Download e-book for kindle: A Course in Algebra (Graduate Studies in Mathematics, Volume by E. B. Vinberg

By E. B. Vinberg

ISBN-10: 0821833189

ISBN-13: 9780821833186

It is a entire textbook on smooth algebra written through an the world over well known professional. It covers fabric typically present in complex undergraduate and uncomplicated graduate classes and provides it in a lucid type. the writer comprises nearly no technically tricky proofs, and reflecting his perspective on arithmetic, he attempts anywhere attainable to exchange calculations and tough deductions with conceptual proofs and to affiliate geometric pictures to algebraic gadgets. the hassle spent at the a part of scholars in soaking up those principles can pay off after they flip to fixing difficulties open air of this textbook.
Another very important function is the presentation of so much issues on numerous degrees, permitting scholars to maneuver easily from preliminary acquaintance with the topic to thorough research and a deeper figuring out. easy themes are integrated, reminiscent of algebraic constructions, linear algebra, polynomials, and teams, in addition to extra complicated issues, comparable to affine and projective areas, tensor algebra, Galois thought, Lie teams, and associative algebras and their representations. a few purposes of linear algebra and team conception to physics are mentioned.
The e-book is written with severe care and comprises over two hundred routines and 70 figures. it truly is perfect as a textbook and in addition appropriate for self sufficient examine for complex undergraduates and graduate scholars.

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Additional info for A Course in Algebra (Graduate Studies in Mathematics, Volume 56)

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Vector spaces V and U over a field K are called isomorphic if there exists a bijective map cp:V -U such that (i) cp(a + b) = W(a) + Wp(b) for any a, b E V; (ii) w(Aa) = Acp(a) for any A E K, a E V. If so, the map cp is called an isomorphism between V and U. 26 1. 2 that it is quite easy to describe vector spaces up to isomorphism. In particular, we will mostly concern ourselves in this book with the so-called finite-dimensional vector spaces; and all of them are isomorphic to spaces K. The key notion for this theory is the notion of a basis.

Bm). Thus, equivalent systems of vectors have the same rank. The definition of an elementary transformation implies that rows of a matrix A' obtained from another matrix A using an elementary transformation can be expressed as a linear combination of the rows of A. But as A can be obtained from A' using the inverse transformation, its rows can be expressed as a linear combination of the rows of A'. Therefore, the systems of rows of A and A' are equivalent and the ranks of these matrices are equal.

Determine the number of k-dimensional subspaces of an n-dimensional vector space over a field of q elements. The next theorem provides a complete description of all finite-dimensional vector spaces. 42. Finite-dimensional vector spaces over the same field are isomorphic if and only if their dimensions are the same. Proof. If f : V - U is an isomorphism of vector spaces and {el, e2, ... , f (en)} is a basis of U, hence dim V = 2. Elements of Linear Algebra 50 dim U. 66, every n-dimensional vector space over a field K is isomorphic to Kn; therefore, all such spaces are isomorphic.

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A Course in Algebra (Graduate Studies in Mathematics, Volume 56) by E. B. Vinberg


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