Product Description
There are a number of very good books available on linear algebra. From this one might deduce that the existing books contain all that one needs to know in the best possible form and that any new book would just repeat material in the old ones. However, new results in linear algebra appear constantly, as do new, simpler, and better proofs of old results. Many linear algebra results obtained in the past thirty years are accessible to undergraduate mathematics majors, but are usually ignored by textbooks. In addition, more than a few interesting old results are not covered in many books. In this book, Prasolov provides the basics of linear algebra, with an emphasis on new results and on nonstandard and interesting proofs. The book features about 230 problems with complete solutions. It would be a fine supplementary text for an undergraduate or graduate algebra course. ... Read more

Customer Reviews (1)

A good book on finite dimentional linear algebra
The book opens with determinants ... excepted for this point, I think this book is worth the money it costs.

All you need to know in linear algebra at undergraduate level is here. ... Read more

Product Description
The FUNDAMENTAL THEOREM OF ALGEBRA states that any complex polynomial must have a complex root. This basic result, whose first accepted proof was given by Gauss, lies really at the intersection of the theory of numbers and the theory of equations, and arises also in many other areas of mathematics. The purpose of this book is to examine three pairs of proofs of the theorem from three different areas of mathematics: abstract algebra, complex analysis and topology. The first proof in each pair is fairly straightforward and depends only on what could be considered elementary mathematics. However, each of these first proofs lends itself to generalizations, which in turn, lead to more general results from which the fundamental theorem can be deduced as a direct consequence. These general results constitute the second prooof in each pair. To arrive at each of the proofs, enough of the general theory of each relevant area is developed to understand the proof. In addition to the proofs and techniques themselves, many applications such as the insolvability of the quintic and the trascendence of e and pi are presented. Finally, a series of appendices give six additional proofs including a version of Gauss' original first proof.

The book is intended for junior/senior level undergraduate mathematics students or first year graduate students. It is ideal for a "capstone" course in mathematics. It could also be used as an alternative approach to an undergraduate abstract algebra course. Finally, because of the breadth of topics it covers it would also be ideal for a graduate course for mathmatics teachers. ... Read more

Customer Reviews (3)

Will please some and disappoint others
This book will please some and disappoint others, depending on what you seek. Incidentally it supplies 12 proofs of the theorem including Gauss' original proof in his doctoral thesis, half of these proofs being in the appendices.

The authors do a superb job of showing how disparate areas of mathematics can be brought to bear on a single problem from different directions. Many problems would have been suitable for such a purpose but the Fundamental Theorem of Algebra is particularly well chosen due to its importance and the fact that none of the proofs are long.

Yet despite my eagerness to read this book I was disappointed. While utilizing disparate areas of mathematics I wanted more insight into the theorem from almost every one of the proofs. The best proof I have seen of this theorem is in an appendix of G. H. Hardy's book A Course of Pure Mathematics. That proof gives insight as to why the theorem is true because it is a constructive proof (it actually constructs a root) that is easily understood, something not true of all constructive proofs. I was surprised to find Hardy's proof not present in this book in this form but only in a much weaker form (proof five) that is not constructive.

Asking for such clarity of insight is asking too much of most proofs of this theorem but I wanted more in that direction. As a specific example, this theorem can not be proved without appealing somewhere to continuity, which makes the proof itself partake of analysis and not purely algebra. The book does not point out in every proof (e.g., proof four) where such an appeal is made and I suspect it will often be hidden from students who may not grasp that continuity is an essential.

Nevertheless I did enjoy this book and definitely recommend it for the stated purpose of showing how disparate areas of mathematics can be applied to the same problem, for it accomplishes this purpose well. I applaud the authors for a superb effort although, like a literary critic reviewing a play but who can not write one, I wish there had been more. And of course, the authors may read my comments with astonishment and disbelief.

Finally, both mathematical comments in the two-star review by "A Customer" are incorrect; the book is actually correct on these points nor did I find others errors in it.

Too Poorly Written
To be sure, the aim of this book is interesting, but it's too poorly written. There are so many serious mistakes that I wonder why the students who took the course of these authors didn't point them out.
For example, the latter half of Lemma 4.1.5 is wrong for it lacks (-i), and Theorem 7.3.3 is wrong for |K:F| is not equal to the number of automorphisms of K fixing F, but is equal to the number of injective isomorphisms form K to algebraic closure of F fixing F.
I recommend you to read more reliable books.

A wonderful text!
This text gives a number of proofs of the Fundamental Theorem of Algebra (FTA).The proofs come from diverse areas of mathematics including complex analysis, abstract algebra (Galois theory) and topology.A version ofGauss's original proof is even given.The background information leadingup to each proof makes the book readable for a math major who has taken afirst course in abstract algebra.A student who uses this text will haveseen a lot of really interesting mathematics by the end and will have asense of how the mathematics can be used. ... Read more

Product Description
Various types of approximation theorems are frequently used ingeneral commutative algebra, and they have been found to be usefultools in valuation theory, the theory of Abelian lattice orderedgroups, multiplicative ideal theory, etc.
Part 1 of this volume is devoted to the investigation of approximationtheorems from a classical point of view. The chapters of this partdeal with fields and rings, partly ordered groups, and with multiringsand d-groups.
Part II investigates approximation theorems from a general,categorical point of view. This part is essentially self-contained andrequires only a basic knowledge of category theory and first-orderlogic.
For researchers and graduate students of commutative algebra, categorytheory, as well as applications of logic.
... Read more

Product Description
Chapters: C*-Algebras, Von Neumann Algebras, Approximately Finite Dimensional C*-Algebra, Commutation Theorem, Direct Integral, Spectrum of a C*-Algebra, Crossed Product, Hilbert C*-Module, Abelian Von Neumann Algebra, Compact Quantum Group, Tomita-takesaki Theory, Baum-connes Conjecture, Subfactor, Central Carrier, Wold Decomposition, Affiliated Operator, Multiplier Algebra, Dixmier Trace, Schröder-bernstein Theorems for Operator Algebras, Von Neumann Bicommutant Theorem, Bratteli Diagram, Reflexive Operator Algebra, Cuntz Algebra, Hyperfinite Type Ii Factor, Partial Isometry, Finite Dimensional Von Neumann Algebra, Uniformly Hyperfinite Algebra, Nuclear C*-Algebra, Universal C*-Algebra, Baer Ring, Ultrastrong Topology, Russo-dye Theorem, Nest Algebra, Hereditary C*-Subalgebra, Calkin Algebra, Ultraweak Topology, Octacube, Temperley-lieb Algebra, Toeplitz Algebra, Naimark's Problem, Enveloping Von Neumann Algebra, Kaplansky Density Theorem, Dirichlet Algebra, Locally Compact Quantum Group, Noncommutative Measure and Integration, Approximately Finite-Dimensional, Projectionless Algebra, Hilbert Algebra. Source: Wikipedia. Pages: 187. 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 von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann, motivated by the study of single operators, group representations, ergodic theory and quantum mechanics. His double commutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. Two basic examples of von Neumann algebras are as follows. The ring L(R) of essentially bounded measurable functions on the real line ...More: http://booksllc.net/?id=294185 ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

Product Description
Purchase includes free access to book updates online and a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: System of Imprimitivity, Kazhdan's Property, Tannaka-krein Duality, Group Algebra, Peter-weyl Theorem, Unitary Representation, Gårding Domain, Principal Series Representation, Quasiregular Representation, Representative Function, Mautner's Lemma. Excerpt:In mathematics , the group algebra is any of various constructions to assign to a locally compact group an operator algebra (or more generally a Banach algebra ), such that representations of the algebra are related to representations of the group. As such, they are similar to the group ring associated to a discrete group. Group algebras of topological groups: C c ( G ) For the purposes of functional analysis , and in particular of harmonic analysis , one wishes to carry over the group ring construction to topological groups G . In case G is a locally compact Hausdorff group , G carries an essentially unique left-invariant countably additive Borel measure called Haar measure . Using the Haar measure, one can define a convolution operation on the space C c ( G ) of complex-valued functions on G with compact support ; C c ( G ) can then be given any of various norms and the completion will be a group algebra. To define the convolution operation, let f and g be two functions in C c ( G ). For t in G , define The fact that f * g is continuous is immediate from the dominated convergence theorem . Also C c ( G ) also has a natural involution defined by: where is the modular function on G . With this involution, it is a *-algebra . Theorem . If C c ( G ) is given the norm it becomes is an involutive normed algebra with an approximate identity . The approximate identity can be indexed on a neighborhood basis of the identity consisting of compact sets. Indeed if V is a compact neighborhood of the identity, let f V be a non-negative con... ... Read more

Product Description
Chapters: Von Neumann Algebra, Commutation Theorem, Direct Integral, Crossed Product, Abelian Von Neumann Algebra, Tomita-takesaki Theory, Subfactor, Central Carrier, Affiliated Operator, Dixmier Trace, Schröder-bernstein Theorems for Operator Algebras, Von Neumann Bicommutant Theorem, Hyperfinite Type Ii Factor, Finite Dimensional Von Neumann Algebra, Baer Ring, Ultrastrong Topology, Ultraweak Topology, Octacube, Temperley-lieb Algebra, Kaplansky Density Theorem, Approximately Finite-Dimensional, Hilbert Algebra. Source: Wikipedia. Pages: 94. 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 von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann, motivated by the study of single operators, group representations, ergodic theory and quantum mechanics. His double commutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. Two basic examples of von Neumann algebras are as follows. The ring L(R) of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, which acts by pointwise multiplication on the Hilbert space L(R) of square integrable functions. The algebra B(H) of all bounded operators on a Hilbert space H is a von Neumann algebra, non-commutative if the Hilbert space has dimension at least 2. Von Neumann algebras were first studied by von Neumann (1929); he and Francis Murray developed the basic theory, under the original name of rings of operators, in a series of papers written in the 1930s and 1940s (F.J. Murray ... Read more

Product Description
Chapters: Fundamental Theorem of Algebra, Fundamental Theorem of Arithmetic, Finitely Generated Abelian Group, Fundamental Theorem on Homomorphisms, Thermodynamic Potential, Fundamental Theorem of Calculus, Fundamental Theorems of Welfare Economics, Fundamental Theorem of Combinatorial Enumeration, Fundamental Theorem of Galois Theory, Helmholtz Decomposition, Collineation, Fundamental Theorem of Cyclic Groups, Fundamental Theorem of Riemannian Geometry, Fisher's Fundamental Theorem of Natural Selection, Fundamental Theorem of Poker, Fundamental Lemma of Calculus of Variations, Fundamental Theorem of Linear Algebra, Fundamental Theorem of Arbitrage-Free Pricing, Fundamental Theorem of Curves, Fundamental Theorem of Linear Programming. Source: Wikipedia. Pages: 109. 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, the fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. Equivalently, the field of complex numbers is algebraically closed. Sometimes, this theorem is stated as: every non-zero single-variable polynomial with complex coefficients has exactly as many complex roots as its degree, if each root is counted up to its multiplicity. Although this at first appears to be a stronger statement, it is a direct consequence of the other form of the theorem, through the use of successive polynomial division by linear factors. In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use the completeness of the reals (or some other equivalent formulation of completeness), which is not an algebraic concept. Additionally, it is not fundamental for modern algebra; its name was given at a time in which algebra was mainly about solving polynomial equations with...More: http://booksllc.net/?id=51414 ... Read more

Product Description
In mathematics, the fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. Equivalently, the field of complex numbers is algebraically closed. Sometimes, this theorem is stated as: every non-zero single-variable polynomial, with complex coefficients, has exactly as many complex roots as its degree, if each root is counted up to its multiplicity. Although this at first appears to be a stronger statement, it is a direct consequence of the other form of the theorem, through the use of successive polynomial division by linear factors. ... Read more

Product Description
Chapters: Vector Space, Group, Linear Map, Polynomial, Euclidean Vector, Cauchy Sequence, Fundamental Theorem of Algebra, Power Set, Dimension, Binary Operation, Isomorphism, Tensor Product, Monoid, Universal Algebra, Algebraic Topology, Algebraic Number, Linear Combination, Linear Independence, Identity Element, Union, Permutation, Algebraic Structure, Formal Power Series, Idempotence, Associativity, Algebraically Closed Field, List of Group Theory Topics, Linear Span, Groupoid, Inverse Limit, Binary Function, Fundamental Theorem on Homomorphisms, Direct Product, Arity, Commutator, Unique Factorization Domain, Isomorphism Class, General Linear Group, Zero Divisor, Embedding, Distributivity, Magma, Homological Algebra, Row and Column Spaces, Complement, Inverse Element, Identity Matrix, Cross Product, Orthogonality, Special Linear Group, Additive Inverse, Eigenvalue, Eigenvector and Eigenspace, List of Algebraic Structures, Algebraic Number Field, Special Classes of Semigroups, Emmy Noether Bibliography, Problems in Loop Theory and Quasigroup Theory, Inverse Semigroup, Interior Algebra, Order of Operations, Light's Associativity Test, Information Algebra, Algebraic K-Theory, Module, Operad Theory, Seifert-van Kampen Theorem, Esquisse D'un Programme, Hecke Algebra, Commutativity, Cycle Graph, Zero-Product Property, Free Object, Valuation Ring, Higher-Dimensional Algebra, Multiplicative Inverse, Pseudo-Ring, Group Extension, Semiring, Semigroup With Two Elements, Cancellative Semigroup, Transpose, Direct Limit, Grothendieck Group, Closure With a Twist, Irreducible Polynomial, Polarization Identity, Involution, List of Abstract Algebra Topics, Variety, Scalar, Absolute Value, Double Groupoid, Cokernel, Dimension Theorem for Vector Spaces, Intersection, Homomorphic Secret Sharing, Near-Field, Racks and Quandles, Goodman-nguyen-van Fraassen Algebra, Conditional Event Algebra, Nearring, Pseudogroup, Anticommutativity, Composition R...More: http://booksllc.net/?id=32370 ... Read more

Product Description
Chapters: Field, P-Adic Number, Fundamental Theorem of Algebra, Hyperreal Number, Galois Theory, Finite Field, Algebraically Closed Field, Galois Group, Field of Fractions, Local Field, Number System, Algebraic Number Field, Valuation, Rational Number, Archimedean Property, Real Closed Field, Fundamental Theorem of Galois Theory, Glossary of Field Theory, Eisenstein's Criterion, Tensor Product of Fields, Dirichlet's Unit Theorem, Cubic Field, Rational Variety, P-Adically Closed Field, Exponential Field, Iwasawa Theory, Characteristic, Transcendence Degree, Construction of Splitting Fields, Quadratic Field, Kummer Theory, Differential Galois Theory, Polynomial Basis, Perfect Field, Primitive Polynomial, Thin Set, Primitive Element Theorem, Stark Conjectures, Additive Polynomial, Brumer-stark Conjecture, Grothendieck's Galois Theory, Separable Polynomial, Quasi-Finite Field, Generic Polynomial, Quasi-Algebraically Closed Field, Global Field, Serre's Conjecture Ii, Field Norm, Multiplicative Group, Embedding Problem, Conjugate Element, Pseudo Algebraically Closed Field, Levi-Civita Field, Tschirnhaus Transformation, List of Number Fields With Class Number One, Normal Basis, Minimal Polynomial, Adjunction, Superreal Number, Field Trace, Change of Bases, Totally Real Number Field, Cm-Field, Krasner's Lemma, All One Polynomial, Formally Real Field, Strassmann's Theorem, Norm Form, Equivariant L-Function, Function Field Sieve, Isomorphism Extension Theorem, Euclidean Field, Composite Field, Equally Spaced Polynomial, Complete Field. Source: Wikipedia. Pages: 335. 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, an algebraic number field (or simply number field) F is a finite (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q ...More: http://booksllc.net/?id=223788 ... Read more

Product Description
High Quality Content by WIKIPEDIA articles! Symbolic computation or algebraic computation, relates to the use of machines, such as computers, to manipulate mathematical equations and expressions in symbolic form, as opposed to manipulating the approximations of specific numerical quantities represented by those symbols. Such a system might be used for symbolic integration or differentiation, substitution of one expression into another, simplification of an expression, etc. Symbolic computation is also sometimes referred to as symbolic manipulation, symbolic processing, symbolic mathematics, or symbolic algebra, but these terms also refer to non-computational manipulation. ... Read more