Customer Reviews

Linear Algebra: An Introduction to Abstract Mathematics

5 star:		(5)
4 star:		(0)
3 star:		(0)
2 star:		(0)
1 star:		(1)

 

 

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.

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 ....

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 !!

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.

Great book. As a former college math professor who taught linear algebra to both math majors and non-majors, I wish this book had been available for use with the math majors. Linear algebra is often taught as a series of cookbook exercises involving using matrices to solve systems of equations, but that approach misses the beauty of the subject. Math majors should see linear algebra as a building block for abstract algebra, and this book performs that task very well.

I have just had a glimpse of all the three reviews here, and wonder whether the reviews are really true. I am now an undergraduate and taking a course which uses this textbook. Frankly speaking, I think this book sucks. If you carefully compare this book with Friedberg, Insel and Spencer's Linear Algebra, you will know the difference. This book misses out many things I consider important. Also, it does not contain many interesting and essential results, like all infinite-dimensional vector space has a basis. I don't think Zorn's lemma should not be mentioned in a book like this one.
Also, I can't really accept that it starts matrices like that. We all know that matrices come from linear transformations. Matrix multiplication comes from the composition of linear transformatiosn. This book does not mention this and start it right away. This approach is quite bad in my opinion.
Finally, I think that this book is messy. I think it is quite difficult to include introduction of linear algebra and abstract algebra at the same time, but if it's so difficult to do it, why don't you study it separately? I think the author tried hard to combine these two stuff, but failed to do that successfully. All my friends who have linear algebra and abstract algebra background agree with me. I definitely wont' recommend this book.