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Review of Gilbert Strang's *Linear Algebra and Its
Applications*, 3rd edition, 1988.

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Reviewed by Tom Hagedorn

Strang's textbook has been widely used for many years (the author
himself learned linear algebra from this text) and not surprisingly,
its organization is fairly traditional. There are eight chapters in
the book, but a standard first course in linear algebra will probably
cover only the first five. The topics of these chapters are, in
order: Matrices and Gaussian Elimination, Vector Spaces and Linear
Equations, Orthogonality, Determinants, and Eigenvalues and
Eigenvectors. As the context for most of Strang's text is
**R**^{n}, and he spends time explaining its geometry, the
placement of orthogonality before that of eigenvalues is quite
natural. Overall, Strang's text contains many applications from
science and engineering and his text seems well-suited for an audience
with a good scientific knowledge. Many of the worked examples, some
of which take several pages, require a good familiarity with science.
Students, and instructors, without such knowledge may have difficulty
with the text. Also, though most results are justified by an informal
discussion in the text, students may have a hard time determining what
is and is not proven, and so this text does not seem suitable for an
course which is intended to introduce students to proofs.
It is interesting to study Strang's development of linear algebra
in his first two chapters. Strang's textbook begins by a thorough
discussion of solving linear equations and the geometry of their
solutions. Matrices and their operations are then introduced.
Elimination theory then leads to a discussion of the LDU decomposition
followed by a discussion of the inverse of a matrix and how to compute
them via the Gauss-Jordan method and how to determine which matrices
are invertible via the use of non-zero pivots. Transposes are also
introduced and discussed. The last section of Chapter 1 introduces an
application of using linear algebra to find an approximate solution to
a differential equation. Though I had fond memories of this text, in
comparison with some of the other texts on the market (e.g. Lay) this
part of the text seemed awkward. While the introduction of matrices
is well-motivated, the introduction of matrix multiplication in the
context of row operations, in the absence of any discussion of linear
transformations, seems unnatural. Similarly, the presentation of
inverses and transposes seems perfunctory and unrelated to the other
ideas of the chapter. Other texts seem better able to present these
concepts in a unified manner.
The real strength of Strang's text, I had often thought, was in
Strang's development of vector spaces in Chapter 2. Strang
immediately introduces the definition of abstract vector spaces and
presents many examples in addition to the space of n-dimensional
vectors. He then discusses the solution of m equations in n unknowns
in terms of the dimension of subspaces and the rank of matrices. He
then develops the concepts of linear independence, basis, and
dimension and discusses the four fundamental subspaces in the context
of Rn. After a section on an application of linear algebra to graphs
and networks, in the final section linear transformations are
discussed. For all these concepts, they are discussed in the context
of Rn and the examples in the context of other vector spaces are not
usually given. Again, as for the first chapter, I had misgivings
about the exposition. For example, I have always found Strang's
discussion of the four fundamental spaces very wordy and ultimately
confusing to students. And the application of linear algebra to
graphs and networks, while interesting to someone who knows linear
algebra, seems to unnecessarily break up the unity of the chapter.
Overall, Strang's *Linear Algebra and Its Applications*
contains a wealth of applications and examples of how to use linear
algebra in science and engineering. I would heartily recommend it as
a reference book for anyone who has already taken a course in linear
algebra, but I would have trouble recommending it as a text for a
first course in linear algebra. Beginning students will be confused
by it and while more advanced students may be able to see the big
picture, linear algebra can and should be presented as a more unified
subject than it is here.