Download e-book for kindle: Finite-Dimensional Vector Spaces (Undergraduate Texts in by Paul R. Halmos

By Paul R. Halmos

ISBN-10: 0387900934

ISBN-13: 9780387900933

From the reports: "The conception is systematically constructed by way of the axiomatic approach that has, on the grounds that von Neumann, ruled the overall method of linear sensible research and that achieves the following a excessive measure of lucidity and clarity....The publication comprises approximately 350 good positioned and instructive difficulties, which disguise a substantial a part of the topic. All in all this is often a very good paintings, of both excessive worth for either scholar and teacher." --ZENTRALBLATT FÜR MATHEMATIK

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Example text

It is important to observe that the theorem shows not only that '0 and '0" are isomorphic-this much is trivial from the fact that they have the same dimension-but that the natural correspondence is an isomorphism. This property of vector spaces is called reflexivity; every finite-dimensional vector space is reflexive. It is frequently convenient to be mildly sloppy about '0": for finitedimensional vector spaces we shall identify '0" with '0 (by the natural isomorphism), and we shall say that the element Zo of '0" is the same as the element Xo of '0 whenever zo(Y) = [xo, y] for all yin '0'.

There is, nevertheless, something unique about the factorization, namely, whether the number of transpositions needed is even or odd. We proceed to state this result precisely, and to prove it. Assume, for simplicity of notation, that k = 4. Let f be the polynomial (in four variables tl, t2, t 3, t4) defined by (In the general case f is the product of all. ) Each permutation 1r in S4 converts f into a new polynomial, denoted by 1rf; by definition (1rf)(tl, t 2 , ta, t4) = fCt,,(1)' t,,(2), t r (3)' t,,(4»).

9 eR", then there exist scalars (3i; such that (in the notation of (a» z(w) = L. Li a;;(3,j for every w. The scalars (3;; are uniquely determined by z. 2. 9 '0 is degenerate if, as a function of one of its two arguments, it vanishes identically for some non-zero value of its other argument; otherwise it is non-degenerate. 9 e2• (b) Give an example of a non-degenerate bilinear form on e2 ffi e2• SPACES 38 SEC. 24 3. If w is a bilinear form on 'l1EB '0, if Yo is in '0, and if a function y is defined on 'l1 by y(x) = w(x, Yo), then y is a linear functional on'll.

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Finite-Dimensional Vector Spaces (Undergraduate Texts in Mathematics) by Paul R. Halmos

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