Saturday, November 22, 2008

Review of Basic Linear Algebra by Blyth and Robertson

Review of Basic Linear Algebra by Blyth and Robertson
The book gives a thorough and rigorous treatment of linear algebra which is what a first year student will expect to see on a linear algebra course from a British university.
There are a number of numerical examples which lead nicely to the theory of linear algebra. The authors have hit the right balance between proofs of theorems and techniques to apply such theorems.
The ordering of the chapters is sensible with the first 4 chapters on matrices and linear equations before the more abstract work on vector spaces. The theory and manipulations on eigenvalues and eigenvectors is left towards the end of the book.
A great asset of the book is that it is portable and reasonably cheap at around £16 for students to buy and carry around in lectures and library.
It is also good to see that brief solutions to most problems are at the back of the book.
The only solutions omitted are the assignment problems which the lecturer can set as part of the coursework.
Additionally there are sufficient exercises with good progression and it is good to see a whole chapter devoted to a computer algebra package.
However I have following reservations:
In the introduction to the book it is important to state why linear algebra is critical to the student’s mathematical studies. It should say something like “after calculus the most useful mathematical tool ever developed is linear algebra because it brings the physical world within the scope of mathematics”.
A book on linear algebra should have plenty of illustrations so that the student can envisage what is going on and these illustrations can be used to motivate him or her. This book has a severe lack of diagrams.
More words are required to motivate the student and soften the blow. Each chapter should have an introduction, a list of objectives and a summary. I follow the maxim ‘Tell them, at great length, what you are going to do. Do it, and then tell them what you have done’.
The authors do not write in a way that will appeal to weaker students. It is far too succinct.
The word ‘basic’ in the title is not appropriate for this book. A number of A level students cannot divide 10^(-7) by (1/2x10^4) even with a calculator. I can’t see how students will cope with this book without a serious input by a tutor.
Another issue is that the book is not interactive in any way. It seems to be a one way delivery from the authors to the student. A book like this should include some questions which will make the student think and arouse his/her anxiety. I could not find a single question in the text of the book for the reader. Clearly there are a number of problems for the student to tackle but I am referring to questions such as:
1. Why are matrices important?
2. How can we prove this theorem?
3. What approach are we going to use to solve this problem?
A more serious issue is that once the authors have covered a particular concept they expect the student to fully digest it. This is not my experience of students. I think a particular concept used in chapter 9 which was covered in chapter 2 say, needs to be signposted so that the student knows exactly where the idea was defined earlier in the book.
A less serious issue is that the authors use some very compact and complicated notation. It will difficult for first year students to follow some of this compact notation unless they have seen it before.
The authors use mathematical software, MAPLE 7, but it would have been better to integrate this into each chapter rather than bolt on a chapter at the end. Students will be more confident in using the software if it is used throughout the book.