Talk:Linear algebra

Content

We should cover:

High school topics:

• Matrix representation of linear equations
• Resolution: Gauss and Crammer. Determinants.
• Matrix multiplication.
• Matrix basic operations (addition, traspose, ...)

High school / University ?:

• Bilinear forms
• Quadratic forms

University topics:

• Properties of a linear system
• Vector spaces and subspaces
• Matrix inversion
• Bases. Orthogonal and orthonormal bases.
• Elementary matrices
• Similarity transformations
• Gram-Schmidt orthonormalization
• ...?
• Diagonalization
  • ...
  • ...
  • Finding orthogonal and orthonormal bases
  • Diagonalization of bilinear/quadratic forms
    • Positive-definite, negative-definite, semidefinite and indefinite matrices. Scalar product
  • Matrix exponentiation by diagonalizing
• Jordan canonical form
• Factorization methods
  • LU factorization
  • Cholesky
  • Singular value decomposition
• Applications
  • Matrix differential equations
  • Least squares fitting
  • Principle components analysis

Feel free to edit the list. It isn't complete at all, but I'm tired... --Jorge 18:51, 12 April 2007 (UTC)

I added some :-) --HappyCamper 15:21, 13 April 2007 (UTC)

They thaught you quadratic forms in high school? Wow. Where are you from? --Jorge 22:50, 30 April 2007 (UTC)

Teachers make all the difference :-) Anyway, should we keep on adding more items to this list? Or just start collecting materials? --HappyCamper 05:42, 2 May 2007 (UTC)

In general, I think we should avoid duplicating information already available on wikipedia. We can just add the links to this information, instead. What we can do though, is add some good exercises/examples. This is stuff that shouldn't be on wikipedia, but should be on wikiversity. I think I'll begin working on the discussion of eigenvalues and eigenvectors, since it is such a powerful concept in mathematics.132.235.38.246 13:11, 28 April 2009 (UTC)

I respectfully disagree. This resource should be self-contained and not merely an addendum to Wikipedia's article on linear algebra (which is still god-awful, twelve years later). I'll try to help flesh it out a bit. AP295 (discuss • contribs) 14:51, 11 March 2021 (UTC)

@AP295: Please note that Wikiversity is not a place for duplication of other Wikimedia projects. Wikiversity has adopted a "learn by doing" model for online education and encourages the creation of learning activities for Wikiversity participants. A self-contained resource would necessarily duplicate content that belongs at Wikipedia or Wikibooks. -- Dave Braunschweig (discuss • contribs) 21:33, 11 March 2021 (UTC)

@Dave Braunschweig: It was my understanding that a learning project should contain more or less the same material as a college course on the subject, and be similarly organized. Wikipedia's summary style would not jibe with this, it's apples-to-oranges. I don't intend to copy any material off of Wikipedia, but attempting to maintain totally disjoint sets of information between the two projects would be awkward and nonsensical. Wouldn't you agree? AP295 (discuss • contribs) 22:14, 11 March 2021 (UTC)

@AP295: "Contain" is a relative term. For example, see Internet Fundamentals. This is a college-level course on the subject which a Google search shows as the definition of Internet fundamentals. The course contains very little content. Instead, it links to content and engages users in ways they can learn and apply that content themselves. Many of our top resources follow this style effectively. And by effectively, I mean that they draw a relatively large number of users for both personal and academic use with very little disruption and very little maintenance required. IT Fundamentals is another example. Not only is it a top Google search result, but CompTIA recommends it to their users who can't afford to purchase proprietary materials on the subject.

If you're convinced that you have a better way to present this information that is both effective and engaging, go for it. Just don't underestimate the amount of effort required to build an effective course. Even with linking content, it typically takes me four to six months to develop a course. The benefit of Creative Commons licenses is being able to build on the works of others rather than having to reinvent those wheels. -- Dave Braunschweig (discuss • contribs) 22:32, 11 March 2021 (UTC)

@Dave Braunschweig: Sorry, I'm not sure what you're trying to say. Again, I'm not going to copy any of Wikipedia's content or even use a similar organization/style. I will link to content when I feel it's appropriate, I'm not here to reinvent the wheel. AP295 (discuss • contribs) 22:47, 11 March 2021 (UTC)

And actually, that's part of the problem. There are many topics for which there does not exist high-quality material under Creative Commons. If I feel that a Wikipedia article is relevant and of acceptable quality, I will certainly link it. If I cannot find a quality source elsewhere on Wikimedia or if Wikipedia's summary style is not appropriate in the context, I will write it myself. I understand that developing a course takes time, and I don't have that much time to spare, so it will be a slow, incremental process unless there are others who can contribute more often. I don't see a problem with that. AP295 (discuss • contribs) 22:58, 11 March 2021 (UTC)

@Dave Braunschweig: One last thing, just so you and I are clear. I want Wikipedia and Wikiversity to both be reliable sources of high-quality information, and I have no problem referencing content from Wikipedia. While it may seem like I have an axe to grind from what's on my user page, I very much want to contribute to the body of high-quality free/"open-source" information on the internet. However, I cannot in good conscious contribute to any Wikimedia project without first stating what I've written on my user page. To do so would condone and reinforce the status quo, and my efforts would ultimately be fruitless. Education comes first, so I have no recourse but to participate in a high-profile project like this one, otherwise whatever content I contribute would likely fade into obscurity. I was blocked from Wikipedia because censorship is the only nonviolent method to silence persistent malcontents. I do not bear any real grudge, I knew exactly what Wikipedia was before getting involved. Here, I'll try to stay away from politics outside my user page so they don't have any excuse to block me from this project also. I hope you and I can get along. AP295 (discuss • contribs) 23:39, 11 March 2021 (UTC)

Content planning

Since it has been almost twelve years since anyone has commented on this talk page, and the article is still very much incomplete, I'd like to put forth a new game plan.

Undergraduate courses (at least where I went to school) put an emphasis on computations with real-valued matrices, for example, matrix/vector operations and algorithms like gaussian elimination. Somewhat more advanced courses treat the problem more generally, with a focus on linear maps and bases, but are not necessarily harder and tend to cover a similar range of topics. I'd like to strike a middle ground with this resource, slightly leaning toward the latter style and putting more emphasis on the fundamental ideas rather than rote computation. Thoughts? AP295 (discuss • contribs) 14:35, 12 March 2021 (UTC)

To familiarize the reader with basic matrix operations, I'll probably refer them to this video: https://www.youtube.com/watch?v=EFApWAl3NJw

From there, I'd like to briefly go over vector spaces and emphasize the difference between a linear map and a matrix, and how a pair of bases for the domain and codomain determine the matrix of a linear map. To avoid cluttering the article with lots of computational examples, I'd also like to link this video on gaussian elimination rather than cover it here. https://www.youtube.com/watch?v=2j5Ic2V7wq4

In other words, I'll probably draw on video resources for the computational aspects and explain fundamental ideas in a more general context here. I believe this is a natural way to organize the content, make the resource very succinct, and give readers a "big picture" overview while drawing on other resources for the computational details. AP295 (discuss • contribs) 14:56, 12 March 2021 (UTC)

One last thing, I'd like to remove the other content outside the first two sections that I recently created. That is exactly the sort of thing I'm trying to avoid putting directly in this resource, and they aren't part of any coherent lesson plan as far as I can tell. Perhaps we can put them in another resource/page as an addendum. AP295 (discuss • contribs) 15:00, 12 March 2021 (UTC)

@Dave Braunschweig: Can you help me move all but the first two sections in the resource to their own page? My rationale is explained in this section above. I'm not sure what to title it since the content is quite disjointed, maybe something like "Linear algebra : Basic examples and computations". I'd just as soon delete them but on the outside chance that editor comes back I don't want to get into an edit war when they see their work is missing. AP295 (discuss • contribs) 15:12, 12 March 2021 (UTC)

@AP295: Yes and no. Consistent with Wikiversity's approach to multiple authors working on the same subject, we use subpages to organize that content. I have moved all of the content to a corresponding subpage. Please feel free to edit and retitle from here, but continue to use a subpage structure for the overall course organization. Thanks! -- Dave Braunschweig (discuss • contribs) 19:35, 12 March 2021 (UTC)

@Dave Braunschweig: Thanks, I will. Are there sub-subpages? I don't really have a place for the other content right now, but I suppose I can put them in a collapsible table if not. Can I use numbers instead of bullet points? I'd prefer to number them so they don't appear as a random collection of links, which might be slightly confusing to novices. AP295 (discuss • contribs) 19:49, 12 March 2021 (UTC)

Nevermind, I think I'm getting the hang of it. Can I rename a page? For example, I've written an intro, and I think it would be a good place for those youtube links on matrix operations and some other info as well, so I'd like to rename "Matrix Operations" page to "Introduction"AP295 (discuss • contribs) 20:11, 12 March 2021 (UTC)

@AP295: In Wikimedia language, we move pages rather than rename them. You should have rights to move pages. It's on the More tab. -- Dave Braunschweig (discuss • contribs) 21:43, 12 March 2021 (UTC)

Thanks again. AP295 (discuss • contribs) 22:12, 12 March 2021 (UTC)

Actually, I don't seem to have a "more" tab. AP295 (discuss • contribs) 22:15, 12 March 2021 (UTC)

Nevermind again. I think it will appear once I'm autoconfirmed.