Algebras of Linear Transformations by Douglas R. Farenick (auth.)

By Douglas R. Farenick (auth.)

The goal of this booklet is twofold: (i) to offer an exposition of the elemental thought of finite-dimensional algebras at a levelthat isappropriate for senior undergraduate and first-year graduate scholars, and (ii) to supply the mathematical origin had to arrange the reader for the complex examine of an individual of a number of fields of arithmetic. the topic below research is in no way new-indeed it really is classical­ but a publication that provides an easy and urban remedy of this concept turns out justified for numerous purposes. First, algebras and linear trans­ formations in a single guise or one other are commonplace positive aspects of assorted elements of recent arithmetic. those contain well-entrenched fields similar to repre­ sentation thought, in addition to more moderen ones corresponding to quantum teams. moment, a learn ofthe uncomplicated conception offinite-dimensional algebras is especially precious in motivating and casting gentle upon extra subtle issues comparable to module conception and operator algebras. certainly, the reader who acquires an exceptional knowing of the elemental conception of algebras is wellpositioned to ap­ preciate ends up in operator algebras, illustration conception, and ring thought. In go back for his or her efforts, readers are rewarded by way of the consequences themselves, numerous of that are basic theorems of notable elegance.

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L"I, then the vectors ~ and TJ are orthogonal. 1. If A E A~ and 1. LINEAR ALGEBRA 22 PROOF. To prove (1) observe that (T - A1)* = T* - Xl and that T - Al is normal ; hence, o= IIT~ - A~II = II(T - A1)~11 = To prove (2), use the fact that A - I-l II(T - t- 0 and (A -I-l)(~ , 1]) = (A~ , 1]) - (Cii1]) = (T~ , 1]) - (~ , T*1]) A1)*~1I = IIT*~ - X~II · compute (by statement (1)) = (T~, 1]) - (T~ , 1]) =0. Hence, (~, 1]) = O. 0 Thus , if ~ is an eigenvector of a normal transformation T , then ~ is also an eigenvector of T* .

In- IC- 1 : Ij E IF, for all j }, where ( satisfies th e relation n L aj(j = o. j=O We observe that th e equat ion above is simply f(() = 0, presuming that it indeed makes sense to evaluate f at ( . 31, IF [xll(f) is a field extension of IF; it is, in addition, a finite-dimensional vector space over IF. Evidently, {I , ( , ( 2, .. , ( n- I} is a spanning set for IF [xli (f) . Moreover, 1, ( , (2, . ,(n-I are linearly independent over IF because, otherwise, f would admit nonconstant polynomial factors of lesser degree.

Answer the question for any field IF of characteristic zero. b. Answer the question for the field IF = 7l/271 of integers modulo 2 and n = 2. 3. Let 23 = {VI,VZ,V3} be a basis for C3 defined by VI = el - e3, Vz = ez + e3, V3 = el + e3. Find the basis of the dual space (C 3)* that is dual to the basis 23. 4. Prove that in a finite-dimensional vector space V, VI = Vz (for vectors VI,VZ E V) if and only if cp(VI) = cp(vz) for every cp E V*. 5. Let V be a finite-dimensional vector space, and let I' : V --t V** be the map such that, for every V E V, the action of fv on each cp E V* is fv(cp) = cp(v) .

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