2008-02-14 18:54MATH 880 PROSEMINAR JT SMITHOUTLINE 9 SPRING 20081. Assignmenta. Continue formulating questions about the paper we’re outlining in class.b. Consider whether and when you might like to discuss material you’re workingon.c. Continue formulating questions about the social organization of mathematics.2. Events. The date for the “social” I announced will be at my house, Wednesday 27February, 6 to about 8 PM. See you there!3. In-class outline. We considered the question of further generalization of the scalars.a. We agreed that although some of these results can be generalized to apply tomodules over rings, where the scalars could be integers, integers modulo anonprime, or even polynomials, that theory diverges considerably fromstandard linear algebra.b. I noted that noncommutative scalars can be accommodated, provided determi-nants are left out of the mix. Some of the fundamental results you learn aboutdeterminants don’t work with noncommutative scalars, and I don’t think it’sclear even now exactly what results should take their place.c. I also noted that scalars with finite characteristic are okay, except that again,some fundamental results about determinants fail for characteristic 2.d. I suspect this means we should follow Axler’s lead: Down with Determinants!4. Current state of the outlinea. Introductionb. ??Should we explain the basics?c. Every vector space has a basis.d. Rank & nullity theoreme. Extension of the theory to infinite-dimensional casesf. ??What features of the definition of “vector space” are really required for thistheory?i. We can accommodate fields, perhaps noncommutative, of any character-istic. We don’t want to do modules over rings.g. ReferencesBaer, Reinhold. 1952. Linear Algebra and Projective Geometry. Pure andapplied mathematics, 2. New York: Academic Press. [Cited in ??.]5. Lemmas, propositions, theorems. Someone asked what’s the difference?a. Lemmas are usually uninteresting in themselves. If you see an argumentrepeated with minor variation in a proof, you might want to formulate it asa lemma with a separate proof, so that you can refer to it in a single lineinstead of repeating it.b. Sometimes a lemma acquires great interest a posteriori, though: for example,Zorn’s. It’s due not to Zorn but to Hausdorff and Kuratowski. But Zorn usedit particularly effectively.Page 2 MATH 880 SPRING 2008 OUTLINE 92008-02-14 18:54c. Sometimes a result is a compendium of a bunch of steps that each requirerather independent proofs. In that case it’s sometimes called a theorem, whilethe constituent parts are enunciated as propositions.d. But in other contexts, proposition may merely mean the meaning of somesentence.e. There’s no standard, but an author should be consistent within a single work,else be guilty of suggesting distinctions where there are