Algebra and Geometry [Paperback]

Alan F. Beardon (Author)

Book Description

ISBN-10: 0521890497 | ISBN-13: 978-0521890496 | Publication Date: June 20, 2005 | Edition: 1

This text gives a basic introduction, and a unified approach, to algebra and geometry. Alan Beardon covers the ideas of complex numbers, scalar and vector products, determinants, linear algebra, group theory, permutation groups, symmetry groups, and various aspects of geometry including groups of isometries, rotations, and spherical geometry. The emphasis is on the interaction among these topics. The text is divided into short sections, with exercises at the end of each section.

Editorial Reviews

Review

"Beardon has written a fine introductory text...[he] outlines a beautiful course."
MAA Reviews, Stephen Ahearn

“Beardon has prepared a book that this reviewer will keep on his shelf as a reference. Highly recommended.”
Choice

Book Description

This text gives a basic introduction, and a unified approach algebra and geometry, two of the cornerstones of undergraduate mathematics. The emphasis is always on the interaction between these topics, and each topic is constantly illustrated by using it to describe and discuss the other topics. Many of the ideas are developed gradually throughout the book. For ease of use by students, the text is divided into short sections, with exercises at the end of each section.

Product Details

• Paperback: 338 pages
• Publisher: Cambridge University Press; 1 edition (June 20, 2005)
• Language: English
• ISBN-10: 0521890497
• ISBN-13: 978-0521890496
• Product Dimensions: 8.9 x 6.7 x 0.6 inches
• Shipping Weight: 1.6 pounds (View shipping rates and policies)
• Average Customer Review: 4.3 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #735,053 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

14 of 14 people found the following review helpful:

5.0 out of 5 stars Excellent first glimpse into the algebraic world, August 3, 2007

By 

Justin Curry - See all my reviews
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This review is from: Algebra and Geometry (Paperback)

This book has become a bible of sorts to first-year students studying mathematics at Cambridge. Its quality as a text cannot be doubted, although its usefulness for further years of algebra is limited. This is precisely the book to study from if you are doing vector calculus and differential equations, but still aren't sure about doing mathematics seriously. If you have not taken a course in linear algebra or abstract algebra, buy the paperback copy of this book and start reading right away. Beardon starts with (what I believe is the best way) the study of permutations (think about shuffling a deck of cards) to develop an intuition of the basic notions of a group. From here the fundamentals for further study in mathematics is laid. I won't repeat the table of contents here, as you can look for yourself, but believe me when I say that mastering the concepts in this book will serve you very well.

I truly wish I had a course which devoted itself to the complete digestion of this book. I used it for self-study and found that it served me very well. It does not fall easily into the structure of most American math sequences, as these departments are often forced to "modularize" mathematics into semester-bite-sized pieces. I believe that this often has a negative impact on the appreciation of mathematics as a whole, especially at the nexus between doing basic calculus and appreciating proof, rigor and beauty in mathematical structures.

The book may not go into the same depth as, say, Artin's "Algebra", but rather the foundational concepts for the study of algebra and geometry are emphasized in a variety of settings. This is very important as the study of "abstract algebra" is precisely that if you do not have a wide-selection of examples and contexts to draw from. This book has plenty of exercises of varying difficulty, and everything in this book is accessible to the beginning student of mathematics.

Bottom line: If you are someone interested in learning linear algebra, geometry, group theory, Mobius transformations, complex variables all in a rigorous yet introductory level, this is the book for you. Developing a robust mental model for mathematics requires building several thin layers at a time. This means not going too deep too quickly, but rather snorkeling around the entire reef, before you gear up for further exploration.

3 of 3 people found the following review helpful:

5.0 out of 5 stars Excellent explanation of how algebra can be used to describe and manipulate objects in space, July 26, 2008

By 

Charles Ashbacher (Marion, Iowa United States) - See all my reviews
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This review is from: Algebra and Geometry (Paperback)

This book is an introduction to group theory, complex numbers and linear algebra. In an unusual approach, the first topic covered is group theory, although it is only a small part of a small toe placed in the water. Chapters covering the set of real numbers, complex numbers and three-dimensional vectors follow this. Vector spaces, systems of linear equation, matrices, and eigenvectors are the topics of the next chapters. The final chapters are "Linear maps of Euclidean space", "Groups", "Mobius transformations", "Group actions" and "Hyperbolic geometry."
The overall theme is to unify the two areas of algebra and geometry by showing that space itself can be described algebraically by repeating the use of real numbers to create multiple dimensions. Large numbers of exercises are included at the ends of the sections, making it easy to assign homework or to further study by working longer and deeper. However, no solutions to the problems are included.
While this book certainly is a good one in these areas, I am puzzled as to where it would fit into an undergraduate curriculum. Abstract algebra and linear algebra are separate courses in the curriculum and having a course covering both would certainly be an oddity. However, that criticism aside, the author does an excellent job in explaining how space can be described and transformed by first representing it as a set of vectors and then transforming those vectors.

1 of 1 people found the following review helpful:

3.0 out of 5 stars Excellent book, bad scan into Kindle, December 27, 2011

By 

Schriftsteller - See all my reviews

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This review is from: Algebra and Geometry (Kindle Edition)

Other reviewers have addressed the merits of the book's content, which is excellent.

However, the Kindle edition is scanned so badly from print that it's almost unusable.

For example, in Section 4.2, the scalar product of two vectors x and y is defined thusly:

x dot y = x1<scanner left this mostly blank> + x2y2 + x3y3

Yes, an intelligent person can figure out what's missing. But you shouldn't pay $35 for a book and then need to "fill in the blanks." The entire content should be there.

Any proofreader would have caught that omission, and the eBook contains other similar omissions.

It is a disgrace that Cambridge UP would issue an eBook in such poor condition.