By Douglas R. Farenick (auth.)
The objective of this ebook is twofold: (i) to provide an exposition of the fundamental concept of finite-dimensional algebras at a levelthat isappropriate for senior undergraduate and first-year graduate scholars, and (ii) to supply the mathematical beginning had to arrange the reader for the complex learn of an individual of a number of fields of arithmetic. the topic less than research is not at all new-indeed it's classical but a ebook that provides an easy and urban remedy of this thought turns out justified for numerous purposes. First, algebras and linear trans formations in a single guise or one other are normal good points of varied elements of contemporary arithmetic. those contain well-entrenched fields similar to repre sentation concept, in addition to more moderen ones reminiscent of quantum teams. moment, a examine ofthe common concept offinite-dimensional algebras is very necessary in motivating and casting mild upon extra subtle themes comparable to module conception and operator algebras. certainly, the reader who acquires a superb figuring out of the fundamental idea of algebras is wellpositioned to ap preciate ends up in operator algebras, illustration thought, and ring idea. In go back for his or her efforts, readers are rewarded by way of the consequences themselves, numerous of that are primary theorems of amazing elegance.
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Extra resources for Algebras of Linear Transformations
The purpose of this section is to prove that, yes, every vector space, whether it is finite- or infinite-dimensional, does indeed have a basis. 35 PROPOSITION . Let ~ be a nonempty subset of a nonzero vector space V. The following statements are equivalent. 1. ~ is a basis of V. 2. The set ~ is a maximal set of linearly independent vectors, where the word maximal is used in the sense that if ~ <:;;; ~I, and the elements of ~' are also linearly independent, then ~' = ~ . If statement (1) is true, then assume that ~' is a set in which the elements of ~' are linearly independent and ~ <:;;; ~' .
5 Fields and Field Extensions Let JK be any field containing IF as a subfield. Then JK can be considered as a vector space over IF, where the operations of scalar multiplication and vector sum are the operations of product and sum in the field K Very often extensions JK of a field IF arise from algebraic extensions, as described in what follows. Let IF [x] denote the ring of all polynomials zt. with indeterminant x, coefficients 0:0, . . , O:g E IF, and arbitrary 9 E We recall from the theory of polynomial rings the following important theorem.
LINEAR ALGEBRA 24 The second item indicates that if T is normal, then the action of T on any vector ~ is simply multiplication of each Fourier coefficient (~ ,