##
**Linear algebra done right.**
*(English)*
Zbl 0843.15002

New York, NY: Springer-Verlag. xvii, 238 p. (1995).

The present book offers another text for a second course in linear algebra at American colleges or universities. Being of this sort, it is aimed at math majors and readers familiar with the basics of matrix calculus, and its main subject is the theory of real or complex vector spaces and their homomorphisms.

As there are very good textbooks on linear algebra in plentiful supply, on the one hand, and as the author has given his new text such an audacious title, on the other hand, one might ask the questions: “What is radically new (and finally done right) in the author’s approach? What is wrong with all the other famous texts on linear algebra, which seemingly served their purpose for generations of students and teachers?”

The answer is that the author has chosen an unusual, partially novel methodical approach to the subject, indeed, whose features, advantages and disadvantages will be described in the sequel. However, it should be mentioned already here that the title of the book is a humorous extravagance, in my opinion, quite so arisen from the author’s factual viewpoint of how to teach linear algebra at this level, and motivated by his intention to emphasize the peculiarity of his approach already by the title.

Now, as to the special features of the text, its main characteristic is that one of the traditionally most exploited methods, namely the determinantal calculus, is on principle not used for developing the theory of vector spaces and linear operators. Actually, determinants of operators and matrices are banished to the end of the text, and appear (as independent objects) only after the discussion of eigenvalues, generalized eigenspaces, and the Jordan normal form of a trigonalizable operator on a (complex or real) vector space. This is certainly a very interesting and original approach to the study of linear operators, which forced the author to develop a rather unconventional, methodically well-considered and well-arranged manner of handling the subject. Being convinced, as the author says in the preface to the instructor, that determinants are “difficult, nonintuitive, and often defined without motivation”, he tried to spare the student the “tortous (torturous?) path” through determinants in order to learn the deep theorems on linear operators. Instead, he presents what he calls the “clean, determinant-free methods” of linear algebra, and these really constitute the text’s major difference from the tradition in teaching linear algebra.

The author’s opinion about the right place for determinants in teaching linear algebra is surely debatable. On the other hand, he (just as surely) realizes his teaching program in a very consequent, utmost careful and methodically masterful manner. Many theorems, whose proofs usually use determinantal calculus, especially those dealing with eigenvalues, characteristic polynomials and minimal polynomials, normal forms for special operators, etc., are presented here with smart and easier ad-hoc proofs, sometimes at the expense of brevity and elegance. Anyway, it is highly interesting to see how much of advanced linear algebra can be covered, and what depth can be reached, without using determinants. Up to the level of linear algebra presented in the text, the author’s approach is certainly very comfortable for beginners, and admittedly very enlightening and useful for teachers, too. As for the deeper study of linear algebra (and also of its related geometry), determinants are a really indispensible tool (for example, in the theory of linear groups, in multilinear algebra and projective geometry, in the advanced theory of Euclidean or unitary vector spaces, etc.), and this material must then be taken from other textbooks. Certainly, determinants are finally introduced at the end of the present textbook, and this makes the author’s book into a solid, helpful basis for further studies in linear algebra and geometry.

The contents of the book are arranged in ten chapters: 1. Vector Spaces; 2. Finite-Dimensional Vector Spaces; 3. Linear Maps; 4. Polynomials; 5. Eigenvalues and Eigenvectors (of trigonalizable operators); 6. Inner-Product Spaces; 7. Operators on Inner-Product Spaces; 8. Operators on Complex Vector Spaces (including the Jordan normal form); 9. Operators on Real Vector Spaces; 10. Trace and Determinant.

Each chapter comes with a set of carefully selected exercises. The reader is assumed to bring with him no prerequisites other than some maturity in mathematical thinking and the basic knowledge of matrix operations. Side-notes to the text help students to understand definitions, proofs, and cross-relations. The preface provides the author’s valuable methodical comments for instructors and students.

Altogether, the text is a didactic masterpiece. However, some experienced teachers and experts may feel that it would have been more appropriate to choose a title like “Linear Algebra Done Differently”, or simply “Linear Algebra”. Those people might be right, somewhere.

In any case, this text on linear algebra is highly recommendable for students and teachers, and it represents a very welcome (alternative) addition to the existing textbooks on linear algebra.

As there are very good textbooks on linear algebra in plentiful supply, on the one hand, and as the author has given his new text such an audacious title, on the other hand, one might ask the questions: “What is radically new (and finally done right) in the author’s approach? What is wrong with all the other famous texts on linear algebra, which seemingly served their purpose for generations of students and teachers?”

The answer is that the author has chosen an unusual, partially novel methodical approach to the subject, indeed, whose features, advantages and disadvantages will be described in the sequel. However, it should be mentioned already here that the title of the book is a humorous extravagance, in my opinion, quite so arisen from the author’s factual viewpoint of how to teach linear algebra at this level, and motivated by his intention to emphasize the peculiarity of his approach already by the title.

Now, as to the special features of the text, its main characteristic is that one of the traditionally most exploited methods, namely the determinantal calculus, is on principle not used for developing the theory of vector spaces and linear operators. Actually, determinants of operators and matrices are banished to the end of the text, and appear (as independent objects) only after the discussion of eigenvalues, generalized eigenspaces, and the Jordan normal form of a trigonalizable operator on a (complex or real) vector space. This is certainly a very interesting and original approach to the study of linear operators, which forced the author to develop a rather unconventional, methodically well-considered and well-arranged manner of handling the subject. Being convinced, as the author says in the preface to the instructor, that determinants are “difficult, nonintuitive, and often defined without motivation”, he tried to spare the student the “tortous (torturous?) path” through determinants in order to learn the deep theorems on linear operators. Instead, he presents what he calls the “clean, determinant-free methods” of linear algebra, and these really constitute the text’s major difference from the tradition in teaching linear algebra.

The author’s opinion about the right place for determinants in teaching linear algebra is surely debatable. On the other hand, he (just as surely) realizes his teaching program in a very consequent, utmost careful and methodically masterful manner. Many theorems, whose proofs usually use determinantal calculus, especially those dealing with eigenvalues, characteristic polynomials and minimal polynomials, normal forms for special operators, etc., are presented here with smart and easier ad-hoc proofs, sometimes at the expense of brevity and elegance. Anyway, it is highly interesting to see how much of advanced linear algebra can be covered, and what depth can be reached, without using determinants. Up to the level of linear algebra presented in the text, the author’s approach is certainly very comfortable for beginners, and admittedly very enlightening and useful for teachers, too. As for the deeper study of linear algebra (and also of its related geometry), determinants are a really indispensible tool (for example, in the theory of linear groups, in multilinear algebra and projective geometry, in the advanced theory of Euclidean or unitary vector spaces, etc.), and this material must then be taken from other textbooks. Certainly, determinants are finally introduced at the end of the present textbook, and this makes the author’s book into a solid, helpful basis for further studies in linear algebra and geometry.

The contents of the book are arranged in ten chapters: 1. Vector Spaces; 2. Finite-Dimensional Vector Spaces; 3. Linear Maps; 4. Polynomials; 5. Eigenvalues and Eigenvectors (of trigonalizable operators); 6. Inner-Product Spaces; 7. Operators on Inner-Product Spaces; 8. Operators on Complex Vector Spaces (including the Jordan normal form); 9. Operators on Real Vector Spaces; 10. Trace and Determinant.

Each chapter comes with a set of carefully selected exercises. The reader is assumed to bring with him no prerequisites other than some maturity in mathematical thinking and the basic knowledge of matrix operations. Side-notes to the text help students to understand definitions, proofs, and cross-relations. The preface provides the author’s valuable methodical comments for instructors and students.

Altogether, the text is a didactic masterpiece. However, some experienced teachers and experts may feel that it would have been more appropriate to choose a title like “Linear Algebra Done Differently”, or simply “Linear Algebra”. Those people might be right, somewhere.

In any case, this text on linear algebra is highly recommendable for students and teachers, and it represents a very welcome (alternative) addition to the existing textbooks on linear algebra.

Reviewer: W.Kleinert (Berlin)

### MSC:

15-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra |

15A03 | Vector spaces, linear dependence, rank, lineability |

15A63 | Quadratic and bilinear forms, inner products |

15A04 | Linear transformations, semilinear transformations |

15A21 | Canonical forms, reductions, classification |

15A18 | Eigenvalues, singular values, and eigenvectors |

15A15 | Determinants, permanents, traces, other special matrix functions |