Linear Algebra: An Introduction to Abstract Mathematics [Hardcover]

Robert J. Valenza (Author)

Product Details

• Hardcover: 237 pages
• Publisher: Springer; 3rd edition (1993)
• Language: English
• ISBN-10: 3540940995
• ISBN-13: 978-3540940999
• Product Dimensions: 9.3 x 6.4 x 0.8 inches
• Shipping Weight: 1.2 pounds
• Average Customer Review: 4.3 out of 5 stars  See all reviews (6 customer reviews)

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35 of 36 people found the following review helpful:

5.0 out of 5 stars A Great Structural Approach to Linear Algebra!, December 8, 2000

By 

Brian Maurizi (Claremont, CA) - See all my reviews

This review is from: Linear Algebra: An Introduction to Abstract Mathematics (Undergraduate Texts in Mathematics) (Hardcover)

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! For instance, the fact that only square matrices can be inverted is a trivial consequence of a property of function mappings called "bijectivity." (a mapping from three- to two- dimensional space can't be bijective, for example) Many seemingly complex linear system problems can be simplified to a trivial questions by, for example, investigating the "span" of the column vectors of a matrix. There are countless problems that simply can't be understood without the kind of structural knowledge that Prof. Valenza's book gives.

Understanding the basic properties that underlie so many mathematical objects has been a true delight for me, and anyone who wants to know what is really going on "behind the scenes" with linear equations would be wise to investigate Prof. Valenza's book. It's no accident that he also wrote a book on Fourier Analysis; understanding structure is simply the key to higher math.

9 of 9 people found the following review helpful:

5.0 out of 5 stars excellent, June 30, 2004

By A Customer

This review is from: Linear Algebra: An Introduction to Abstract Mathematics (Undergraduate Texts in Mathematics) (Hardcover)

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 ....

6 of 6 people found the following review helpful:

5.0 out of 5 stars math is art ..., February 6, 2003

By A Customer

This review is from: Linear Algebra: An Introduction to Abstract Mathematics (Undergraduate Texts in Mathematics) (Hardcover)

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 !!