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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 linear algebra texts at this level, it introduces group theory early in the discussion, both as mathematical preparation for vector spaces and as a preliminary example of the rigorous development of informal axiomatic systems.

Topics covered include: sets and functions; groups and group homomorphisms; vector spaces and linear transformations; basis and dimension; matrices; representation of linear transformations; inner product spaces; determinants; eigenvectors and eigenvalues; and triangulation of endormorphisms and reduction to normal forms. A supplementary chapter discusses the connection between linear algebra and multivariate calculus and introduces categories and functors. ... Read more

Customer Reviews (5)

the ideal book for beginning undergraduate students
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.

Is this really good?
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.

excellent
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 ....

math is art ...
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 !!

A Great Structural Approach to Linear Algebra!
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 withoutthe 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. ... Read more

Product Description
This book is concerned with the study of algebras of linear transformation acting on finite-dimensional vector spaces, which in one guise or another arise in various parts of modern mathematics. The book's aims are: (i) to give an exposition of the basic theory of finite-dimensional algebras at a level that is appropriate for senior undergraduate and first-year graduate students, and (ii) to provide the mathematical foundation that prepares the reader for the advanced study of any one of several fields of mathematics. Indeed, the most important features of the book are the novelty of the selection of topics and the level of accessibility in which these topics are treated. The book features a relatively new approach to finite-dimensional operator algebras which is based on algebraic rather than functional-analytic methods. The reader who acquires a good understanding of the basic theory of algebras is well-positioned to appreciate results in operator algebras, representation theory, linear algebra, and ring theory. ... Read more

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Der vorliegende Band wurde für die Neuauflage von Aloys Krieg, einem Schüler von Herrn Koecher, ergänzt und aktualisiert. Wichtigste Ergänzungen sind der Spektralsatz für selbstadjungierte Endomorphismen in euklidischen und unitären Vektorräumen sowie die Anwendung der Jordanschen Normalform auf Differentialgleichungen. Auch sind neue Übungsaufgaben hinzugekommen.

Aus den Rezensionen: "... ein erfreulicher Lichtblick. Ohne die klare theoretische Linie zu verwirren, versteht der Autor Querverbindungen zur Geometrie, Algebra, Zahlentheorie und (Funktional-) Analysis immer wieder aufzuhellen. Zwischenkommentare helfen dabei ebenso wie die eingehenden historischen Notizen und Einschübe, insbesondere über Graßmann, Hamilton und Cayley sowie die Geschichte der Determinanten. Besondere Kapitel über die Elementargeometrie der Ebene des Raumes kommen endlich einmal auch auf nichttriviale Sätze zu sprechen; Feuerbachkreis und Euler-Gerade, Spiegelungspunkte und Sphärik. ... Studenten und Dozenten kann diese Buch wärmstens empfohlen werden." Zentralblatt für Mathematik

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Auf nahezu allen Gebieten der aktuellen Mathematik spielen heute Techniken der Linearen Algebra eine wichtige Rolle. Das Lehrbuch bietet eine systematische Einführung und entspricht dem Lehrstoff einer zweisemestrigen Einführungsveranstaltung. Besonderen Wert legt der Autor auf eine sorgfältige Entwicklung der in der Linearen Algebra gebräuchlichen Begriffsbildungen, wobei jedes Kapitel mit einer Darlegung der zugehörigen geometrischen Ideen beginnt. Ergänzend bietet der Band umfangreiches und themenbezogenes Übungsmaterial.

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Das Buch bietet einen idealen Einstieg in die Mathematik: Jedes Kapitel beginnt mit konkreten und vertrauten Begriffen oder Situationen. Davon ausgehend abstrahieren die Autoren schrittweise bis sie zu den Begriffen der modernen Mathematik kommen. Dabei wird auf Anwendungen mit einem engen Bezug zur Informatik wie etwa Routenplaner oder Codierungstheorie eingegangen. Das Buch ist so angelegt, dass jeder der drei Teile (Algebra, Analysis und Diskrete Strukturen) unabhängig voneinander gelesen und verstanden werden kann.

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Around 1978, a mutation of associative algebras was introducedto generalize the formalism of classical mechanics as well as quantummechanics.
This volume presents the first book devoted to a self-contained anddetailed treatment of the mathematical theory of mutation algebras,which is based on research in this subject over the past fifteenyears. The book also deals with a broader class of algebras, mutationsof alternative algebras, which are a natural generalization ofmutations of associative algebras. A complete structure theory,including automorphisms, derivations and certain representations, isgiven for mutations of artinian alternative algebras, and, inparticular, of Cayley--Dickson algebras. Since the mutation algebrasdo not form a variety, the structure theory explored in this volumetakes quite a different approach from the standard theory ofnonassociative algebras and provides an important interplay with thetheory of noncommutative (associative) algebras through mutationparameters. New simple algebras and open problems presented in thisbook will stimulate additional research and applications in the area.
This book will be valuable to graduate students, mathematicians andphysicists interested in applications of algebras.
... Read more

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Das vorliegende Buch bietet eine auf die Belange der mathematischen Grundausbildung der Informatiker zugeschnittene Einführung in die Lineare Algebra, die den Leser bis hin zu den Euklidischen Vektorräumen und der Hauptachsentransformation führt. Besonders interessant sind Anwendungen der Vektorrechnung in der Codierungstheorie, Anwendungen der Matrizenrechnung auf lineare Gleichungssysteme und elementare Rechenmethoden zur Invertierung und Zerlegung von Matrizen und zur Bestimmung von Eigenwerten. Dem Teil über Lineare Algebra geht ein breit angelegter Teil über Grundlagen der Mathematik und diskrete Mathematik voraus. Neben der Mengenlehre und der Einführung der Zahlen (mit einem Abschnitt über Rekursion) enthält das Buch Kapitel über Graphentheorie, algebraische Grundstrukturen (bis hin zum Rechnen in Booleschen Algebren), über Wahrscheinlichkeitsrechnung und eine Einführung in Fuzzy-Mengen. Mit vielen Beispielen und Anwendungen auch bestens zum Selbststudium geeignet. ... Read more

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This volume presents a course in linear algebra forundergraduate mathematics students. It is considerably wider in itsscope than most of the available methods and prepares the students foradvanced work in algebra, differential equations, and functionalanalysis. Therefore, for example, it is transformation-oriented ratherthan matrix oriented, and whenever possible results are proved forarbitrary vector spaces and not merely for finite-dimensional vectorspaces. Also, by proving results for vector spaces over arbitraryfields, rather than only the field of real or complex numbers, itprepares the way for the study of algebraic coding theory, automatatheory, and other subjects in theoretical computer science.
Topics are dealt with thoroughly, including ones that normally do notfeature in undergraduate textbooks, and many novel and challengingexercises are given. The fact that most students are computer-literateis taken into account, not so much by emphasizing computationalaspects of linear algebra which are best left to the computer, but byconcentrating on the theory behind it.
Audience: Recommended for a one-year undergraduate course inlinear algebra.
... Read more

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Text heavily biased toward functional methods. Discusses the Hankel norm approximation for the case of scalar rational functions. Paper. DLC: Algebras, Linear. ... Read more

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This book covers recent results in linear algebra with indefinite inner product. It includes applications to differential and difference equations with symmetries, matrix polynomials and Riccati equations. These applications are based on linear algebra in spaces with indefinite inner product. The latter forms an independent branch of linear algebra called indefinite linear algebra. This new subject is presented following the principles of a standard linear algebra course.

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This book provides a self-contained and accessible introduction to linear and multilinear algebra. Besides the standard techniques for linear and multilinear algebra many advanced topics are included. Emphasis is placed on the Kronecker product and tensor product. The Kronecker product has widespread applications in signal processing, discrete wavelets, statistical physics, computer graphics, fractals, quantum mechanics and quantum computing. All these fields are covered in detail. A key feature of the book is the many detailed worked-out examples. Computer algebra applications are also given. Each chapter includes useful exercises. The book is well suited for pure and applied mathematicians as well as theoretical physicists and engineers. New topics added to the second edition are: braid-like relations, Clebsch-Gordan expansion, nearest Kronecker product, Clifford and Pauli group, universal enveloping algebra, computer algebra and Kronecker product. ... Read more

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Basic Algebra and Advanced Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Together, the two books give the reader a global view of algebra and its role in mathematics as a whole, presenting the subject matter in a forward-looking way that takes into account its historical development.

Three prominent themes recur and blend together at times: the analogy between integers and polynomials in one variable over a field, the interplay between linear algebra and group theory, and the relationship between number theory and geometry. The exposition proceeds from the particular to the general, often providing examples well before a theory that incorporates them. The presentation includes blocks of problems that introduce additional topics and applications to science and engineering to guide further study. Many examples and hundreds of problems are included, along with separate sections giving hints or complete solutions for most of the problems.

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Chapters: Bivector. Source: Wikipedia. Pages: 208. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In mathematics, a bivector is a quantity in geometric algebra that generalises the idea of a vector. If a scalar is considered a zero dimensional quantity, and a vector is a one dimensional quantity, then a bivector can be thought of as two dimensional. Bivectors have applications in many areas of mathematics and physics. They are related to complex numbers in two dimensions and to both pseudovectors and quaternions in three dimensions. They can be used to generate rotations in any dimension, and are a useful tool for classifying such rotations. They also are used in physics, tying together a number of otherwise unrelated quantities. Bivectors are generated by the exterior product on vectors given two vectors a and b their exterior product is a bivector. But not all bivectors can be generated this way, and in higher dimensions a sum of exterior products is often needed. More precisely a bivector that requires only a single exterior product is simple; in two and three dimensions all bivectors are simple, but in higher dimensions this is not generally the case. The exterior product is antisymmetric, so negates the bivector, producing a rotation with the opposite sense, and is the zero bivector. Parallel plane segments with the same orientation and area corresponding to the same bivector .Geometrically, a simple bivector can be interpreted as an oriented plane segment, much as vectors can be thought of as directed line segments. Specifically for the bivector , its magnitude is the area of the parallelogram with edges a and b, its attitude that of any plane specified by a and b, and its orientation the sense of the rotation that would align a with b. It does not have a definite location or positi...More: http://booksllc.net/?id=884040 ... Read more

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Engineers and scientists need to have an introduction to the basics of linear algebra in a context they understand. Computer algebra systems make the manipulation of matrices and the determination of their properties a simple matter, and in practical applications such software is often essential. However, using this tool when learning about matrices, without first gaining a proper understanding of the underlying theory, limits the ability to use matrices and to apply them to new problems. This book explains matrices in the detail required by engineering or science students, and it discusses linear systems of ordinary differential equations. These students require a straightforward introduction to linear algebra illustrated by applications to which they can relate. It caters of the needs of undergraduate engineers in all disciplines, and provides considerable detail where it is likely to be helpful.According to the author the best way to understand the theory of matrices is by working simple exercises designed to emphasize the theory, that at the same time avoid distractions caused by unnecessary numerical calculations. Hence, examples and exercises in this book have been constructed in such a way that wherever calculations are necessary they are straightforward. For example, when a characteristic equation occurs, its roots (the eigenvalues of a matrix) can be found by inspection.The author of this book is Alan Jeffrey, Emeritus Professor of mathematics at the University of Newcastle upon Tyne. He has given courses on engineering mathematics at UK and US Universities. ... Read more

Customer Reviews (1)

Ray M. Bowen
At first, I found this book to be difficult reading. However, after a very serious attempt to discover the Author's intent, I truely began to cheerish the information presented in this book. One day I plan to contact Mr.Bowen, and maybe study with him for a time. ... Read more

Product Description

Diese Einführung in die lineare Algebra bietet einen sehr anschaulichen Zugang zum Thema. Die englische Originalausgabe wurde rasch zum Standardwerk in den Anfängerkursen des Massachusetts Institute of Technology sowie in vielen anderen nordamerikanischen Universitäten. Auch hierzulande ist dieses Buch als Grundstudiumsvorlesung für alle Studenten hervorragend lesbar. Darüber hinaus gibt es neue Impulse in der Mathematikausbildung und folgt dem Trend hin zu Anwendungen und Interdisziplinarität.

Inhaltlich umfasst das Werk die Grundkenntnisse und die wichtigsten Anwendungen der linearen Algebra und eignet sich hervorragend für Studierende der Ingenieurwissenschaften, Naturwissenschaften, Mathematik und Informatik, die einen modernen Zugang zum Einsatz der linearen Algebra suchen. Ganz klar liegt hierbei der Schwerpunkt auf den Anwendungen, ohne dabei die mathematische Strenge zu vernachlässigen. Im Buch wird die jeweils zugrundeliegende Theorie mit zahlreichen Beispielen aus der Elektrotechnik, der Informatik, der Physik, Biologie und den Wirtschaftswissenschaften direkt verknüpft. Zahlreiche Aufgaben mit Lösungen runden das Werk ab.

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Zum Anfang des Studiums sind Studierende der Ingenieurwissenschaften hauptsächlich mit Grundlagen beschäftigt, zu denen wesentlich die Mathematik gehört. Hier sind insbesondere die Analysis (in einer Variablen) und Lineare Algebra zu nennen, die zu oft eine große Hürde darstellen.

Mit unserem Buch wollen wir den Weg ebnen, indem wir Sie ausführlich – und ohne Umwege – mit dem genannten Stoff vertraut machen.

In einem verbindlichen, aber dennoch entspannten Stil, bringen wir Ihnen die wichtigen Methoden und Begriffe bei.

Besonderheiten:

• Zahlreiche Bilder und Beispiele.
• Viele begleitende Aufgaben mit vollständigen Lösungen.
• Klausuraufgaben mit kompletten Lösungen.
• Motivation und Verständnisfragen für jedes Kapitel.
• "Erste-Hilfe-Kurs" für Prüfungen.