By Douglas R. Farenick (auth.)

ISBN-10: 0387950621

ISBN-13: 9780387950624

ISBN-10: 1461300975

ISBN-13: 9781461300977

The goal of this e-book is twofold: (i) to provide 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 starting place had to arrange the reader for the complex examine of a person of a number of fields of arithmetic. the topic less than examine is on no account new-indeed it truly is classical but a ebook that gives a simple and urban therapy of this concept turns out justified for numerous purposes. First, algebras and linear trans formations in a single guise or one other are general positive factors of varied elements of recent arithmetic. those comprise well-entrenched fields corresponding to repre sentation thought, in addition to more moderen ones resembling quantum teams. moment, a research ofthe hassle-free idea offinite-dimensional algebras is especially beneficial in motivating and casting gentle upon extra refined themes similar to module concept and operator algebras. certainly, the reader who acquires a superb knowing of the fundamental concept of algebras is wellpositioned to ap preciate ends up in operator algebras, illustration conception, and ring conception. In go back for his or her efforts, readers are rewarded by means of the implications themselves, a number of of that are basic theorems of impressive elegance.

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39 D. R. Farenick, Algebras of Linear Transformations © Springer-Verlag New York, Inc. 2001 40 An algebra 2. ALGEBRAS ~ is commutative if for all a, b E ab = ba ~; othe rwise, ~ is said to be noncommutative. We do not impose commutativity as one of t he axioms for an algebra , for doing so would exclude many interestin g mathematical objects t hat are by nature noncommutative. It is of course reasonable to question why one ought to retain t he associative law if the commutative law is so readily jettisoned.

Banach's work demonstrated profoundly the value of studying abstract vector spaces in their own right , independent of the particular disciplines to which theorems about vector spaces might apply. As an analyst, Banach worked mostly with the field lR.. But a wide array of different fields are of interest in algebra. More generally, E. Artin and E. 8 EXERCISES 33 vector space concept, and what resulted was the notion of a module over a ring, which contained vector spaces as a special case. Thus, when in the early 1930s linear algebra at last arose as an independent subject area, it was devoted to the study of modules and their homomorphisms.

PROOF. The proof of (1) is left to the reader (Exercise 13b). To prove 2(i), note that, for every ~ E 9Jl, and therefore (1). 0 11m is hermitian. Statement 2(ii) is a direct consequence of To prove the Spectral Theorem for normal linear transformations, we first establish it for hermitian linear transformations. 29 THEOREM. 1fT E ~(Sj) is hermitian, then Sj has an orthonormal basis consisting of eigenvectors of T . PROOF. Let all . . , ak be the distinct (real) eigenvalues of T . For each j the subspace ker(T - aj 1) consists of all eigenvectors of T corresponding to the eigenvalue aj; hence each ker(T - ajl) is invariant under T.

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

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