Which one will you choose?


Michele Intermont is Associate Professor of Mathematics at Kalamazoo College.

  • Dedication
  • Preface for the Instructor
    • Philosophy of the Text
    • Major Changes for the Fifth Edition
    • Plans for Coverage
    • Prerequisite Chart for Later Sections
    • Acknowledgments
  • Preface to the Student
  • A Light-Hearted Look at Linear Algebra Terms
  • Symbol Table
  • Computational & Numerical Techniques, Applications
  • Chapter 1: Vectors and Matrices
    • Abstract
    • 1.1 Fundamental Operations with Vectors
    • 1.2 The Dot Product
    • 1.3 An Introduction to Proof Techniques
    • 1.4 Fundamental Operations with Matrices
    • 1.5 Matrix Multiplication
    • Review Exercises for Chapter 1
  • Chapter 2: Systems of Linear Equations
    • Abstract
    • 2.1 Solving Linear Systems Using Gaussian Elimination
    • 2.2 Gauss-Jordan Row Reduction and Reduced Row Echelon Form
    • 2.3 Equivalent Systems, Rank, and Row Space
    • 2.4 Inverses of Matrices
    • Review Exercises for Chapter 2
  • Chapter 3: Determinants and Eigenvalues
    • Abstract
    • 3.1 Introduction to Determinants
    • 3.2 Determinants and Row Reduction
    • 3.3 Further Properties of the Determinant
    • 3.4 Eigenvalues and Diagonalization
    • Review Exercises for Chapter 3
  • Chapter 4: Finite Dimensional Vector Spaces
    • Abstract
    • 4.1 Introduction to Vector Spaces
    • 4.2 Subspaces
    • 4.3 Span
    • 4.4 Linear Independence
    • 4.5 Basis and Dimension
    • 4.6 Constructing Special Bases
    • 4.7 Coordinatization
    • Review Exercises for Chapter 4
  • Chapter 5: Linear Transformations
    • Abstract
    • 5.1 Introduction to Linear Transformations
    • 5.2 The Matrix of a Linear Transformation
    • 5.3 The Dimension Theorem
    • 5.4 One-to-One and Onto Linear Transformations
    • 5.5 Isomorphism
    • 5.6 Diagonalization of Linear Operators
    • Review Exercises for Chapter 5
  • Chapter 6: Orthogonality
    • Abstract
    • 6.1 Orthogonal Bases and the Gram-Schmidt Process
    • 6.2 Orthogonal Complements
    • 6.3 Orthogonal Diagonalization
  • Chapter 7: Complex Vector Spaces and General Inner Products
    • Abstract
    • 7.1 Complex n-Vectors and Matrices
    • 7.2 Complex Eigenvalues and Complex Eigenvectors
    • 7.3 Complex Vector Spaces
    • 7.4 Orthogonality in
    • 7.5 Inner Product Spaces
    • Review Exercises for Chapter 7
  • Chapter 8: Additional Applications
    • Abstract
    • 8.1 Graph Theory
    • 8.2 Ohm’s Law
    • 8.3 Least-Squares Polynomials
    • 8.4 Markov Chains
    • 8.5 Hill Substitution: An Introduction to Coding Theory
    • 8.6 Rotation of Axes for Conic Sections
    • 8.7 Computer Graphics
    • 8.8 Differential Equations
    • 8.9 Least-Squares Solutions for Inconsistent Systems
    • 8.10 Quadratic Forms
  • Chapter 9: Numerical Techniques
    • Abstract
    • 9.1 Numerical Techniques for Solving Systems
    • 9.2 LDU Decomposition
    • 9.3 The Power Method for Finding Eigenvalues
    • 9.4 QR Factorization
    • 9.5 Singular Value Decomposition
  • Appendix A: Miscellaneous Proofs
    • Proof of Theorem 1.16, Part (1)
    • Proof of Theorem 2.6
    • Proof of Theorem 2.10
    • Proof of Theorem 3.3, Part (3), Case 2
    • Proof of Theorem 5.29
    • Proof of Theorem 6.19
  • Appendix B: Functions
    • Functions: Domain, Codomain, and Range
    • One-to-One and Onto Functions
    • Composition and Inverses of Functions
    • New Vocabulary
    • Highlights
    • Exercises for Appendix B
  • Appendix C: Complex Numbers
    • New Vocabulary
    • Highlights
    • Exercises for Appendix C
  • Appendix D: Elementary Matrices
    • Prerequisite: Section 2.4, Inverses of Matrices
    • Elementary Matrices
    • Representing a Row Operation as Multiplication by an Elementary Matrix
    • Inverses of Elementary Matrices
    • Using Elementary Matrices to Show Row Equivalence
    • Nonsingular Matrices Expressed as a Product of Elementary Matrices
    • New Vocabulary
    • Highlights
    • Exercises for Appendix D
  • Appendix E: Answers to Selected Exercises
    • Section 1.1 (p. 1–19)
    • Section 1.2 (p. 19–34)
    • Section 1.3 (p. 34–52)
    • Section 1.4 (p. 52–65)
    • Section 1.5 (p. 65–81)
    • Chapter 1 Review Exercises (p. 81–83)
    • Section 2.1 (p. 85–105)
    • Section 2.2 (p. 105–118)
    • Section 2.3 (p. 118–134)
    • Section 2.4 (p. 134–147)
    • Chapter 2 Review Exercises (p. 148–151)
    • Section 3.1 (p. 153–166)
    • Section 3.2 (p. 166–177)
    • Section 3.3 (p. 177–187)
    • Section 3.4 (p. 188–206)
    • Chapter 3 Review Exercises (p. 206–210)
    • Section 4.1 (p. 213–225)
    • Section 4.2 (p. 225–238)
    • Section 4.3 (p. 238–250)
    • Section 4.4 (p. 250–267)
    • Section 4.5 (p. 268–281)
    • Section 4.6 (p. 281–292)
    • Section 4.7 (p. 292–311)
    • Chapter 4 Review Exercises (p. 311–317)
    • Section 5.1 (p. 319–335)
    • Section 5.2 (p. 336–353)
    • Section 5.3 (p. 353–365)
    • Section 5.4 (p. 365–373)
    • Section 5.5 (p. 374–387)
    • Section 5.6 (p. 388–406)
    • Chapter 5 Review Exercises (p. 406–412)
    • Section 6.1 (p. 413–428)
    • Section 6.2 (p. 428–445)
    • Section 6.3 (p. 445–460)
    • Chapter 6 Review Exercises (p. 460–463)
    • Section 7.1 (p. 465–473)
    • Section 7.2 (p. 473–480)
    • Section 7.3 (p. 480–483)
    • Section 7.4 (p. 484–491)
    • Section 7.5 (p. 492–509)
    • Chapter 7 Review Exercises (p. 509–512)
    • Section 8.1 (p. 513–527)
    • Section 8.2 (p. 527–530)
    • Section 8.3 (p. 530–540)
    • Section 8.4 (p. 540–552)
    • Section 8.5 (p. 552–557)
    • Section 8.6 (p. 557–564)
    • Section 8.7 (p. 564–581)
    • Section 8.8 (p. 581–590)
    • Section 8.9 (p. 591–598)
    • Section 8.10 (p. 598–605)
    • Section 9.1 (p. 607–620)
    • Section 9.2 (p. 621–629)
    • Section 9.3 (p. 629–635)
    • Section 9.4 (p. 636–644)
    • Section 9.5 (p. 644–666)
    • Appendix B (p. 675–685)
    • Appendix C (p. 687–691)
    • Appendix D (p. 693–700)
  • Index
  • Inside Back Cover
    • Equivalent Conditions for Linearly Independent and Linearly Dependent Sets
    • Kernel Method (Finding a Basis for the Kernel of L)
    • Range Method (Finding a Basis for the Range of L)
    • Equivalent Conditions for One-to-One, Onto, and Isomorphism
    • Dimension Theorem
    • Gram-Schmidt Process

Dummy View - NOT TO BE DELETED