Advanced Linear Algebra

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Springer Science & Business Media, Sep 20, 2007 - Mathematics - 525 pages
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This is a graduate textbook covering an especially broad range of topics. The first part of the book contains a careful but rapid discussion of the basics of linear algebra, including vector spaces, linear transformations, quotient spaces, and isomorphism theorems. The author then proceeds to modules, emphasizing a comparison with vector spaces. A thorough discussion of inner product spaces, eigenvalues, eigenvectors, and finite dimensional spectral theory follows, culminating in the finite dimensional spectral theorem for normal operators. The second part of the book is a collection of topics, including metric vector spaces, metric spaces, Hilbert spaces, tensor products, and affine geometry. The last chapter discusses the umbral calculus, an area of modern algebra with important applications.

For the third edition, the author has:

* added a new chapter on associative algebras that includes the well known characterizations of the finite-dimensional division algebras over the real field (a theorem of Frobenius) and over a finite field (Wedderburn's theorem);

* polished and refined some arguments (such as the discussion of reflexivity, the rational canonical form, best approximations and the definitions of tensor products);

* upgraded some proofs that were originally done only for finite-dimensional/rank cases;

* added new theorems, including the spectral mapping theorem and a theorem to the effect that , dim(V)=dim(V*) with equality if and only if V is finite-dimensional;

* corrected all known errors;

* the reference section has been enlarged considerably, with over a hundred references to books on linear algebra.

From the reviews of the second edition:

"In this 2nd edition, the author has rewritten the entire book and has added more than 100 pages of new materials. ... As in the previous edition, the text is well written and gives a thorough discussion of many topics of linear algebra and related fields. ... the exercises are rewritten and expanded. ... Overall, I found the book a very useful one. ... It is a suitable choice as a graduate text or as a reference book."

- Ali-Akbar Jafarian, ZentralblattMATH

"This is a formidable volume, a compendium of linear algebra theory, classical and modern ... . The development of the subject is elegant ... . The proofs are neat ... . The exercise sets are good, with occasional hints given for the solution of trickier problems. ... It represents linear algebra and does so comprehensively."

-Henry Ricardo, MathDL


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Part IBasic Linear Algebra
Part IITopics

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Page 9 - Two mxn matrices are equivalent if and only if they have the same rank. The...
Page 13 - Zn = {0,l,...,n— 1}, for some nonnegative integer n. A set that is not finite is infinite. The cardinal number (or cardinality,) of a finite set is just the number of elements in the set. 2) The cardinal number of the set N of natural numbers is NO (read "aleph nought"), where N is the first letter of the Hebrew alphabet. Hence, |N| = |Z| = |Q| = No 3) Any set with cardinality NO is called a countably infinite set and any finite or countably infinite set is called a countable set.
Page 13 - T The first part of the next theorem tells us that this is also true for infinite sets. The reader will no doubt recall that the power set P(S) of a set S is the set of all subsets of S. For finite sets, the power set of S is always bigger than the set itself. In fact, The second part of the next theorem says that the power set of any set 5 is bigger (has larger cardinality) than 5 itself. On the other hand, the third part of this theorem says that, for infinite sets S, the set of...
Page 12 - A' be a partially ordered set in which every chain has an upper bound. Then there exists a maximal element in X.
Page 10 - C or a finite field), this is relatively easy to do, but for other base fields (such as Q), it is extremely difficult. D Zorn 's Lemma In order to show that any vector space has a basis, we require a result known as Zorn's lemma. To state this lemma, we need some preliminary definitions. Definition A partially ordered set...

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About the author (2007)

Dr. Roman has authored 32 books, including a number of books on mathematics, such as Introduction to the Finance of Mathematics, Coding and Information Theory, and Field Theory, published by Springer-Verlag. He has also written Modules in Mathematics, a series of 15 small books designed for the general college-level liberal arts student. Besides his books for O'Reilly, Dr. Roman has written two other computer books, both published by Springer-Verlag.

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