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Linear algebra for computational sciences and engineering (2nd ed.)
Neri F., Springer International Publishing, New York, NY, 2019. 574 pp. Type: Book (978-3-030213-20-6)
Date Reviewed: Aug 10 2021

This is supposedly an improved and expanded version of the first edition, which I have not seen, so I am considering this text purely on its individual merits.

This is a substantial book, intended to be reasonably comprehensive for its extended audience: people who are users of mathematics, rather than being mathematicians themselves. The text is presented in two parts: “Foundations of Linear Algebra” and “Elements of Linear Algebra.” Given that the terms “foundations” and “elements” are so often used with the same meaning, I don’t know why more descriptive titles were not used.

The first part starts with an introductory chapter discussing axiomatic systems, sets, and functions; chapters on matrices, linear equations, geometric vectors, complex numbers, geometric algebra, and conic sections follow. The “elements” part starts with algebraic structures (groups, rings, fields, homomorphisms), followed by chapters on vector spaces, inner product spaces, linear mappings, computational complexity, graph theory, and electrical networks.

This might be a fine book, but its usefulness is severely compromised by a confusion with the level of rigor, some strange definitions, an ordering of topics that inhibits rather than promotes understanding, and some curious solecisms. The book has all the hallmarks of a lack of proper editing, or even an initial reviewing of sample chapters.

Consider the first chapter. The author seems unclear whether sets should be introduced as an axiomatic system, or simply from a naive perspective. The latter would be vastly preferable--it would enable this chapter to be shortened and simplified, and at the same time increase its understanding. Even the basic (naive) definition of a set on page 4 is incorrect: ”a collection of objects that share a common feature“; the only common feature is that they are elements of the same set. Because the author does not distinguish between finite and infinite sets immediately, the definition of a proper subset includes that the cardinality of that subset is less than the cardinality of the outer set. Much of the material could have been improved with the inclusion of a few Venn or Euler diagrams. Then there is an issue of notation: the author uses the symbol ⪯ for non-reflexive ordering; surely the underline in the symbol includes equality. This happens many times in the book, and I think it is a great source of confusion. The author is also wedded to the use of the first-order logic predicates ∀ and ∃ even in written text, where they do not belong.

The second chapter starts by introducing a vector as a list of real numbers, which is not a bad place to start. But then the author immediately introduces a matrix as an array and describes operations on matrices in terms of the matrix elements. This is bad pedagogy, as it is deficient in intuition and doesn’t provide insight. Many beginning students must wonder, for example, why matrix addition is defined by corresponding elements, but multiplication by the scalar product of rows and columns. A little care introducing matrices by way of some geometric intuition (for example, as a transformation between coordinate systems, although there are many other methods) will provide a much cleaner introduction. The author strangely fails to mention the conditions under which operations of addition and multiplication are defined. Possibly the worst offender in this chapter is the definition of determinants using permutations. This merely enhances the notion that linear algebra is a mass of fiddly calculations without any intuitive meaning. A major omission is (reduced) row echelon form--surely a natural part of linear equations, as well as any discussion of linear independence and matrix rank.

The chapter on linear equations is adequate, though let down by some horrendous notation such as superscripted indices (which are also used, unaccountably, for indexing column vectors in matrix multiplication), and calling the Gauss-Seidel iterative method ”Gauss-Seidel’s method“ as though there was a mathematician called Gauss-Seidel. One serious omission is of the permutation matrix from the discussion of LU factorization: in general, A = LU is impossible; a permutation matrix is needed so that A = PLU.

The chapters on geometric vectors could have been split, with half of the material covered in chapter 1 to provide some geometric intuition to start building a theory of linear algebra. This is a lost opportunity. The complex number chapter is an odd mix; much is explained, but just as much is not, for example, Euler’s theorem. A curious omission is the complex conjugate right at the beginning, where it could have been used to describe division as well as the norm. When it turns up, it is notated by a dot, which is almost impossible to see, instead of the more standard bar.

The last chapter in this section, on conics, is possibly in the wrong spot, as it would benefit enormously from the use of diagonalization. The chapter on eigensystems (which is in Part 2) could also benefit from some discussion of numerical methods: the power method, Rayleigh quotients, and the QR method could be included.

Leaving aside Part 2, the chapters on graph theory and electrical networks are poorly chosen, as very little of the previous theory is used. This is especially true of graph theory. Better choices for these chapters might be search engine ranking, linear programming, error-correcting codes, computer graphics, and statistical regression or principal component analysis.

It would be impossible in a short review such as this to detail all the deficiencies of this text, but it is not wholly a lost cause. Some careful editing, better attention to pedagogy and the building of insight, and the insertion of some missing topics (such as those mentioned above) will go a long way to greatly increasing its usability.

Reviewer:  Alasdair McAndrew Review #: CR147329 (2111-0265)
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