DOC PREVIEW
SF State MATH 880 - Outline 15

This preview shows page 1-2 out of 5 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 5 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 5 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 5 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

2008-02-28 11:08MATH 880 PROSEMINAR JT SMITHOUTLINE 15 SPRING 20081. Assignmenta. Continue formulating questions about the paper we’re outlining in class.i. Where should examples go?ii. What about history?iii. What general cleanup is appropriate?b. Continue formulating questions about the social organization of mathematics.2. In-class outlinea. I reviewed the informal paper I wrote on this subject in 1968 as a graduatestudent and found that the basic properties of dependence of vectors werestated in Van der Waerden [1931] 1953, volume 1, 99 ff. That was in fact thetext in the undergraduate course I took in 1959 that was analogous to Math335. Its first edition is considered the first comprehensive modern-algebratext. Here are the properties:i. All points in a finite set F of vectors depend on F.ii. If E, F are finite sets of vectors and each member of E depends on F,then each vector dependent on E depends on F.iii. If E is a finite set of vectors, q is a vector, and p is a vector dependenton F c {q} but not on F, then q depends on F c { p}.iv. A vector depends on a set X of vectors if and only if it depends on somefinite subset of X.b. Van der Waerden didn’t actually state property (iv), because he only consid-ered dependence on finite sets. But many authors after him noticed that hisrestriction wasn’t necessary. Van der Waerden proceeded to derive the maintheorems of linear algebra from those properties, and applied them to thestudy of algebraic dependence in fields.c. We revised the outline, incorporating this framework just after the introduc-tion. I massaged it a little further, to clean up some minor language andorganization flaws.d. This will cause us to rethink the introduction, later.e. We also organized the long list of remaining definitions and results into twogroups, on bases and on dimension.f. Finally, we inserted a placeholder for a reference to Van der Waerden [1937]1953, and I have constructed that reference.g. The current state of the outline is posted online and repeated at the end ofthe current document.3. Compound surnamesa. I learned recently that Dutch surnames beginning with Van should be alpha-betized under Van. So I’ve had to change my practice.b. This is confusing, because German names such as John von Neumann arealphabetized as Neumann, John von. And then Italian surnames such as DePaolis are alphabetized under De.Page 2 MATH 880 SPRING 2008 OUTLINE 152008-02-28 11:08c. Such details make a difference, because in a large bibliography, De Paolis andPaolis might be many pages apart!d. Punctuation marks in surnames such as Whiting-O’Keefe are equally perplex-ing. Moreover, there are blanks in the middle of Van der Waerden. Mycurrent practice is to treat all punctuation marks and blanks as though theyweren’t there. That causes Van der Bilt and Vanderbilt to be treated identi-cally, which is what you want.e. Occasionally names occur that don’t fit this scheme, or exceptions must bemade to accord with widespread public use. The goal of the rules in a manualsuch as Chicago 1993 is to minimize the number of ad hoc exceptions.4. Van der Waerden [1937] 1953a. I include both dates because I used the 1953 book, while the 1937 version maybe historically significant. We might want to discuss when this generalframework arose. Actually it might have been in the original 1931 addition,in which case I’d use the dates [1931] 1953. I’d also do that if we wished todiscuss the influence of this book on the development of modern-algebracourses. It might also be better to cite an English edition more recent than1953, with an ISBN number.b. I included the language about Artin and Noether, and about the translation,because it’s historically significant. I used exactly the language on the titlepage, rather than spend my time editing it.c. Should first names be used? For me that has been a perplexing question.i. Sometimes they’re required to distinguish people. For example, thereare two famous mathematicians named Noether, and two named Artin!ii. Sometimes they’re hard to find: Emil Artin and Emmy Noether areeasy. It’s rather harder to find Bartel Leendert van der Waerden!iii. Sometimes that search leads to distracting results, such as Julius Wil-helm Richard Dedekind, whom everyone knows merely as Richard.d. I included the original publication information as a favor to readers who mightwant to find out how something was originally phrased.e. I omitted some interesting lore from this reference, because it’s probablyirrelevant. The title and copyright pages identify the book as published andsold in the public interest as property confiscated from Springer during WorldWar II!5. Here’s the current state of the in-class outline.a. Introduction. THIS NEEDS RETHINKING.i. ??Should we explain the basics?(1) vector space(2) independent subset(3) spanning subsetii. The idea of the following arrangement is to base all the initial results onthe ideas in the introduction, and on the followingb. Van der Waerden’s frameworkMATH 880 SPRING 2008 OUTLINE 15 Page 32008-02-28 11:08i. 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. Note that each v 0 S depends on S.iv. Show that v depends on S if and only if it depends on a finite subsetof S.v. Show that if each member of a set R of vectors depends on S, theneach vector that depends on R also depends on S.vi. 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 nonzero, so has aninverse, and u would thus be a linear combination of vectors inS c { v}.vii. SHOULD THERE BE A RECAPITULATION HERE, THAT WE’LL BASE ALL FUR-THER RESULTS ONLY ON THE NOTION OF DEPENDENCE AND THE PROPERTIESJUST PROVED? c. Basesi. Span and independence (1) Define span S– of a set S of vectors.(2) State and prove some properties of the mapping S ² S–.(3) Define a set S of vectors to be independent if no v 0 S dependson S – {v}.ii. Theorem. The following properties of a set B of vectors are equivalent:(1) S is independent and span the whole vector space V.(2) B is a maximal independent subset of V.(3) B is a minimal spanning subset of V. iii.


View Full Document

SF State MATH 880 - Outline 15

Download Outline 15
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Outline 15 and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Outline 15 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?