Linear Algebra: Examples and Applications

This is a textbook for a course in the basic theory of linear algebra. It contains the usual topics, starting from linear systems and culminating with the spectral theorem for normal operators, but omitting the Jordan normal form. On the other hand, the organization of the book has a more personal flavor than most other books at the same level. This was a pleasant surprise.

As the author says in the preface, the book tries to use examples and applications to justify the abstract language of linear algebra. This is a laudable goal, but I'm not sure it has been fully attained in this book. The applications presented in the book vary in interest and depth. They do not all work equally well; which parts do work will probably depend on the reader.

The intended audience is the "curious and motivated student" in scientific areas, not just mathematics. While the curious mathematics student will most probably benefit from reading this book, other science students may have problems: some parts of the book may be too abstract; they are not usually present in other books at the same level, perhaps for a good reason.

The author says in the preface that he has tried to avoid routine exercises. In this, I think he has succeed. I found several interesting and unusual problems that I'll probably use in exams.

In summary, I recommend this book as side reading for anyone interested in Linear Algebra, specially students looking for a change from their textbook. Instructors will find here some interesting material that they can use in class and in exams. I did.

Let's now take a closer look at the book.

The terminology used in the book is not always standard. The author speaks of "compatible linear system" when "consistent" seems to be the favored term (which he also uses). He also seems intent on unifying the terminology used for matrices and for operators. Thus, he talks about symmetric operators instead of self-adjoin operators, transpose operators instead of adjoint operators. This makes some sense, because matrices of symmetric operators are symmetric and the matrix of the adjoint is the transpose, and the double terminology can be confusing to the student. But perhaps it is too late to change that. (Just as it is too late to speak about "orthonormal" matrices instead of the less accurate "orthogonal" matrices.)

Theorems and propositions are not always numbered. This is confusing, even to the author, who mentions "Theorem 1" on page 211 when he apparently mean the (unnumbered) theorem in Section 8.2.2. Likewise, there are no figure numbers or even captions and the pictures are rarely discussed in the text. (The book also contains several hand-drawn cartoons by the author whose point I have missed.)

Each chapter ends with brief notes, references to accessible literature, such as papers in the American Mathematical Monthly and Mathematics Magazine, and frequently also keywords for internet search.

The book is well written and there are very few typos. I found only two.

The book begins in Chapter 1 with a leisurely discussion of linear systems and the elimination method. It contains an interesting motivational example: finding decompositions of the sphere into pentagons and hexagons. A general solution for the number of pentagons and hexagons is found, but the author says that not all configurations may be realizable and deciding which are is a difficult problem. However, no references are given and I was left wishing for the full story.

Chapter 2 introduces the basic abstract concepts: vector spaces, generators, linear independency, bases, dimension. It contains a longer-than-usual discussion of infinite-dimensional vector spaces, including an interesting exposition of the space of rational functions (the explicit basis from the theory of partial fractions is given without proof and without even mentioning the term "partial fractions"!). Chapter 2 also contains an appendix on set theory that I think would be better placed at the end of the book.

Chapter 3 is about matrix multiplication, whose definition is motivated by the composition of linear fractional transformations and linear change of variables. The elementary operations introduced in Chapter 1 for the solution of linear systems are interpreted as matrix products and the LU factorization is deduced. The appendix contains a brief presentation of Strassen's algorithm, not an usual topic in other books. Chapter 3 does not use the language introduced in Chapter 2 and would fit much more naturally right after Chapter 1.

A minor historical quibble is that Chapter 3 begins by suggesting that the word "matrix" comes from typography. However, both MathWorld and Wikipedia say that the term was first used by Sylvester in the sense of "an enclosure within which something originates or develops" ("matrix" is Latin for "womb").

Another arguable historical point appears on page 155. The author claims that Galois proved that there are no algebraic formulas for solving polynomial equations of degree greater than 4. While Galois's work does include this result, most historians would say that it was first proved by Abel.

Chapter 4 introduces linear transformations. The main result is, of course, the rank-nullity theorem, from which the equality of row-rank and column-rank is deduced. This is further discussed in Chapter 5. Chapter 6 introduces eigenvalues and eigenvectors. The usual application to difference equations is given, in particular to Fibonacci numbers. Chapter 7 introduces inner products. The law of cosines for triangles is obtained by solving a linear system, which I found natural and unlike the way it is done in school.

Chapter 8 is dedicated to a proof of the spectral theorem for symmetric operators. The existence of eigenvectors for symmetric operators is proved using a continuity argument that could be adapted for proving the singular value decomposition, but is not. (The SVD is proved later in the book as a consequence of the spectral theorem.) Chapter 12 contains a proof of the complex spectral theorem based on the fundamental theorem of algebra. (The author proves in Chapter 11, on applications of the determinant, that every operator has a complex eigenvalue. He then concludes that every matrix is "trigonalizable", another non-standard term.)

Chapter 9 contains a long introduction to duality, including the duality of Platonic solids and projective duality, that seems a little out of place. Chapter 10 contains a long but nice discussion of determinants.

Luiz Henrique de Figueiredo is a researcher at IMPA in Brazil. His main interests are numerical methods in computer graphics, but he remains an algebraist at heart. He is also one of the designers of the Lua language.