PROJECT 2 - GENERAL VECTOR SPACES DISTRIBUTED: 27 MARCH 2017 DUE: 14 APRIL 2017 My original intention was to provide you with a paper describing vector spaces over fields other than the real numbers R. Vectors spaces over fields C, Q, F4 , etc. are extremely interesting and I'd be remiss to let you complete a linear algebra course without even a glancing encounter. However with the end of the semester approaching so quickly I simply do not have the time to complete such a paper and then it hit me; I should have YOU write the paper! What better way to learn about general vector spaces than to write a paper developing the analogous properties from real vector spaces? I am including the skeleton of my paper which includes some definitions and examples that must be included in your paper. In addition, you must include definitions for: subspace, linear independence, span, basis, and dimension along with appropriate examples (examples that are also real vector spaces are not sufficient). You do not have to include any proofs of theorems not included in my skeleton. The papers must be entirely typed and include both introduction and conclusion sections relating general vector spaces to real vector spaces. All references to our text should be appropriately cited and any Mathematica code used in relation to an example should be cited and included as an appendix. There should be NO mathematica code in your paper itself. Use of outside websites, professors, classmates, friends, or any other breathing or inanimate object is strictly prohibited and constitutes cheating. Use of other text books found in Andruss library however is permitted but proper footnote citations (including call number & page number) are required. Papers must be submitted via Dropbox by 17:00 on 14 April 2017. Guidelines: A paper that includes exactly the above requested information and all of worked out material from the attached sheet (all correct) and nothing else will earn a 70. Papers with all of this work included and correct but in addition, marginal exposition will earn a 73. To earn a higher grade, the paper should include additional examples, at least one after each definition, property, and theorem requested in the above description. The paper must also include exposition tying the concepts together while both relating the material back to real vector spaces and highlighting the differences. The paper should be a self-contained exposition on general vector spaces. Imagine that a friend was going to take a high level physics course and you wanted to give her a handbook on vector spaces that she could reference throughout the semester. This paper should present itself as such a reference. Please double-space your papers and use font size either 12 or 14. As these projects are submitted through Dropbox, there is no incentive to using a smaller font. Submissions must be in pdf format. Note: If you like the concepts studied in this project, courses you may want to talk to a professor about are Abstract Algebra I & II, Linear Algebra II, or independent studies in Representation Theory or Complex Variables. 1 Fields Definition. A field F is a set of elements called scalars, equipped with two operations, addition (a + b) and multiplication (ab), that satisfy the following axioms. Addition (A1) a + b F for all a, b F (A2) a + b = b + a for all a, b F (A3) (a + b) + c = a + (b + c) for all a, b, c F (A4) There exists a unique scalar 0 F , call zero such that a 0 = 0 a = 0 for all a F . (A5) For each a F , there exists a unique scalar a F called the negative of a such that a + (a) = 0. Multiplication (M1) ab F for all a, b F (M2) (a + b)c = ac + bc for all a, b, c F . (M3) ab = ba for all a, b F . (M4) (ab)c = a(bc) for all a, b, c F . (M5) There exists a unique scalar 1 F called the multiplicative identity such that for all a 1 = a for all a F . (M6) For each a F , where a 6= 0, there exists a unique scalar a1 called the multiplicative inverse of a such that a a1 = 1. Insert Theorem and Proof here: Show that in a field F , if ab = 0, then a = 0 or b = 0. Insert Example here: Show that the set of rational numbers Q is a field. 2|a, b Q} is a field. Insert Example here: Show that the set Q[ 2] = {a + b Note to Eric: I think the multiplicative inverse of a + b 2 may be: ab 2 . (a b 2)(a + b 2) Insert Example here: Show that the set Z is not a field. General Vector Spaces Definition. A vector space V over a field F is a nonempty set equipped with two operations, addition and scalar multiplication that satisfy the ten axioms of a vector space Note:(Include axioms) same axioms as given for a real v.s. on page 184 but with scalars in F and not R. The elements are called vectors. If F = R then V is called a real vector space, if F = Q then V is called a rational vector space, if F = C then V is called a complex vector space. We denote by F n an ordered n-tuple (a1 , a2 , ..., an ) where ai F for 1 i n. The operations of addition and scalar multiplication are defined component wise. Insert Insert Insert Insert Example Example Example Example Here: Here: Here: Here: Show that Q4 is a vector space over Q. Show that Qnis a vector space over Q for any n Z+ . Show that Q[ 2] is a vector space over Q. Determine a basis and the dimension of Q[ 2] over Q. 2