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Linear Algebra: An Introduction to Abstract Mathematics (Undergraduate Texts in Mathematics) [Hardcover]

Robert J. Valenza
4.4 out of 5 stars  See all reviews (7 customer reviews)

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Book Description

February 1, 1999 0387940995 978-0387940991 Corrected
Based on lectures given at Claremont McKenna College, this text constitutes a substantial, abstract introduction to linear algebra. The presentation emphasizes the structural elements over the computational - for example by connecting matrices to linear transformations from the outset - and prepares the student for further study of abstract mathematics. Uniquely among algebra texts at this level, it introduces group theory early in the discussion, as an example of the rigorous development of informal axiomatic systems.

Product Details

  • Series: Undergraduate Texts in Mathematics
  • Hardcover: 237 pages
  • Publisher: Springer; Corrected edition (February 1, 1999)
  • Language: English
  • ISBN-10: 0387940995
  • ISBN-13: 978-0387940991
  • Product Dimensions: 0.6 x 6.1 x 9.1 inches
  • Shipping Weight: 1.1 pounds (View shipping rates and policies)
  • Average Customer Review: 4.4 out of 5 stars  See all reviews (7 customer reviews)
  • Amazon Best Sellers Rank: #1,705,176 in Books (See Top 100 in Books)

Customer Reviews

Most Helpful Customer Reviews
39 of 40 people found the following review helpful
5.0 out of 5 stars A Great Structural Approach to Linear Algebra! December 8, 2000
As a person who has a healthy interest in mathematics and has taken many classes, this is definatley one of the best! Professor Valenza taught it (he has been teaching this Linear Algebra class at CMC for ten years) and his book is essentially an excellent compilation of the lecture notes from his class. It takes a very different tack from most linear algebra texts: Usually, a linear algebra text begins by inroducing matrices and solving simultaneous equations, teaching computational methods. Prof. Valenza starts with the structure BEHIND all of that math however: Sets, Groups, and Vector Space properties. This structure is absolutely essential to knowing what's going on: My father took a (less superior) linear algebra class many years ago, and he never understood the concepts behind the mathematical manipulations; I actually sat down with him and taught him the things that I learned in Prof. Valenza's class. I really think that the knowledge in this book is invaluable to someone who wants to know what Linear Algebra is really about.
Just a few examples of the truly deep knowledge that this book communicates follows. For instance (this will ring a bell for those who have taken calculus) the "constant of integration" that must be added when doing an antiderivative is actually a property of group homomorphisms. The "absolute value" that must be introduced when taking square roots is structurally THE SAME property of group homomorphisms. Also, we all know that you can't divide by zero; it's just not allowed. But, the reason for that is ultimatley rooted in group theory; namely, the real numbers are NOT a group under multiplication. This type understanding has EVERYTHING to do with matrices and systems of equations!
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10 of 10 people found the following review helpful
5.0 out of 5 stars excellent June 30, 2004
By A Customer
This book treats the basic principles of abstract algebra.
It is targeted to graduate students that need a more theoretical approach to mathematics (instead of the usual calculus courses)
This book is the best introduction to abstract algebra for the following reasons
-its style : good introduction in each chapter, making the reader curious to read further.
-its rigor : everything is well explained in full details with proof.
-its elegance : This book treats the abstract structural aspects of algebra and then suddenly shows how more concrecte applications follow from these abstract results. This is the kind of elegance and style that makes mathematics an art : build a very abstract theory and then see how more concrete stuff follows immediately as special case of this abstract framework. This way, new things can be discovered and most of the time (as in this book), you can explain practical calculation rules in a short and rigorous way.
Definitely the finest there is ....
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6 of 6 people found the following review helpful
5.0 out of 5 stars math is art ... February 6, 2003
By A Customer
Great book !
If you read this book, you will not only gain knowledge of abstract algebra, but also understand clearly why mathematics is art. It was a real fun reading this book. The topics are presented in such a way that the author leads you to a climax, making you curious to read further, and help you to explore the beauty in all the ideas of abstract mathematics. This book is the best book I ever read on abstract algebra.

The emphasis in on rigor and abstract structural concepts. It is nice to see that the more practical applications follow as a special result from the abstract structural concepts. This is a very elegant approach !!
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6 of 6 people found the following review helpful
5.0 out of 5 stars the ideal book for beginning undergraduate students October 7, 2005
By xxx
I fully disagree with the one star review...

This is a beautiful book though you have to belong to a certain reader segment to appreciate it.

The readers that will like this book probabely are beginning undergraduate students that want to build a mathematical career and want a first and quick introduction to abstract mathematics. The reader is not overwhelmed by exotic topics that are rarely used, but is introduced to abstract basic principles needed to understand other courses like for instance quantum mechanics,more advanced graduate courses in algebra or functional analysis.

The power of this book is that it covers just enough material to have a solid foundation of algebra for other abstract courses like functional analysis,

When I compare it for instance with the book of Shilov, I strongly prefer this book since it is better organised, covers less topics, but enough to know the basics. This book succeeds in providing shorter proofs compared to Shilov without sacrifying rigor and clarity. How is this possible ?? Ah my friend, this is a reward coming from abstract reasoning as illustrated by this book.
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