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Elementary linear algebra. A matrix approach. (English) Zbl 0960.15002

Upper Saddle River, NJ: Prentice Hall. xiv, 477 p. (2000).
At first sight this appears to be yet another of the many linear-algebra-via-matrices books available on the market. But when one looks closer, one quickly realizes that this is one with some important differences. The authors have chosen to write a text that is matrix-oriented in order to make it accessible to a wide variety of students, but they have not at the same time become verbose and forfeited a succinct and precise mathematical style. On the contrary, their book is admirably structured in terms of examples, definitions, theorems with proofs, and discussions. Moreover, they have emphasized something that is extremely important, but not easy for students to grasp: the many alternative ways of seeing concepts such as, for example: linear independence, rank, consistency of equations, orthogonality of a matrix, etc.
The chapter headings are: 1. Matrices, vectors and systems of linear equations, 2. Matrices and linear transformations, 3. Determinants, 4. Subspaces and their properties, 5. Eigenvalues, eigenvectors, and diagonalization, 6. Orthogonality, 7. Vector spaces. Apart from these core topics, the book also includes a number of more specialized topics such as LU decomposition, singular value decomposition, spectral decomposition for symmetric matrices, Lagrange interpolation, complex numbers, block multiplication, the Moore-Penrose inverse, and quadratic forms. The authors have also included a number of interesting applications: the Leontief input-output model, current flow in electrical circuits, the Leslie matrix and population change, traffic flow, adjacency matrices, Markov chains, systems of differential equations, harmonic motion, difference equations, and least squares approximation including the use of trigonometric polynomials for periodic functions. The chapters are interspersed with many practice problems and exercises, with answers to the practice problems and the odd-numbered exercises.
The authors also stress the geometrical aspects of linear algebra and, wherever possible, draw attention to the usefulness of computer software in linear algebra, without, however, presuming its use. Throughout the book, they have adopted the recommendations of the 1993 Linear Algebra Curriculum Study Group.
This book stands out from among the legion of linear algebra books as one that is eminently suited as a course book and for personal study. It not only introduces the subject in a way suitable for beginners, but at the same time provides a rigorous and yet not too dry or technical treatment and includes more advanced topics as well, which adds to its attractiveness. It can be thoroughly and enthusiastically recommended for use worldwide.

MSC:

15-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra
15A06 Linear equations (linear algebraic aspects)
15A04 Linear transformations, semilinear transformations
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
65F20 Numerical solutions to overdetermined systems, pseudoinverses
41A05 Interpolation in approximation theory
15A09 Theory of matrix inversion and generalized inverses
15B36 Matrices of integers
93D25 Input-output approaches in control theory
90B10 Deterministic network models in operations research
92D25 Population dynamics (general)
39A10 Additive difference equations
34A30 Linear ordinary differential equations and systems
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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