Linear algebra. Historical notes by Victor J. Katz. 3rd ed. (English) Zbl 0949.15002

Reading, MA: Addison-Wesley. xii, 538 p. (1995).
[For the first edition (1987) see Zbl 0667.15001, for the second edition (1990) see Zbl 0724.15001.]
The book is designed to serve as the basis for a first undergraduate course in linear algebra. It includes all standard parts: Systems of linear equations, matrices, vector spaces, systems of linear inequalities, determinants, eigenvalues and eigenvectors, diagonalization of matrices and some applications of this process, Euclidean spaces, orthogonality, complex vector spaces, linear transformations and similarity, complex matrices, the Jordan canonical form of a matrix, some methods of linear programming, solving large linear systems. The historical notes of Victor Katz are very interesting and useful.
The present third edition has some new features. In previous editions fundamental notions of linear algebra were first introduced in the context of axiomatically defined vector spaces. Having easily mastered matrix algebra and techniques for solving linear systems, many students are unable to adjust to discontinuity in difficulty. In an attempt to eradicate this abrupt jump the authors extensively revised the first portion of the text to introduce these fundamental ideas gradually, in the context of \({\mathbb R}^n\). Most of the reorganization of the text of the new edition is driven by the authors’ desire to tackle this problem. Vector geometry (geometric addition, the dot product, length, the angle between vectors) is now presented for vectors in \({\mathbb R}^n\) in the first two sections of Chapter 1. This provides a geometric foundation for the notions of linear combinations and subspaces.
Also applications of matrix algebra to binary linear codes now appear at the end of Chapter 1. The professional PC software MATLAB is widely used for computations in linear algebra. Throughout the text of the new edition the authors include optional exercises to be done using the Student Edition of MATLAB. Each exercise set includes an explanation of the procedures and commands in MATLAB needed for the exercises. The PC software LINTEK by the first author, designed explicitly for this book, is revised and upgraded and is free to student using the text of the reviewed book. The authors place the applications at the end of the chapters where the pertinent algebra is developed, unless the applications are so extensive that they merit a chapter by themselves. For example, Chapter 1 concludes with applications to population distribution (Markov chains) and to binary codes. There are abundant pencil-and-paper exercises as well as computer exercises. Most exercise sets include a ten-part true-false problem. That exercise gives students valuable practice in deciding whether a mathematical statement is true, as opposed to asking for a proof of a given true statement.
So this new edition of the book is an excellent textbook for students.


15-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra
15-03 History of linear algebra
15A03 Vector spaces, linear dependence, rank, lineability
15A04 Linear transformations, semilinear transformations
15A06 Linear equations (linear algebraic aspects)
15A15 Determinants, permanents, traces, other special matrix functions
15A18 Eigenvalues, singular values, and eigenvectors
15A21 Canonical forms, reductions, classification
65F05 Direct numerical methods for linear systems and matrix inversion
90C05 Linear programming
15-04 Software, source code, etc. for problems pertaining to linear algebra
68W30 Symbolic computation and algebraic computation