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Linear Algebra: Ideas and Applications

Richard C. Penney
Publisher: 
John Wiley
Publication Date: 
2008
Number of Pages: 
480
Format: 
Hardcover
Edition: 
3
Price: 
110.00
ISBN: 
9780470178843
Category: 
Monograph
[Reviewed by
P. N. Ruane
, on
09/15/2008
]

To say that a particular book is ‘readable’ or ‘student-friendly’ is, of course, a matter of subjective comparison. For example, for the very first treatise on linear algebra, by Hermann Grassmann (1844), comparison was obviously impossible. But such was the inscrutable style of his exposition that many of the greatest mathematicians of his generation found it incomprehensible.

A century later, however, Paul Halmos’ book Finite Dimensional Vector Spaces (1947) became the very first university textbook devoted solely to linear algebra. And yet, although easygoing compared to Grassmann’s innovative book, most postgraduate students would find it very hard going as an introductory text (for instance, chapter 10 deals with calculus on linear manifolds, and there is hardly an exercise in the book).

Linear algebra was a rare component of undergraduate mathematics until the 1960s, and the publication of Finite Mathematical Structures, by Kemeny, Snell, Thompson and Merkel, (1959) may have precipitated its emergence as one of the most commonly occurring components in undergraduate mathematics. Appropriately, in 1965 the CUPM recommendation suggested that linear algebra be a course in the undergraduate curriculum and gave guidelines for course content.

Fifty years later, a quick search on the Barnes and Noble web site will come up with over 700 books with the phrase ‘linear algebra’ included in the title (perhaps with duplications, such as the listing of both hard and soft back versions of a particular book). Consequently, the process of choosing a book on this subject may be a more complex process than reading that by Paul Halmos.

It also seems to me that books on linear algebra are getting thicker and thicker as years go by — which is partly due, no doubt, to the wider range of themes they now cover. For instance, coverage of matrices has long been integrated into books like this one by Richard Penney, and in some depth at that. Also, many books cover a wide range of applications; in this book these include graph theory, circuit theory, computing, traffic flow, predator-prey problems, harmonic oscillators, geometry, numerical techniques, economics, and many more.

In the preface, the author says that this book not just about techniques, theorems and proofs, but about ideas. What this means is that the central concepts of linear algebra are not suddenly unloaded upon the reader, but they emerge gradually due to the careful structuring of the text. Another way of saying this is that the material is developed heuristically by means of examples and exercises that constitute about one third of the book’s 480 pages. Not only that, but Richard Penney writes very clearly, and there are no overly extended sections of prose in the whole book.

As for the exercises, they vary greatly in their nature and difficulty. Many sets are preceded by a True/False collection. Others consist of problems that are graded from routine to longer and more open-ended questions. In addition, there are about twenty computer projects dispersed across the books eight chapters, which are designed so that students undertake them on an independent basis using MATLAB (or Maple).

Regarding the contents, each of the chapters concludes with a summary of its main ideas, and, with the exception of duality, they cover nearly every topic that one would meet in an undergraduate introductory course on linear algebra. In fact, the amount of material here is possibly sufficient for a two-semester course. Moreover, due to the book’s readability and the organisation of the material, it is also highly suited to self-study.

Being a third edition, this book may be familiar to many readers of MAA reviews, so here’s a summary of the main changes:

  • New chapter on generalized eigenvectors and chain bases, with coverage of the Jordan form and the Cayley-Hamilton theorem.
  • New chapter on numerical techniques
  • New section on Hermitian symmetric and unitary matrices.
  • Computational approaches to finding eigenvalues (forward and reverse iteration).

Finally, I think it was Bertrand Russell who said something like ‘the bigger the book the greater its malign influence’, but that doesn’t apply to this one by Richard Penney — although I do I feel that it more readily applies to book reviews, so I’ll conclude by saying that, if Barnes and Noble have better books on linear algebra than this one, then the process of choosing from their vast collection is far more complex than I thought.


Peter Ruane was introduced to linear algebra by means of the book by Kemeny, Snell et al in 1964. However, he also referred to the relevant chapters of Birkhoff and MacLane, whose book A Survey of Modern Algebra (1940) contained the first undergraduate treatment of linear algebra.

Preface.

Features of the Text.

1. Systems of Linear Equations.

1.1 The Vector Space of m x n Matrices.

The Space Rn.

Linear Combinations and Linear Dependence.

What Is a Vector Space?

Why Prove Anything?

True-False Questions.

Exercises.

1.1.1 Computer Projects.

Exercises.

1.1.2 Applications to Graph Theory I.

Self-Study Questions.

Exercises.

1.2 Systems.

Rank: The Maximum Number of Linearly Independent Equations.

True-False Questions.

Exercises.

1.2.1 Computer Projects.

Exercises.

1.2.2 Applications to Circuit Theory.

Self-Study Questions.

Exercises.

1.3 Gaussian Elimination.

Spanning in Polynomial Spaces.

Computational Issues: Pivoting.

True-False Questions.

Exercises.

Computational Issues: Flops.

1.3.1 Computer Projects.

Exercises.

1.3.2 Applications to Traffic Flow.

Self-Study Questions.

Exercises.

1.4 Column Space and Nullspace.

Subspaces.

Subspaces of Functions.

True-False Questions.

Exercises.

1.4.1 Computer Projects.

Exercises.

1.4.2 Applications to Predator-Prey Problems.

Self-Study Questions.

Exercises.

Chapter Summary.

2. Linear Independence and Dimension.

2.1 The Test for Linear Independence.

Bases for the Column Space.

Testing Functions for Independence.

True-False Questions.

Exercises.

2.1.1 Computer Projects.

2.2 Dimension.

True-False Questions.

Exercises.

2.2.1 Computer Projects.

Exercises.

2.2.2 Applications to Calculus.

Self-Study Questions.

Exercises.

2.2.3 Applications to Differential Equations.

Self-Study Questions.

Exercises.

2.2.4 Applications to the Harmonic Oscillator.

Self-Study Questions.

Exercises.

2.2.5 Computer Projects.

Exercises.

2.3 Row Space and the Rank-Nullity Theorem.

Bases for the Row Space.

Rank-Nullity Theorem.

Computational Issues: Computing Rank.

True-False Questions.

Exercises.

2.3.1 Computer Projects.

Chapter Summary.

3. Linear Transformations.

3.1 The Linearity Properties.

True-False Questions.

Exercises.

3.1.1 Computer Projects.

3.1.2 Applications to Control Theory.

Self-Study Questions.

Exercises.

3.2 Matrix Multiplication (Composition).

Partitioned Matrices.

Computational Issues: Parallel Computing.

True-False Questions.

Exercises.

3.2.1 Computer Projects.

3.2.2 Applications to Graph Theory II.

Self-Study Questions.

Exercises.

3.3 Inverses.

Computational Issues: Reduction vs. Inverses.

True-False Questions.

Exercises.

Ill Conditioned Systems.

3.3.1 Computer Projects.

Exercises.

3.3.2 Applications to Economics.

Self-Study Questions.

Exercises.

3.4 The LU Factorization.

Exercises.

3.4.1 Computer Projects.

Exercises.

3.5 The Matrix of a Linear Transformation.

Coordinates.

Application to Differential Equations.

Isomorphism.

Invertible Linear Transformations.

True-False Questions.

Exercises.

3.5.1 Computer Projects.

Chapter Summary.

4. Determinants.

4.1 Definition of the Determinant.

4.1.1 The Rest of the Proofs.

True-False Questions.

Exercises.

4.1.2 Computer Projects.

4.2 Reduction and Determinants.

Uniqueness of the Determinant.

True-False Questions.

Exercises.

4.2.1 Application to Volume.

Self-Study Questions.

Exercises.

4.3 A Formula for Inverses.

Cramer’s Rule.

True-False Questions.

Exercises 273.

Chapter Summary.

5. Eigenvectors and Eigenvalues.

5.1 Eigenvectors.

True-False Questions.

Exercises.

5.1.1 Computer Projects.

5.1.2 Application to Markov Processes.

Exercises.

5.2 Diagonalization.

Powers of Matrices.

True-False Questions.

Exercises.

5.2.1 Computer Projects.

5.2.2 Application to Systems of Differential Equations.

Self-Study Questions.

Exercises.

5.3 Complex Eigenvectors.

Complex Vector Spaces.

Exercises.

5.3.1 Computer Projects.

Exercises.

Chapter Summary.

6. Orthogonality.

6.1 The Scalar Product in Rn.

Orthogonal/Orthonormal Bases and Coordinates.

True-False Questions.

Exercises.

6.1.1 Application to Statistics.

Self-Study Questions.

Exercises.

6.2 Projections: The Gram-Schmidt Process.

The QR Decomposition 334.

Uniqueness of the QR-factoriaition.

True-False Questions.

Exercises.

6.2.1 Computer Projects.

Exercises.

6.3 Fourier Series: Scalar Product Spaces.

Exercises.

6.3.1 Computer Projects.

Exercises.

6.4 Orthogonal Matrices.

Householder Matrices.

True-False Questions.

Exercises.

6.4.1 Computer Projects.

Exercises.

6.5 Least Squares.

Exercises.

6.5.1 Computer Projects.

Exercises.

6.6 Quadratic Forms: Orthogonal Diagonalization.

The Spectral Theorem.

The Principal Axis Theorem.

True-False Questions.

Exercises.

6.6.1 Computer Projects.

Exercises.

6.7 The Singular Value Decomposition (SVD).

Application of the SVD to Least-Squares Problems.

True-False Questions.

Exercises.

Computing the SVD Using Householder Matrices.

Diagonalizing Symmetric Matrices Using Householder Matrices.

6.8 Hermitian Symmetric and Unitary Matrices.

True-False Questions.

Exercises.

Chapter Summary.

7. Generalized Eigenvectors.

7.1 Generalized Eigenvectors.

Exercises.

7.2 Chain Bases.

Jordan Form.

True-False Questions.

Exercises.

The Cayley-Hamilton Theorem.

Chapter Summary.

8. Numerical Techniques.

8.1 Condition Number.

Norms.

Condition Number.

Least Squares.

Exercises.

8.2 Computing Eigenvalues.

Iteration.

The QR Method.

Exercises.

Chapter Summary.

Answers and Hints.

Index.

Dummy View - NOT TO BE DELETED