Download PDF by E. B. Vinberg: A Course in Algebra (Graduate Studies in Mathematics, Volume

By E. B. Vinberg

ISBN-10: 0821833189

ISBN-13: 9780821833186

This can be a complete textbook on glossy algebra written via an across the world well known professional. It covers fabric commonly present in complex undergraduate and uncomplicated graduate classes and offers it in a lucid sort. the writer contains nearly no technically tricky proofs, and reflecting his perspective on arithmetic, he attempts anywhere attainable to exchange calculations and hard deductions with conceptual proofs and to affiliate geometric photos to algebraic items. the hassle spent at the a part of scholars in soaking up those rules pays off after they flip to fixing difficulties outdoor of this textbook.
Another very important function is the presentation of so much themes on a number of degrees, permitting scholars to maneuver easily from preliminary acquaintance with the topic to thorough learn and a deeper figuring out. easy subject matters are integrated, equivalent to algebraic buildings, linear algebra, polynomials, and teams, in addition to extra complicated themes, corresponding to affine and projective areas, tensor algebra, Galois conception, Lie teams, and associative algebras and their representations. a few functions of linear algebra and crew conception to physics are mentioned.
The booklet is written with severe care and includes over two hundred workouts and 70 figures. it's perfect as a textbook and likewise appropriate for self sufficient learn for complex undergraduates and graduate scholars.

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

Example text

A square matrix has two diagonals. The one leading from the upper left corner to the lower right corner is called the main diagonal or simply the diagonal and the other one the secondary diagonal. A square matrix is called diagonal if all its entries outside of the (main) diagonal are zero. Multiplication by a diagonal matrix looks especially simple: a1 0 ... 0 b11 0 a2 ... 0 b21 o o ... an b1 b12 ... b22 ... b1p b2 bnP = aibll a1b12 ... a2b21 a2b22 ... anbnl anbn2 ... albla a2b2p I (every row of the second matrix is multiplied by the respective diagonal entry of the first matrix).

Vector Spaces In elementary geometry, we not only add vectors but also multiply them by numbers. Analyzing properties of these two operations, we come to the notion of a vector space. 24 1. Algebraic Structures Before actually defining it, we should note that here we step away from what we previously meant by an operation. Multiplication of a vector by a number is not an operation on two elements of the same set. It is an operation that assigns to each pair (number, vector) another vector. This is how things also are in the general definition of a vector space; however, there elements of an arbitrary (fixed) field replace real numbers.

The linearly independent system {el, e2i... , eh l can be completed to a basis of V. Thus, if U # V, dim V > k. 41. 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, ...

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

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