2008-03-14 14:16MATH 880 PROSEMINAR JT SMITHIN-CLASS OUTLINE SPRING 2008This WordPerfect document is the current state of an outline, composed in class, of amodel expository paper. During each discussion I’ll edit it as appropriate, in a newtemporary copy. After the discussion, I’ll update this document from that, and includea “snapshot” of its current state in the corresponding class outline.Where to place historical information is undecided. In this document there are tentativeplaceholders where it might be appropriate to include that. But this strategy may resultin too many digressions, particularly if no coherent story line should be discovered to tiethe history together. Should that be the case, these items should go instead in the historysuggested near the end of the paper.1. This line is merely a placeholder to make the indentation correspond to that in thesnapshots.Dimension in Linear Algebra and Related Theoriesa. Introductioni. This paper presents the main features of the theory of linear depend-ence.(1) This theory includes several of the major concepts and theoremsof linear algebra,(2) commonly introduced in first courses in linear algeba.ii. But its context is considerably more general than that,(1) to permit a much broader scope of application. iii. The theory is presented in the context of a vector space V over a divi-sion ring K of scalars.(1) Footnote: an alternative term for division ring is skew field.(2) Postulates for a division ring.(3) Examples: real numbers, complex numbers, rational numbers.(Others later.) (4) Postulates for a vector space V over a division ring K, whoseelements are called scalars.(5) Examples: real and complex n-space. (Others later.)(6) Definition of a subspace of V.(7) Examples: {0}, V, lines and planes given by parametric equationsin any n-space, hyperplanes and hyperlines given by linear equa-tions in any n-space.iv. The presentation follows that of Van der Waerden [1937] 1953.(1) In this framework, (a) dependence of a vector on a set of vectors is defined first,(b) and few basic theorems (¿ › A BETTER TERM ?) are proved.Page 2 MATH 880 SPRING 2008 IN-CLASS OUTLINE2008-03-14 14:16(c) All subsequent definitions and theorems are based solely onthese.(2) This permits all these definitions and theorems to be carried intactinto the development of any theory that includes an analogousconcept of dependence that satisfies those few basic theorems.(3) Moreover, by eliminating distracting use of methods specific tovector spaces, the framework encourages presentation of the sim-plest proofs.v. The present context for this theory is in fact considerably more generalthan that of a first linear-algebra course.(1) Rather than requiring scalars to be real or complex numbers,(2) it permits use of scalars in some finite division rings, such as pfor prime p. The two-element ring 2 = { 0, 1} is often used in thetheory of computation.(3) And it even permits the scalars to lie in a noncommutative divisionring, such as the quaternions. This permits use of linear algebraat an earlier stage in the study of the foundations of geometry thanwould be possible otherwise, since commutativity of scalars is hardto derive from geometric axiom systems.(4) Rather than requiring the vector space to be finite-dimensional, theproofs presented here apply even to infinite-dimensional spaces.This permits use of linear algebra in studying the spaces of polyno-mials, formal power series, and continuous functions, for example.The examples could also include spaces of sequences of any finitelength, of all sequences, of all convergent sequences, etc.vi. MAYBE AN EXPLANATION IS NEEDED HERE THAT THIS PAPER WILL WILL NOT ATTEMPT TOCOVER THE THEORY OF MODULES OVER RINGS, WHICH SATISFY ALL THE POSTULATES FORVECTORS AND SCALARS EXCEPT ... .vii. A PARAGRAPH IS NEEDED HERE TO ENUMERATE THE SUBSEQUENT SECTIONS OF THE PAPER.b. Some historyi. The notion of a vector space was first codified by Giuseppe Peano in[1888] 2000, section 72. That work was an introduction to vector calcu-lus, under intense development at that time to support a myriad ofapplications in physics and engineering. Peano based his presentationon the pioneering but unwieldy [1844] 1994 system of Hermann Grass-mann, which never saw much direct use.ii. MAYBE A SPECIFIC EXAMPLE FROM PEANO [1888] 2000 COULD BE MENTIONED HERE.iii. The strategy of identifying certain algebraic structures as worthy ofintensive study—such as groups, rings, fields, and vector spaces—andgathering their properties into algebraic theories emerged during the1910–1930 decades through the publications and lectures of ErnstSteinitz, Emmy Noether, Emil Artin, and others. They were attemptingto make comprehensible a vast array of earlier algebraic studies, andprovide a framework for their extension. This organization was popular-ized in the 1931 first edition of Van der Waerden [1937] 1953.MATH 880 SPRING 2008 IN-CLASS OUTLINE Page 32008-03-14 14:16iv. That book remained for decades the standard introduction to abstractalgebra. But linear algebra, the theory of vector spaces, took a slightlydifferent turn. It took form later, but had reached its present form inBirkhoff and Mac Lane 1941, the standard English text in that subjectat least through the 1960s.v. The approach in Van der Waerden [1937] 1953, section 33, on which thepresent paper is based, was developed for presenting the theory oftranscendental extensions of fields: for example, the field obtained fromthe rational field by adjoining all algebraic expressions involving rationalnumbers and π.vi. HISTORICAL STUDIES ABOUT THOSE BOOKS ARE AVAILABLE. IT MIGHT BE POSSIBLE TO USETHEM AND REPORT MORE ABOUT THEIR AUTHORS’ MOTIVATIONS.c. Van der Waerden’s frameworki. Explain about the framework, in Van der Waerden [1937] 1953, section33.ii. Define dependence of a vector v on a set S of vectors.iii. Example from 3.iv. Note that each v 0 S depends on S.v. S h o w t h a t v depends on S if and only if it depends on a finite subsetof S.vi. Show that if each member of a set R of vectors depends on S, theneach vector that depends on R also depends on S.vii. Exchange Lemma. If a vector v is not dependent on a set S of vectors,but is dependent on S c { u } for some vector u, then u is dependenton S c { v}.(1) Proof. v would be the sum of a linear combination of vectors inX and tu for some scalar t. But t must be