2008-02-08 19:44MATH 880 PROSEMINAR JT SMITHOUTLINE 7 SPRING 20081. Assignmenta. Continue formulating questions about the social organization of mathematics.b. Continue formulating questions about the paper we’re outlining in class.2. Topics for in-class outline of an expository paper. We considereda. dependence and independence questions in linear algebrab. the fundamental theorem of finitely generated Abelian groupsc. the fundamental theorem of arithmeticd. Cauchy’s theorems in complex analysis.We chose the first.3. Outline of expository paper. I’ll keep a WordPerfect document posted with thecurrent state of the outline, and edit it live, according to class discussions. Aftereach discussion I’ll include a “snapshot” of its current state in the correspondingclass outline. Here’s the first snapshot.a. Introductionb. ??Should we explain the basics?c. Rank & nullity theoremd. Extension of the theory to infinite-dimensional casese. ??What features of the definition of “vector space” are really required for thistheory?f. References4. The introduction and references came first. (It really helps to establish somerequired parts first, to avoid the “blank page” hurdle.) Then the rank & nullityitem, to establish at least one major topic.5. I’ll use ?? to signal a question about something that might go at the place desig-nated. The question about basics came up last year and caused much discussion.Let’s count on doing that, but not until we get more major items down, so we havean idea of what the basics might be.6. The second major item appeared next, probably because it lies near one student’sown chosen topic. This led me to explain a little about the finite- and infinite-dimensional cases.a. You’re used to seeing an element of a vector space V called dependent on asubset S f V if it can be written as a finite sum of scalar multiples of ele-ments of S. That notion is not sensitive to that of the dimension of V.b. You’re used to classes that require V to be finite-dimensional: it shouldcontain a finite subset S on which every vector in V is dependent. Fromthat assumption, you derive several rather major theorems about bases anddimension.Page 2 MATH 880 SPRING 2008 OUTLINE 72008-02-08 19:44c. Can those theorems and the related definitions be formulated and provedwithout that hypothesis? Yes, for the most part. But you must allow thedimension of V to be an infinite cardinal.d. Is that a useful extension of the theory? To some extent. It is useful instudying dimensional aspects of the foundations of geometry, where you wantto isolate the places where dimension occurs in the basic notions. What betterthan to simply banish it from the assumptions? The infinite-dimensionaltheory is also useful in analyzing some basic notions in higher algebra. Butat least until quite recently, this notion has not seen much use in analysis.I’ve seen a few papers and one monograph in that area.e. In analysis there’s a competing notion of dependence that works for somevector spaces—for example, Hilbert space—that support a notion of conver-gent series. In such a space V, a vector might be called dependent on a subsetS f V if it can be written as the sum of a convergent infinite series of scalarmultiples of elements of S. Much of the theory of bases and dimension canbe modified to fit this situation, and the result is very useful in many applica-tion areas.f. But the results for the two sets of definitions may be different. For example,the dimension of a space is defined to be the cardinal of a minimal set S onwhich all vectors in V depend. For the finite-sum definition of dependence,the dimension of Hilbert space is 2ω, the cardinal of the set of all real num-bers. But for the convergent-infinite-series definition, the dimension is justω, the cardinal of the set of all natural numbers.7. The second question arose from an ill-formed inquiry phrased using the wordindependence. I interpreted it to refer to the question whether portions of thetheory in question are really dependent on all the other requirements for vectorspaces.a. You’re used to seeing vector-space theory developed with real scalars only,then extended by noting that nothing depended on that requirement, and acomparable theory exists for complex scalars. How about relaxing thatrequirement further? Many texts present the theory with an arbitrary fieldof scalars. But you learn in algebra classes about structures that are almostfields, but for which the commutative law fails—noncommutative fields, forexample the quaternions. (Fields, commutative or not, are also called divisionrings or skew fields.) Can vector-space theory be developed for scalars thatform even a noncommutative field?b. Yes, indeed. Reinhold Baer’s Linear Algebra and Projective Geometry developsthe entire theory for arbitrary, finite or infinite dimension, over arbitrary,commutative or skew fields. I’ll bring the book next time so that we canconstruct a reference to it.c. This tale of extensions to more and more general scalars can stretch yetfurther. Do you know where it