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Linear Algebra: Ideas and Applications

Richard C. Penney
John Wiley
Publication Date: 
Number of Pages: 
[Reviewed by
Mindy Capaldi
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I found this textbook very appealing, in large part due to my personal struggles finding the “perfect” text for the course. I have taught Linear Algebra using three different textbooks so far. Although each book had its strengths, none of them was quite right for the level of course and/or teaching style that I employ. What I really want is a readable, proof-inclusive but not proof-intensive, applications-driven textbook.

Readability: In my last two iterations teaching Linear Algebra I used a flipped pedagogy, where students read material before coming to class and then practiced problems during class time. When I first flipped I ended up typing my own 1–2 page readings for each class period because the book did not seem accessible enough. Penney’s self-stated goal (in “Features of the Text”) is that students be able to read and use his Linear Algebra book on their own. I believe he achieved that goal. The writing is relatively conversational and easy to read. A nice example of this is on page 53 in the section on Gaussian Elimination: “Some readers might be tempted to say that we get the same description of the solution because we are, after all, solving the same system. This, however, is not the correct explanation. Suppose, for example…” It feels like the author is explaining the material as he would in person to a student or class.

Another constructive addition to this text are the self-study questions that follow application topics. Recognizing that instructors likely won’t have time to cover every interesting application, Penney wrote them into the text in a manner allowing for individual study. Since I live in a city with an increasing number of roundabouts, I especially enjoyed the systems of equations application using traffic patterns on page 72.

Proofiness: The second time I flipped Linear Algebra I used a free online linear algebra text. The price was right and it included nice applications. To encourage students to learn how to read mathematics texts, I intended to have students largely read from the textbook. Unfortunately, beyond the first couple of chapters the book assumed more student knowledge of proof techniques and notation than I expected. My students were going to take an introduction to proofs course in a subsequent semester. Of course, Linear Algebra is a foundational course in transitioning students from computational to abstract mathematics. Penney’s text introduces abstract material in parallel to the computational. Concepts such as linear independence and span are defined early, on pages seven and ten, and woven throughout the chapters to slowly make students comfortable with their meaning and use.

Most of the more abstract definitions are first motivated through a concrete example. Theorems and proofs are not avoided, but are also not overly-emphasized: the exercises ask for a few proofs, but not every section does so. Instead, there is a wonderful variety of exercise types. Almost every set of problems include computational, geometric, “give an example”, explain your solution, true-false, and MATLAB questions. I ask students to explain/justify or give an example on my exams, and I appreciated seeing so many of those types of problems in the homework sets.

Applications: The number and value of the applications that an upper-level mathematics textbook includes is important, and can mean the difference between a student understanding the usefulness and appeal of abstract mathematics or not. I chose Joseph Gallian’s Contemporary Abstract Algebra as my textbook for that subject largely due to its applications. Penney’s book has many interesting applications, as would be expected based on its title. A few examples are least-squares, wavelets, and the Leontief model from economics. There are also numerous computer projects based on MATLAB. I do wish that some of the computer-driven work used other platforms, such as Microsoft Excel or Geogebra.

Overall, I think that this book has a lot of strengths. It seems especially useful for someone like me who wants their students to read the text more. It wouldn’t necessarily be the best fit for a student population who was experienced with proving theorems, but for students who haven’t made that transition it bridges the gap between computational and theoretical mathematics. I’ll mention one final pro and con of the book. The con is that it is somewhat expensive, as mathematics textbooks often are. The pro is that there is a companion website with a password-protected solutions manual and figures.

See also our review of the third edition.

Mindy Capaldi is an assistant professor at Valparaiso University. Her current research area is mathematics education in the Scholarship of Teaching and Learning realm. Her favorite area of mathematics is Abstract Algebra. She is a fan of reading fiction and doing math, and spends much of her time on these two activities. 






1.1 The Vector Space of m × n Matrices 1

The Space Rn 4

Linear Combinations and Linear Dependence 6

What is a Vector Space? 11

Exercises 17

1.1.1 Computer Projects 22

1.1.2 Applications to Graph Theory I 25

Exercises 27

1.2 Systems 28

Rank: The Maximum Number of Linearly Independent Equations 35

Exercises 38

1.2.1 Computer Projects 41

1.2.2 Applications to Circuit Theory 41

Exercises 46

1.3 Gaussian Elimination 47

Spanning in Polynomial Spaces 58

Computational Issues: Pivoting 61

Exercises 63

Computational Issues: Counting Flops 68

1.3.1 Computer Projects 69

1.3.2 Applications to Traffic Flow 72

1.4 Column Space and Nullspace 74

Subspaces 77

Exercises 86

1.4.1 Computer Projects 94

Chapter Summary 95


2.1 The Test for Linear Independence 97

Bases for the Column Space 104

Testing Functions for Independence 106

Exercises 108

2.1.1 Computer Projects 113

2.2 Dimension 114

Exercises 123

2.2.1 Computer Projects 127

2.2.2 Applications to Differential Equations 128

Exercises 131

2.3 Row Space and the rank-nullity theorem 132

Bases for the Row Space 134

Computational Issues: Computing Rank 142

Exercises 143

2.3.1 Computer Projects 146

Chapter Summary 147


3.1 The Linearity Properties 149

Exercises 157

3.1.1 Computer Projects 162

3.2 Matrix Multiplication (Composition) 164

Partitioned Matrices 171

Computational Issues: Parallel Computing 172

Exercises 173

3.2.1 Computer Projects 178

3.2.2 Applications to Graph Theory II 180

Exercises 181

3.3 Inverses 182

Computational Issues: Reduction versus Inverses 188

Exercises 190

3.3.1 Computer Projects 195

3.3.2 Applications to Economics 197

Exercises 202

3.4 The LU Factorization 203

Exercises 212

3.4.1 Computer Projects 214

3.5 The Matrix of a Linear Transformation 215

Coordinates 215

Isomorphism 228

Invertible Linear Transformations 229

Exercises 230

3.5.1 Computer Projects 235

Chapter Summary 236


4.1 Definition of the Determinant 238

4.1.1 The Rest of the Proofs 246

Exercises 249

4.1.2 Computer Projects 251

4.2 Reduction and Determinants 252

Uniqueness of the Determinant 256

Exercises 258

4.2.1 Volume 261

Exercises 263

4.3 A Formula for Inverses 264

Exercises 268

Chapter Summary 269


5.1 Eigenvectors 271

Exercises 279

5.1.1 Computer Projects 282

5.1.2 Application to Markov Processes 283

Exercises 285

5.2 Diagonalization 287

Powers of Matrices 288

Exercises 290

5.2.1 Computer Projects 292

5.2.2 Application to Systems of Differential Equations 293

Exercises 295

5.3 Complex Eigenvectors 296

Complex Vector Spaces 303

Exercises 304

5.3.1 Computer Projects 305

Chapter Summary 306


6.1 The Scalar Product in RN 308

Orthogonal/Orthonormal Bases and Coordinates 312

Exercises 316

6.2 Projections: The Gram-Schmidt Process 318

The QR Decomposition 325

Uniqueness of the QR Factorization 327

Exercises 328

6.2.1 Computer Projects 331

6.3 Fourier Series: Scalar Product Spaces 333

Exercises 341

6.3.1 Application to Data Compression: Wavelets 344

Exercises 352

6.3.2 Computer Projects 353

6.4 Orthogonal Matrices 355

Householder Matrices 361

Exercises 364

Discrete Wavelet Transform 367

6.4.1 Computer Projects 369

6.5 Least Squares 370

Exercises 377

6.5.1 Computer Projects 380

6.6 Quadratic Forms: Orthogonal Diagonalization 381

The Spectral Theorem 385

The Principal Axis Theorem 386

Exercises 392

6.6.1 Computer Projects 395

6.7 The Singular Value Decomposition (SVD) 396

Application of the SVD to Least-Squares Problems 402

Exercises 404

Computing the SVD Using Householder Matrices 406

Diagonalizing Matrices Using Householder Matrices 408

6.8 Hermitian Symmetric and Unitary Matrices 410

Exercises 417

Chapter Summary 419


7.1 Generalized Eigenvectors 421

Exercises 429

7.2 Chain Bases 431

Jordan Form 438

Exercises 443

The Cayley-Hamilton Theorem 445

Chapter Summary 445


8.1 Condition Number 446

Norms 446

Condition Number 448

Least Squares 451

Exercises 451

8.2 Computing Eigenvalues 452

Iteration 453

The QR Method 457

Exercises 462

Chapter Summary 464