# Get most out of least squares solution

## Linear Algebra - Beginning and Intermediate Algebra. First edition

November 28, 2003

Linear Algebra: An Interactive Approach is an excellent text. It is appropriate for science and engineering undergraduates as users of linear algebra. Its clear style and sections containing proofs of theorems means that it will also be valued by mathematics students as a foundation for this vital area.

The material is standard, with early chapters on linear systems and matrices, the algebra of matrices, subspaces and rank. Later chapters develop the ideas of inverse and LU-decomposition, determinants, eigenvalues, inner product spaces, vector spaces and linear mappings. The important concept of the generalised inverse and its application to the best least-squares solution to a system of equations is well explained and demonstrated.

As well as examples, there are plenty of exercises, end-of-chapter review questions and projects. Answers and hints are given. The book is supported by a CD, containing an electronic version of the text. There is a companion website, which is still under development. This does not really matter for most purposes as the text is largely self-contained, including Matlab-based solutions to "drill" exercises. Sadly, the files for an interesting-looking project on singular value decomposition and image processing, supposedly on the CD and the website, are apparently not in either location.

Beginning and Intermediate Algebra serves to demonstrate just how far the concept of a textbook has developed. This is not so much a book as a learning environment devoted to basic algebra. There is a growing number of students who enter university and discover that an understanding of such material is essential for progress. For one reason or another, they have not acquired it, and this book may well be highly appropriate for them.

The content ranges from a review of arithmetic through linear equations and inequalities to quadratic equations, exponential and logarithmic functions and finally some very basic material on conic sections. Clearly, a vast amount of teaching experience has been incorporated in this work. There are vast numbers of exercises, including "check yourself" exercises, chapter "summary" exercises, followed by a chapter test, and a series of cumulative tests. There are also "writing" and "group" exercises, to encourage verbal explanation and communication.

A considerable variety of support material is available for both students and instructors, including videos and CDs and solutions manuals. As for web-based resources, there is an online learning centre with links to PageOut, McGraw-Hill's course website tool, and NetTutor. Links from WebCT are also possible.

Altogether an impressive work, but there is a concern about such a "weighty" resource. The book itself is a heavyweight volume, and I wonder if this, together with the amount of supplementary material, could reinforce the false impression that the content is excessively difficult?

Nigel Steele is professor of mathematics, Coventry University.

## Linear Algebra: An Interactive Approach. First edition

Author - Surender K. Jain and Ananda D. Gunawardena
Editor - Thomson Brooks/Cole
Pages - 418
Price - £32.99
ISBN - 0 534 40915 6

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