Comparing Linear Algebra Texts: Bretscher, 1st edition, vs. Lay, 2nd edition.

Reviewed by Lorenzo Sadun

The following is a comparative review I did of two linear algebra texts for Prentice-Hall. The organization of the review was determined by the questions Prentice-Hall asked. I apologize for the repetition that this causes.

I. Organization.

The organization of the two books is quite similar, but is very different from other linear algebra texts. The key feature in common is postponing matrix algebra until linear transformations are understood. This is very good, as it avoids the trap of students breezing through a few weeks of easy computations, only to hit a wall when abstract concepts appear. Moreover, the ease of matrix algebra obscures the important concepts of linear transformations that are needed later on. It's much better to do those concepts first, and only later learn the easy way of doing computations.

The key difference between the texts is that Bretscher postpones abstract vector spaces (as opposed to subspaces of Euclidean space) until the very end of the book, but does orthogonality and least-squares in the middle, while Lay does the opposite. I am much more comfortable with Lay's ordering, as the vector spaces I encounter are generally abstract. (Solution sets to ODEs may be finite dimensional, but there's no natural inner product).

Abstract vector spaces are the big conceptual stumbling block, and postponing them until the end is ensuring that students won't really have time to understand them. Furthermore, dealing with orthogonality early on confuses the issue and makes understanding vector spaces without inner products even more difficult.

II. Content

For the most part, Lay has a better topic coverage for my purposes. Bretscher does a wonderful job explaining linear transformations in geometry, and this probably makes the whole notion of a linear transformation easier to grasp, but most physics and engineering problems are not directly geometrical!

When I teach linear algebra, a quintessential example I try to get to (at the end of the term) is the vibrating string. A student who understands the vibrating string is ready to do Fourier Analysis in a future class. This example puts together several difficult concepts. Functions on an interval make an abstract vector space. The vibrational modes are the eigenfunctions of the second derivative operator. They also form an orthonormal set, which makes expanding an arbitrary function in the vibrational modes straightforward. To my mind, these concepts naturally appear in the following order: first vector spaces, then eigenvalues (with applications to 2nd order coupled ODEs), then orthogonality.

Lay does it that way. Bretscher does it in the opposite order. Naturally, I prefer Lay's approach, although I can see the internal consistency of Bretscher's approach.

Bretscher has a more thorough coverage of dynamical systems, but it's all at the end. Lay has examples sprinkled through the text. Of course, once you get to eigenvalues it's easy to find a wealth of interesting applications. The hard thing is to keep the students' attention before that point.

Bretscher devotes far more time and effort than Lay to developing geometrical intuition. This has both advantages and disadvantages. The advantage is that $R^n$ and its subspaces are a simple setting in which to learn about linear transformations, bases, and other vector-space concepts. The disadvantage (aside from the extra time spent) is that students think certain geometrical constructs -- in particular the inner product -- are instrinsic to vector spaces, and find themselves unable to handle spaces that don't have a natural inner product.

A similar issue arises in calculus. Most texts describe a derivative as the slope of a tangent line and an integral as the area under a curve -- two geometrical ideas. Students grasp these ideas quickly, but then stumble on the "rate of change" notion of derivative and the "total accumulated stuff" notion of integral. Geometry is a conceptual crutch, in both the good and bad sense. In my opinion, there is too little geometry in most linear algebra texts, and too much in Bretscher.

Both texts give good motivations for the key concepts. Bretscher's motivations are likely to be more geometric, while Lay's are tied more directly to applications, some of which are a bit contrived.

Bretscher has a lot of historical footnotes -- far more than Lay. To some students, especially those with a strong background in the humanities, these can be interesting and motivational. But to others I'm sure it comes across as pedantry. Depending on your audience, it could be either a big plus or a big minus.

Bretscher de-emphasizes the classroom use of technology, while Lay is more encouraging. My own view is that technology can be a big aid in teaching linear algebra, but only if the instructor is committed to using it as an exploratory tool as opposed to a calculational shortcut. Both texts can be used with or without computers, and both have companion works to allow for effective computerization. Lay has assigned problems geared for using technology to explore, while Bretscher does not.

III. Problems.

Both texts have extensive problems that cover a wide range of difficulty. Bretscher's problems are generally more abstract, while Lay has many more drill problems. Bretscher has a number of deep problems that require a page of explanation before you can begin. This is a very good thing, as it allows an instructor to use the homework to explore new territory. Lay has practice problems, with the solution worked out in detail 2 pages later. This is very helpful, especially to weaker students. Lay has computer-based problems that are designed to use technology to explore.

In general, the difference in problems reflects the fact that Bretscher is pitched to a somewhat different audience than Lay. Bretscher assumes that you already know how to turn the crank, and asks fairly open-ended problems to prompt you to think, and to explore some important applications of the material. Lay gives you some drill work, asks some theoretical questions that, while abstract, typically are not open-ended, and puts the free exploration in the computer-based problems.

Stronger and better motivated students would probably get more from Bretscher's problems, while weaker students would benefit from the step-by-step reinforcement of Lay.

IV. Applications.

Lay has fewer applications, usually directly related to the issued being developed, and usually somewhat contrived to make the calculations easier. Bretscher has more applications, which are more realistic. The downside to this is that it takes considerable effort to understand the physics, enginnering, economics, or biology behind one of Bretscher's applications. When the students are sophisticated and well versed in other subjects, this wealth of applications is a strength. When the students are struggling with linear algebra and just want a few examples to sharpen their intuition, the depth of the examples is a real handicap. Once again, what's right for sophisticated students isn't right for everybody.

V. Writing style.

Both authors write quite clearly. However, there is once again a large difference in the amount of sophistication expected of the reader. Bretscher is writing for students who know a fair amount about other subjects, who have wide-ranging interests, and who consider themselves erudite. Such students not only will appreciate the depth of Bretscher's examples, but they will accept geometry as an intuition-building playground. Lay is written for students who know less, and who just want (or have) to learn linear algebra to proceed with their studies in physics, engineering or economics. Such students find higher-dimensional geometry to be yet another abstraction to be swallowed, and need applications that are more directly "real-world". More sophisticated students may feel that Lay is talking down to them, especially with some oversimplified examples. Less sophisticated students may find Bretscher pompous or pretentious.

VI. Overall evaluation.

Although I have pointed out the differences, there texts are far more alike than they are different. Both Bretscher and Lay are very good texts, MUCH better than the numerous alternatives I have seen. Their organization is quite similar, with an early emphasis on linear transformations, and an initial avoidance of matrix algebra. This is good. There are a few significant differences in approach, with Lay emphasising abstract vector spaces and Bretscher emphasizing the geometry of Euclidean space. My own preference is for Lay's approach, but this may just be an outgrowth of having taught successfully out of Lay's earlier edition.

Ultimately, the biggest difference is the readership at which the book is aimed. Bretscher is aimed at confident scholars who just happen not to know linear algebra yet. Lay is aimed at more timid students, whose mathematical background may be more limited, and who may or may not have an understanding of other sciences.

If I were teaching at an Ivy League school, or if I were teaching an honors section of applied linear algebra, I would probably use Bretscher. However, at a large state school students are likely to react negatively to Bretscher's highbrow style. Furthermore, some students need the guidance of Lay's sometimes simplistic examples. At such a school I feel that Lay is the better choice.