Talk:Linear Algebra/Archive 1

Various
I am working on the section about determinants, but I find this difficult to do without the concepts of rank and matrix inverses. I would like to take this project on, but I think I need to do a lot of reorganizing. Please let me know if there are any objections to this.--Mike -

I was going to turn this into my next project, but then I realized it took me forever to do something in latex when I could write down on a piece of paper in 5 seconds. I gave up after writing the first section of the first chapter, which was exetremly short and still took me 2 hours. -Ghostal

Perhaps it would be a good idea to create pages specifically for linear algebra, rather than linking to the algebra ones? Then one can get further in depth with linear algebra concepts w/o confusing algebra students. I doubt anyone will mind since this project seem[ed] abandoned [as of 2005], but I don't want to start a flame war either (is a wikibooks n00b). --anonymousbob
 * Edits in brackets -- a correction to out-of-date discouraging information. Harold f (talk) 22:16, 12 March 2008 (UTC)

A new direction
I am a college student and has just finished studying linear Algebra I and II. I have found in my own studies, as well as teaching other students that geometric interpretations are the best way to introduce linear algebra concepts. It seems that this book tries to shove the math at the students without trying to teach it to them.--Jon513 12:28, 19 January 2006 (UTC)

Reorganization
I sugests a total reorganization of this book. It isn't very descriptive and does not explain things, it just gives out a lot of information but much of it does not make sense because of the lacking explanation. And I would suggest this book follow the outline provided by Whiteknight at his user page. If nobody from the community objects I will start working on the outline provided by Whiteknight, during next week, the first I'm going to change are the TOC. But if somebody has objections please say so... --Xharze 09:18, 9 October 2006 (UTC)


 * Thank you! I am very happy to find people who are willing to help with this project. I've been so busy lately that I just haven't had the time to actually fix this book and make it good. Please let me know if you need any help with this, I would love to help out any way I can. --Whiteknight (talk) (projects) 21:22, 8 October 2006 (UTC)


 * Just happy to help. I was thinking about starting the change today, as you have given the community a week to object and as far as I can see nobody has... So I'm starting to change the TOC today, and I hope it won't be a problem. --Xharze 09:18, 9 October 2006 (UTC)


 * Please do! thank you for helping with this project! --Whiteknight (talk) (projects) 13:08, 9 October 2006 (UTC)

Change of Basis
I just added the Change of Basis section; if anyone has any criticism (which I'm sure there could be), please feel free to edit! - Évariste 27 June 2007 6:00 PM

Current Development of This Book as of 8/07
Is anyone using this or developing further this book? I will be teaching a web course in the subject this fall and may wish to use parts of this.

George Dorner george@dorners.net

New table of contents
I am planning to further develop this book in the coming days. Firstly here is my plan for the new table of contents (1st 3 chapters) as I think that the current one contains many double entries and does not have an appropriate flow.


 * Chapter 1:Linear Equations
 * Matrices
 * Invertible matrices
 * Elementary row transformations
 * Row reduced echelon matrices
 * Systems of linear equations


 * Chapter 2:Vector Spaces
 * Vector spaces and subspaces
 * Linear dependence and independence
 * Bases and dimensions
 * Null spaces
 * Row and column spaces
 * Coordinates
 * Application:Interpolation
 * Application:The Wronskian


 * Chapter 3:Linear Transformation
 * Linear transformations
 * The algebra of linear transformations
 * Isomorphism
 * Linear functionals
 * The double dual
 * The transpose of a linear transformation

This can be followed by chapters on determinants, canonical forms and inner product spaces etc. I am waiting for comments from anyone interested. Cheers--Shahab 16:54, 1 September 2007 (UTC)

On vectors
I would like to remove all references to vectors as row or column matrices. They should be defined as elements of a vector space.--A (talk) 20:00, 9 December 2007 (UTC)

Moreover, I would like to base the whole book on Determinants. The logical dependence shall go like this: Determinants->Rank->Linear Dependance (vector space introduced here)->Dimension->Linear Transformations->Matrices (since their operations are based upon linear transformations, especially multiplication)->Range, Null space->Invariant space->Eigenvalues, Eigenvectors->... Where dimension and linear dependence play a role in the general solution of a system of m linear equations of n variables.--A (talk) 02:31, 10 December 2007 (UTC)


 * Good idea. I think that proofs of many theorems can be added to The Book of Mathematical Proofs to avoid the flow of the book becoming tedious.--Shahab (talk) 05:22, 11 December 2007 (UTC)

New Book
The uploaded new book is far superior to our existing linear algebra book. If there are no specific objections we should start the process of deleting/merging/redirecting the old pages as per their individual content. Basically only those pages which have topics not present in the new book should be retained. Meanwhile I'll try to properly format the new book.--Shahab (talk) 13:42, 13 March 2008 (UTC)
 * Since no objections have been raised I am going to remove the old table of contents, move the old pages to the pages to be merged section and bring the new table of contents on the top. Our focus should be formatting the book, since most of the content is already there. Cheers--Shahab (talk) 04:59, 19 March 2008 (UTC)


 * I've started changing the LaTeX format to MathML format. Going through the Determinant Property page, I noticed someone put the old content back in without fully reformatting the LaTeX.  It would probably be best to get a page down, and then merge old content. The "new", unformatted content is rather hard to read, and it doesn't make sense to have the same material repeated on the new layout.  --Retropunk (talk) 23:49, 31 January 2009 (UTC)

Order of presentation
In think that it would be in good order to be more cautious while incorporating a new book, in that the best process, instead of trashing the old and bringing in the new, and somehow trying to piece the old bits into the new one, is actually to carefully incorporate the new content.

In any case, while looking through the Linear Algebra book, I find it quite problematic to decide what order to present the following topics in: systems of linear equations, determinants and matrices, vector spaces, linear transformations. In a way, they are all quite dependent on the other. For example, systems of linear equations is dependent on some ideas on dimension, while the foundation determinants and matrices is in some way related to systems of linear equations. Systems of linear equations is also dependent on some ideas about vector spaces, due to dimension, and there is also the connection between determinants and matrices and vector spaces. But the theory of matrices is based on the theory of linear transformations, which in turn is based on vector spaces.

Thus, you get some kind of interdependency of these four topics. I favor the approach in Georgi E. Shilov's book, which introduces Determinants, and gets to matrices last as being based on linear transformations. There is a single drawback to this, and it is that it does not adequately explain the origin of the determinant. However, I think that explaining the origin of determinants could be quite cumbersome, and could be done later.--A (talk) 06:10, 1 August 2008 (UTC)
 * OK but please to note that the donated Jim Hefferon's book didn't follow this sequence and so you might have to do some rewriting.--Shahab (talk) 17:36, 2 August 2008 (UTC)
 * I'll be sure to do that. Regards, A (talk) 23:38, 4 August 2008 (UTC)

Yes, I think we are still somehow in a state of confusion as regards to the organization of algebra-related materials. (The organization of analysis stuff seems adequate, if not perfect.) As for linear algebra, in particular, I don't know any standard order of presentation of introductory materials, and I don't have any concrete suggestion (at least yet), but here are some of my thoughts. There seems to be three ways to view linear algebra. (i) Linear algebra as a standalone topic, with applications in a number of math or non-math related fields (e.g., systems of linear equations or coding theory). (ii) Linear algebra as a part of abstract algebra; i.e., essentially a subtopic in module theory (in this line of thinking, Jordan forms appears very naturally), and (iii) Linear algebra as functional analysis on a finite-dimensional spaces. I assume this book follows the view of (i), because Abstract Algebra should have (ii), and Functional Analysis would contain some of (iii). Therefore, it makes sense if this book emphasizes motivation behind linear algebra and its applications--materials that are hard to include in Abstract Algebra and Functional Analysis. So, the book might start with a discussion of how to solve a system of linear equations, and then introduces matrices, determinants (which can tell if systems of equations has non-trivial solutions), and so forth.

I personally would like to see a non-freshman linear algebra book; something that discusses representation theory and, most importantly, symplectic geometry. But I guess this is something for the future. -- Taku (talk) 13:44, 11 August 2008 (UTC)

Also, I am thinking that, in Functional Analysis, it might make sense to have a chapter on linear algebra after "Ch 3: Geometry of Banach spaces" before a chapter on operator algebras and spectral theory, because some topics in operational theory are simply algebra, requiring no analysis. I am not sure about this yet though. -- Taku (talk) 13:53, 11 August 2008 (UTC)


 * In the above post, I didn't mention one important point: that is, who is the target audience of the book, or put in another manner, what are prerequisites for the book? I believe that this book should be accessible to those with no prior exposure to either real-analysis or topology or even naive set-theory. Denoting the set of eigenvalues of $$T$$ by $$\sigma(T)$$ might not be a good idea, for example, if this is the case. -- Taku (talk) 12:42, 13 August 2008 (UTC)