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

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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.
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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.
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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.
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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.
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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.
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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.
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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.