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Further Linear Algebra

T.S. Blyth and E.F. Robertson
Springer Verlag
Publication Date: 
Number of Pages: 
Springer Undergraduate Mathematics Series
[Reviewed by
Morteza Seddighin
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This book goes beyond the elementary theory of linear algebra and treats some of the more advanced topics in the subject. It is very clearly written and easy to understand. The book starts with a very good review of elementary linear algebra but moves forward very rapidly to advanced material. In chapter 1 the authors treats topics such as Fourier expansion of an element relative to an orthonormal basis and Parseval's identity. The exercises and illustrative examples are very interesting. For instance in chapter 1, after the angle θ between two vectors x and y is defined by θ=/(|x||y|), the reader is asked to prove the relationship

cos(θ1 - θ2) = cos(θ1)cos(θ2) + sin(θ1)sin(θ2)

This relationship, which is the generalization of a formula in elementary trigonometry, is in fact valid even for infinite dimensional linear spaces and its proof is based on Parseval's identity. In chapters 5 and 6 the authors present Jordan, Rational, and Classical Forms of matrices. Many linear algebra books cover the theory of Normal Matrices only over the complex field. By contrast, in chapter 10 of this book the authors cover the theory of Normal Matrices over the real field. This is actually more interesting than the complex theory, since one can not apply the spectral theory which is valid only for Normal Matrices on the complex field.

In chapter 11, the authors focus on computational aspects of linear algebra. Here they show the reader how to use the LinearAlgebra package of Maple to perform computations. There are several advanced Maple procedures written in this chapter to help the reader to perform linear algebra computations.

In chapter 12 of this book the authors present brief biographies of great mathematicians who had made major contributions to the field of linear algebra, including Fourier, Parseval, and Hilbert.

This book will be of interest to anyone who wishes to have a good grasp of linear llgebra and matrix theory. It can also be used as an advanced undergraduate textbook. Although this book does not treat infinite dimensional linear spaces, it provides the reader with a deep understanding of finite dimensional linear spaces. Many aspects of the theory of finite dimensional linear spaces can easily be generalized to the infinite dimensional case. Therefore, this book will also be helpful to those who intend to study infinite dimensional spaces later.

Morteza Seddighin ( is associate professor of mathematics at Indiana University East. His research interests are functional analysis and operator theory.

The Story So Far.- Inner Product Spaces.- Direct Sums of Subspaces.- Primary Decomposition.- Reduction to Triangular Form.- Reduction to Jordan Form.- Rational and Classical Forms.- Dual Spaces.- Orthogonal Direct Sums.- Bilinear and Quadratic Forms.- Real Normality.- Computer Assistance.- But Who Were They?- Solutions to the Exercises.- Index.