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Archive 1 |
Hi everyone,
I promoted this article but I do feel that this is a borderline good article as it is an extremely brief article for such a large branch of mathmatics.
Cedars 07:50, 23 April 2006 (UTC)
I assume the part about "systems of linear equations in finite dimensions" was intended to distinguish the subject of linear algebra from, say functional analysis. However, the distinction lies not in the number of dimensions, but in whether the linear structure is studied as a thing in itself (as opposed to being studied in the context of a topology). In other words, pure vector spaces are the province of linear algebra, while topological vector spaces are the province of functional analysis. Thus, even infinite-dimensional linear phenomena, if studied from a purely algebraic standpoint, are technically part of linear algebra.-- Komponisto 21:36, 26 July 2006 (UTC)
I think it would be helpful if someone clarified the meaning of "over a field" from the first sentence of the fourth paragraph in the 'Elementary Introduction' section. The sentence reads as follows: 'A vector space is defined over a field, such as the field of real numbers or the field of complex numbers.' -- —The preceding unsigned comment was added by DrEricH ( talk • contribs) .
The derivation of the maximum-likelihood estimator of the covariance matrix of a multivariate normal distribution is perhaps surprisingly subtle and elegant, involving the spectral theorem of linear algebra and the fact that it is sometimes better to view a scalar as the trace of a 1×1 matrix than as a mere scalar. See estimation of covariance matrices. Please help contribute a "linear algebraists' POV" to that article. Michael Hardy 20:20, 10 Sep 2004 (UTC)
"However, it has few, if any, applications in the natural sciences and the social sciences, and is rarely used except in esoteric mathematical disciplines."
This is just plain wrong. Linear algebra is used in both the natural and social sciences. Physics and Chemistry are obvious. Biology uses matrices and all that malarkey when looking at coupled ODEs. Social sciences use them in some stats work and in ODEs/PDEs. Anywho, the above statement is misleading and should be removed.-- 137.205.132.41 10:20, 16 January 2007 (UTC)
The passage:
For small systems, ad hoc methods are sufficient. Larger systems require one to have more systematic methods. The modern day approach can be seen 2,000 years ago in a Chinese text, the Nine Chapters on the Mathematical Art ( simplified Chinese: 九章算术; traditional Chinese: 九章算術; pinyin: Jiǔzhāng Suànshù).
Is very similar to:
For small systems, ad hoc methods certainly suffice. Larger systems, however, require more systematic methods. The approach generally used today was beautifully explained 2,000 years ago in a Chinese text, the Nine Chapters on the Mathematical Art (Jiuzhang Suanshu, 九章算術).
Which is taken from Linear Algebra with Applications Third Edition by Otto Bretscher.
To me, it sounded a bit too similar to the original text.
Just something I noticed.
The statement from the article:
"Since vectors, as n-tuples, are ordered lists of n components, it is possible to summarize and manipulate data efficiently in this framework. For example, in economics, one can create and use, say, 8-dimensional vectors or 8-tuples to represent the Gross National Product of 8 countries. One can decide to display the GNP of 8 countries for a particular year, where the countries' order is specified, for example, (United States, United Kingdom, France, Germany, Spain, India, Japan, Australia), by using a vector (v1, v2, v3, v4, v5, v6, v7, v8) where each country's GNP is in its respective position."
is misleading and incorrect. There is a big difference between a tuple and a vector. The tuple of GNP values of 8 countries does not behave like a vector. For example, how would it behave under a linear transformation? What are its basis vectors?
It would improve this article if this statement were removed.
206.169.236.122 20:02, 6 February 2007 (UTC)
A number of editors of linear algebra/vector space articles are uncomfortable about making statements which rely on the axiom of choice without mentioning it. To some extent, I share their unease (or to put it more light-heartedly, "You say every vector space has a basis? Great! Now give me a well-ordering of the real numbers - it might come in handy..."). On the other hand, these are articles about linear algebra, so it is a pity to constantly distract the reader with digressions into logic and set theory. So we have a choice (fortunately a finite choice!): do we mention choice or not? I know this has been discussed on a few talk pages before, but my recent edit of this article suggests a compromise: footnoting references to choice. I imagine the use of footnotes may polarize opinion, but it might be a sensible way forward in this case, so let me know what you think. And I'll make a few other similar edits to stimulate the discussion :) Geometry guy 21:42, 13 February 2007 (UTC)
(Hmmm... I'm quite proud of that split infinitive.) I've footnoted the choices in Dual space. One obvious question that arises is whether it would be better to have just one choice-related note with all the relevant caveats, or several. I'd be inclined to put them together to avoid repetition. Geometry guy 22:18, 13 February 2007 (UTC)
I have to (regretfully) state that for a "top importance" article, this one is remarkably incoherent. Problems are manifold, but just for starters, there is the issue of consistency within the article itself and in wikipedia in general.
And the list goes on, and on, and on. Arcfrk 15:01, 19 March 2007 (UTC)
How about rewriting it? I mean, really. Readable articles for basic mathematics topics shouldn't be _too_ hard for us, should they? I tried turning the intro into grammatically correct english; as for content, however, I came here to learn and my linear algebra, history thereof, etc. is still weak. Please help! User:x14n 10th-ish Oct. 2007
I'm hopefully gonna spend some time in the next few days reworking some pieces of this article. As is, it really is a complete mess. A couple of thoughts that spring to mend:
-There should be a definition of vector space, some substantial mention of module theory and a couple of comments about why vector spaces and modules are different and why they are the same.
-The example given about the GNP of 8 countries is misleading in its triviality. Vectors are much more than just lists of numbers. Towards the end of explaining what linear algebra is and what a vector space really is, this article should have some well developed heuristic explanations of the concept of "linearity".
-I think the section that just lists important theorems should be trashed. It is completely unenlightening to just list off a bunch of results that all involve technical concepts, none of which have been defined.
Its late for me and these comments might be a little bit vague but please respond. Ill try to realize some of this stuff when ive had some sleep. Jrdodge 08:31, 11 November 2007 (UTC)
I was reading my linear Algebra book for class (Otto Bretsher's Linear Algebra with Applications. 3 edition. Upper Saddle River, New Jersey: Pearson Education, 2005) when I came across something interesting on page 8: "When mathematicans in ancient China had to solve a system of simultaneous linear equations such as, they took all the numbers involved in this system and arranged them in a rectangular pattern (Fang Cheng in Chinese as follows:
All the information about this system is conveniently stored in this array of numbers. The entries were respresented by counting rods; [...] the equations were then solved in a hands-on fashion, by manipulating the rods" I did some googling and found out that how Fang Chang is Chapter 8 in a book called Nine Chapters on the Mathematical Art which shows how almost 2000 years ago Chinese had a method similar to Guassian Elimination for solving linear equations even though they didn't call it Linear Alegbra. I thought it would be an interesting side note to add to the history section. the link about the book is here Nine Chapters on wiki and here Nine Chapters on google books —Preceding unsigned comment added by 128.61.43.160 ( talk) 18:05, 12 June 2008 (UTC)
I agree with people elsewhere on this page who feel that this entry is in need of major rewriting and changing. I have just made a dent in this. I began by tweaking the "history" section slightly (to treat the history of linear algebra as synonymous with the history of "abstract", ie, post-1850s linear algebra, is inaccurate) and removing a good chunk of stuff about quaternions. (Although it is certainly related to linear algebra, so are vectors--- which predate quaternions by centuries--- and it seems unusual to give so much attention to quaternions here.)
The statement that the use of Cramer's rule (which dates to the 1700s, mind you, not the 1850s) to solve partial differential equations led to the introduction of linear algebra to the math curriculum is, to this reader, laughable, and as it was not sourced in a way that makes sense to me I took it out. Copson's quote seems to have much more bearing on the difference in education between two universities, one English and one Scottish, in the beginning of the 19th century, than it does on anything specific to linear algebra. (In any case, the quoted portion of Copson does not say anything about the role of Cramer's rule in partial differential equations.)
Frankly, the article in its current form reads like a mish-mash of submissions by beginning students of linear algebra, well-intentioned people relating the observations of third parties (e.g. footnotes in general science works, or introductory paragraphs in introductory textbooks), and people too inclined to include abstract technical detail that is probably better left to more specific entries than "linear algebra". In my very humble opinion. 75.167.204.90 ( talk) 05:08, 7 July 2009 (UTC)
I've started a revamping of the article, which was long overdue. In the first installment, I've replaced a rambling and incoherent section Elementary introduction with a synopsis of the first few chapters in a standard linear algebra text. The section on history needs to go, too: there is no excuse to having such poor quality material, especially since a very good historical account is contained in the article on vector spaces (and, perhaps, elsewhere on wikipedia). It seems unwise to fork the content, especially from the maintenance point of view. I also feel that the list of results doesn't add much, but if anyone has ideas about how to incorporate some of them into the narrative, please, share them here or implement them yourself. Arcfrk ( talk) 05:07, 9 February 2010 (UTC)
This article must speak to the non-mathematical reader and to students. Mathematicians already know linear algebra and do not need to read about it on Wikipedia. Every line of the article should pass two tests. It should be mathematically accurate. And it should be readable by someone who is not a mathematician. The later parts of this article can address students who already know the material in the earlier parts of the article, but there should not be anything in the article that you need a Ph.D. in mathematics to read.
I've watched this article over the years swing back and forth between extremes. Sometimes it is oversimplified, sometimes too technical. I would like to see it at least move out of the start class, and I think there are currently some editors here working toward the same goal.
Rick Norwood ( talk) 16:48, 9 February 2010 (UTC)
Turning now to the picture, I find both the picture and the caption confusing. Were I beginning reader, I might think that all subspaces were lines through the origin. And the three dimensional effect is not clear: I'm not sure where the colored planes intersect. Also, vector subspaces are fundamental to the study of vector spaces, but more important in linear algebra, I think, is the use of a matrix to transform one vector to another. Does anyone have a picture showing this? Rick Norwood ( talk) 17:06, 9 February 2010 (UTC)
A "Matrix" isn't listed in the section on main structures, yet references to matricies are all over this page especially in the "most useful theorems" section. —Preceding unsigned comment added by 65.50.39.118 ( talk) 05:39, 7 September 2010 (UTC)
The statement about definite and semi-definite matrices is not correct as stated. Matrices should be assumed to be symmetric. Moreover, this is slightly off-topic: it is rather part of bilinear algebra rather than linear algebra.
The statement ``A non-zero matrix A with n rows and n columns is non-singular if there exists a matrix B that satisfies AB = BA = I where I is the identity matrix is much more a definition than a theorem
In my opinion, the main non-trivial result of linear algebra says that the Dimension of a vector space is well defined: Theorem: If a vector space has two bases, then they have the same cardinality.
A matrix is orthogonal diagonalizable if and only if it is normal (please check and edit!!!) —Preceding unsigned comment added by Niv.sarig ( talk • contribs) 23:06, 15 December 2009 (UTC)
Clearly, A matrix with only positive eigenvalues is not necessarily positive definite (also not semi for non-negative). These should be erased. (A counter example is M=[2 0;5 1] with eigenvalues 1,2>0 and vector v=[1;-1] but the inner product is v^tMv=-2<0.) —Preceding unsigned comment added by Niv.sarig ( talk • contribs) 20:46, 2 October 2010 (UTC)
The statement "any claim that the concepts of linear algebra were known to mathematicians prior to the end of the nineteenth century is inaccurate, an instance of the historical error of anachronism." seems strange. Herman Grassmann's The theory of Linear Extension (1834) seems to deal with linear algebra. 128.240.229.7 ( talk) 07:35, 21 January 2011 (UTC) Niko
Hi, I tried to re-write the introduction but someone reversed my changes. I think defining linear algebra as a branch of mathematics that studies vectors is not quite correct and also kind of circular. I'll re-write again if some people have suggestions where what I wrote wasn't clear, but I don't feel like writing everything out again only to have it deleted. Loadedsalt ( talk) 22:35, 18 February 2011 (UTC)
Clearly, I agree that linear algebra does not study vectors, and I have just changed it. Your re-write, however, was a bit too radical for a single change and, perhaps, hard for non-experts. Let's see if my revision survives longer :) 2andrewknyazev ( talk) 00:55, 19 February 2011 (UTC)
So what other stuff has the structure of a linear space but has elements that are not real or complex numbers?
You can have a space comprised of, say, all continuous functions or polynomials. In the polynomial case, however, Pn is isomorphic to Rn+1 (Pn being the space of all polynomials of degree at most n).
Veddan ( talk) 10:25, 24 March 2008 (UTC)
Consider the space C([0,1],R) of all continuous real-valued functions over the closed interval [0,1], this is a vector space, since linear combinations of continuous functions are continuous. The vectors in C([0,1],R) do not have "elements" in the same way n-tuples of real or complex numbers do. Also, the fact that polynomials of arbitrarily high degree exist in the space means that it is not finite-dimensional.
However, one way to unify the two ideas is to think about n-tuples as functions from the finite set {1,2,...,n} to R. Now finite-dimensional vector space Rn can be seen as a space of functions whose domain is finite, whereas the infinite-dimensional vector space C([0,1],R) is a space of functions whose domain is infinite.
tobilehman ( talk) 19:58, 21 December 2011 (UTC)
Hi all,
I have been editing this page lately, and here are some of my ideas on how to improve the page. Please feel free to act on these ideas, comment on them, reject them, and add your own thoughts!
1) In the scopes of study section, elaborate on determinants and inner product spaces, since they are important concepts. I am not sure if we should, but we could add some information on Hermitian and normal operators and the fact that they are diagonalizable and have orthonormal basis of eigenvectors.
2) In the applications section, elaborate on the solution of linear equations. I do not know if it is best to introduce this application through an example or through theory. I also don't know if we should use the augmented matrix notation or the equation notation that we are currently using; my inclination is that the augmented matrix notation might be cleaner and might connect better with the rest of the article.
3) Add the section on best-fit lines. Personally, I am not too familiar with this subject, so it might be better if someone else writes that section, but if no one will do it, I could relearn that material and write it up.
4) Add more applications? There are so many, so it is debatable how many we should include.
5) Expand the history section. The history of mathematics is really interesting, and when I read Wikipedia articles on math concepts, I like to read the history section.
6) Flesh out the generalization section. Maybe we should elaborate on what linear algebra theorems remains true in module theory and what becomes false, or put to symbols the concept of multilinear algebra. We should not write a whole expose of the subject, but link the subject to linear algebra.
7) Maybe mention its role in a mathematical education? That it is often used as a bridge to abstract math?
Best,
Majesty of Knowledge ( talk) 22:46, 26 January 2012 (UTC)
I think that that:
OrenBochman ( talk) 00:26, 19 March 2012 (UTC)
The following comment was removed from the eigenvectors and eigenvalues section today:
Perhaps true, but without reference, and inserted into introductory material, the comment is out of place. The topic requires a topology of matrices and an indication of density, beyond the scope of this article. Rgdboer ( talk) 22:16, 7 October 2012 (UTC)
This is inaccurate, for instance a matrix over the integers modulo 4 with a determinant of 2 would be uninvertible.
Was I being too pedantic for a Wikipedia article?
Maybe it should read "A matrix is invertible if and only if its determinant is nonzero (but see Invertible)" ?
I was taught this exact statement in school, and it cost me time and effort when I started trying to work with matrices over rings other than the integers or the reals. I'd rather not see anyone else misled by this implicit assumption. 91.84.221.238 ( talk) 02:24, 15 January 2008 (UTC)
Non-linear algebra redirects to this page! Really!? That's absurd..... someone, should write a new page on the general results of non-linear algebra. Here's a good text: http://arxiv.org/pdf/hep-th/0609022v4.pdf — Preceding unsigned comment added by 99.149.190.128 ( talk) 19:44, 6 May 2012 (UTC)
Can someone knowledgable in the social sciences further develop the sociology section under Applications? Right now it looks very underdeveloped and unprofessional. 134.126.68.80 ( talk) 18:15, 1 October 2014 (UTC)
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Please add some discussion of "lists" as used by Axler in Linear Algebra Done Right. 184.158.0.11 ( talk) 12:53, 10 October 2017 (UTC)
Currently there exists a popular ~2 hour long series of videos on Youtube known as Essence of Linear Algebra which briefly touches upon core topics in linear algebra, providing a visual depiction as well as intuition for why these concepts were created and what they are used for. I believe it is a valuable resource which presents information in a way that a text and image encyclopedia cannot. Upon adding this though it was removed with the person removing it clarifying on their talkpage that it was removed for not meeting the criteria of a reliable source. However, WP:RS specifically states that the reliability of a source is inextricably linked to the statement that it is being used to support. Since external links are not being used to support any statements, this would make consideration of whether or not an external link is a reliable source a moot point. To further support this, WP:EL makes no mention of a link's reliability, only requiring that it be accurate, and WP:ELPEREN (a non-official supplement) explicitly states that reliable sources and external links have different criteria. With this in mind I would like to reintroduce the link into the external links section. Thank you for your time. JustOneMore ( talk) 06:53, 26 July 2018 (UTC)
In computational number theory you sometimes get people doing linear algebra on matrices made out of integers modulo a prime. Often the prime is 2, but larger ones are also used.
My guess is the elements have to be from a ring or maybe a field. Anyway something with a group operation on the whole set, another group operation on the set except for identity of the first group, distributive law between the two group operations. — Preceding unsigned comment added by RoseParks ( talk • contribs) 10:45, 30 June 2001 (UTC)
You can do linear algebra over any field. If you're working with rings, they're called modules. Modules share many of the properties of vector spaces, but certain important basic facts are no longer true (the term dimension doesn't make much sense anymore, as bases may not have the same cardinality.) --Seb — Preceding unsigned comment added by 24.201.184.161 ( talk) 07:51, 25 February 2002 (UTC)
This is not an elegant section: it feels slightly like a dumping ground for a bunch of facts. Did anyone else have the same feeling? — Preceding unsigned comment added by Yiliu60 ( talk • contribs) 23:01, 05 July 2005 (UTC)
Here is my site with linear algebra example problems. Someone please put this link in the external links section if you think it's helpful and relevant. Tbsmith
http://www.exampleproblems.com/wiki/index.php/Linear_Algebra — Preceding unsigned comment added by Tbsmith ( talk • contribs) 19:13, 28 December 2005 (UTC)
Quoted from the main page:
This is truly a striking example :-)
Toby Bartels and I are going to correct this and I think we're also going to write about linear algebra over a rig (algebra) (this is not a typo!). -- Miguel — Preceding unsigned comment added by Miguel~enwiki ( talk • contribs) 01:41, 29 May 2002 (UTC)
@ Miguel @ Toby Bartels No rig? — Preceding unsigned comment added by Lbertolotti ( talk • contribs) 15:05, 06 September 2015 (UTC)
I think the reference to Hüseyin Tevfik Pasha's book should be removed. I looked at the English translation of his book on archive.org and it looks like it is a small textbook on very simple Linear Algebra. It doesn't look like it contains any new research nor does it look like a comprehensive textbook of the state of the art at (or near) the time of publication.
To be frank, it looks like it has only been included due to Turkish nationalism. The only reason why I haven't removed it myself is that I don't want to fight the Turkish nationalists. -- Peter Lund, 2018-08-06 — Preceding unsigned comment added by 80.62.117.202 ( talk) 07:38, 06 August 2018 (UTC)
Quote from the article: "In 1843, William Rowan Hamilton (from whom the term vector stems) discovered the quaternions."
Huh? I didn't find the answer on a quick perusal of the William Rowan Hamilton article either. I didn't see it in quaternions either. It sounds like an interesting story, but what (or where) is the story? Spalding ( talk) 18:25, 4 October 2004 (UTC)
I'm trying to learn something about linear algebra because I want to understand eigenvectors so I can read an article I have. The first part of the article was good; but then we get to the sideways "e" which means identical to? Then couldn't the section "Linear Transformations" be simpler? It's important to write for non-mathematicians. Edlong ( talk) 22:32, 14 August 2018 (UTC)
This section is does not respects
WP:NPOV, as considering only two countries (USA and France). Moreover, it is full of mistakes: The first sentence Linear algebra first appeared in American graduate textbooks in the 1940s
lets the user think that matrices, determinants, and systems of linear equations were not taught in American universities before 1940. The sentence "In France during the 1960s, educators attempted..." hides the fact that in France, the content of courses are fixed at central level (Minister of Education), and that "educators attempts" may only denote educational-research experiments.
This section is also a WP:Original synthesis, as not referring to any historical work, and referring only to WP:Primary sources.
Therefore, although educational history could be an interesting subject, the present content of the section is so far from an encyclopedic content that it is well suited for WP:TNT. Thus I'll remove this section. D.Lazard ( talk) 13:59, 19 August 2018 (UTC)
IMO, section "Applications" is a mess. It contains a list of applications, described in a way that make sense only for readers that are customized with the applied fields. Thus the content of the section is absolutely not useful for most readers. Moreover, most of the items of this list are narrow fields that are subfields of wider areas that use systematically linear algebra in the same way: robotics is a subfield of mechanics, where linear algebra is used for the geometry of the space, forces, ...; robotics has nothing specific from this point of view. Similarly, Fourier series expansion and quantum mechanics are applications of the theory of Hilbert spaces and functional analysis, that is are applications not of linear algebra directly, but of theories that use linear algebra. Therefore, my opinion is that we must apply WP:TNT to this section, for focusing onto the widest areas of applications. What follows is an essay for a section "Application" of encyclopedic nature.
Linear algebra is used in almost all areas of mathematics, and therefore in almost scientific domains that use mathematics. These applications may be divided in several wide categories
.
The modelisation of the ambient space passes through geometry and uses linear algebra for explicit models.(in the present list of applications, robotics and computer graphic are example of this)
Functional analysis studies function spaces that are vector space with additional structures, such as the Hilbert space structure.
Most physical phenomena are modeled by partial differential equations that are generally solved by linear approximations. This lead generally to compute with matrices of very large size. An example is weather forecasting.
More generally, almost all scientific computation involve linear algebra. For this reason, the algorithms of linear algebra are probably the algorithms whose implementation has been best optimized (see BLAS and LAPACK).
Clearly this essay deserve to be expanded, but, at is it, its content is certainly more useful for readers than the present state of the section. Further opinions? D.Lazard ( talk) 17:35, 10 September 2018 (UTC)
![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 |
Hi everyone,
I promoted this article but I do feel that this is a borderline good article as it is an extremely brief article for such a large branch of mathmatics.
Cedars 07:50, 23 April 2006 (UTC)
I assume the part about "systems of linear equations in finite dimensions" was intended to distinguish the subject of linear algebra from, say functional analysis. However, the distinction lies not in the number of dimensions, but in whether the linear structure is studied as a thing in itself (as opposed to being studied in the context of a topology). In other words, pure vector spaces are the province of linear algebra, while topological vector spaces are the province of functional analysis. Thus, even infinite-dimensional linear phenomena, if studied from a purely algebraic standpoint, are technically part of linear algebra.-- Komponisto 21:36, 26 July 2006 (UTC)
I think it would be helpful if someone clarified the meaning of "over a field" from the first sentence of the fourth paragraph in the 'Elementary Introduction' section. The sentence reads as follows: 'A vector space is defined over a field, such as the field of real numbers or the field of complex numbers.' -- —The preceding unsigned comment was added by DrEricH ( talk • contribs) .
The derivation of the maximum-likelihood estimator of the covariance matrix of a multivariate normal distribution is perhaps surprisingly subtle and elegant, involving the spectral theorem of linear algebra and the fact that it is sometimes better to view a scalar as the trace of a 1×1 matrix than as a mere scalar. See estimation of covariance matrices. Please help contribute a "linear algebraists' POV" to that article. Michael Hardy 20:20, 10 Sep 2004 (UTC)
"However, it has few, if any, applications in the natural sciences and the social sciences, and is rarely used except in esoteric mathematical disciplines."
This is just plain wrong. Linear algebra is used in both the natural and social sciences. Physics and Chemistry are obvious. Biology uses matrices and all that malarkey when looking at coupled ODEs. Social sciences use them in some stats work and in ODEs/PDEs. Anywho, the above statement is misleading and should be removed.-- 137.205.132.41 10:20, 16 January 2007 (UTC)
The passage:
For small systems, ad hoc methods are sufficient. Larger systems require one to have more systematic methods. The modern day approach can be seen 2,000 years ago in a Chinese text, the Nine Chapters on the Mathematical Art ( simplified Chinese: 九章算术; traditional Chinese: 九章算術; pinyin: Jiǔzhāng Suànshù).
Is very similar to:
For small systems, ad hoc methods certainly suffice. Larger systems, however, require more systematic methods. The approach generally used today was beautifully explained 2,000 years ago in a Chinese text, the Nine Chapters on the Mathematical Art (Jiuzhang Suanshu, 九章算術).
Which is taken from Linear Algebra with Applications Third Edition by Otto Bretscher.
To me, it sounded a bit too similar to the original text.
Just something I noticed.
The statement from the article:
"Since vectors, as n-tuples, are ordered lists of n components, it is possible to summarize and manipulate data efficiently in this framework. For example, in economics, one can create and use, say, 8-dimensional vectors or 8-tuples to represent the Gross National Product of 8 countries. One can decide to display the GNP of 8 countries for a particular year, where the countries' order is specified, for example, (United States, United Kingdom, France, Germany, Spain, India, Japan, Australia), by using a vector (v1, v2, v3, v4, v5, v6, v7, v8) where each country's GNP is in its respective position."
is misleading and incorrect. There is a big difference between a tuple and a vector. The tuple of GNP values of 8 countries does not behave like a vector. For example, how would it behave under a linear transformation? What are its basis vectors?
It would improve this article if this statement were removed.
206.169.236.122 20:02, 6 February 2007 (UTC)
A number of editors of linear algebra/vector space articles are uncomfortable about making statements which rely on the axiom of choice without mentioning it. To some extent, I share their unease (or to put it more light-heartedly, "You say every vector space has a basis? Great! Now give me a well-ordering of the real numbers - it might come in handy..."). On the other hand, these are articles about linear algebra, so it is a pity to constantly distract the reader with digressions into logic and set theory. So we have a choice (fortunately a finite choice!): do we mention choice or not? I know this has been discussed on a few talk pages before, but my recent edit of this article suggests a compromise: footnoting references to choice. I imagine the use of footnotes may polarize opinion, but it might be a sensible way forward in this case, so let me know what you think. And I'll make a few other similar edits to stimulate the discussion :) Geometry guy 21:42, 13 February 2007 (UTC)
(Hmmm... I'm quite proud of that split infinitive.) I've footnoted the choices in Dual space. One obvious question that arises is whether it would be better to have just one choice-related note with all the relevant caveats, or several. I'd be inclined to put them together to avoid repetition. Geometry guy 22:18, 13 February 2007 (UTC)
I have to (regretfully) state that for a "top importance" article, this one is remarkably incoherent. Problems are manifold, but just for starters, there is the issue of consistency within the article itself and in wikipedia in general.
And the list goes on, and on, and on. Arcfrk 15:01, 19 March 2007 (UTC)
How about rewriting it? I mean, really. Readable articles for basic mathematics topics shouldn't be _too_ hard for us, should they? I tried turning the intro into grammatically correct english; as for content, however, I came here to learn and my linear algebra, history thereof, etc. is still weak. Please help! User:x14n 10th-ish Oct. 2007
I'm hopefully gonna spend some time in the next few days reworking some pieces of this article. As is, it really is a complete mess. A couple of thoughts that spring to mend:
-There should be a definition of vector space, some substantial mention of module theory and a couple of comments about why vector spaces and modules are different and why they are the same.
-The example given about the GNP of 8 countries is misleading in its triviality. Vectors are much more than just lists of numbers. Towards the end of explaining what linear algebra is and what a vector space really is, this article should have some well developed heuristic explanations of the concept of "linearity".
-I think the section that just lists important theorems should be trashed. It is completely unenlightening to just list off a bunch of results that all involve technical concepts, none of which have been defined.
Its late for me and these comments might be a little bit vague but please respond. Ill try to realize some of this stuff when ive had some sleep. Jrdodge 08:31, 11 November 2007 (UTC)
I was reading my linear Algebra book for class (Otto Bretsher's Linear Algebra with Applications. 3 edition. Upper Saddle River, New Jersey: Pearson Education, 2005) when I came across something interesting on page 8: "When mathematicans in ancient China had to solve a system of simultaneous linear equations such as, they took all the numbers involved in this system and arranged them in a rectangular pattern (Fang Cheng in Chinese as follows:
All the information about this system is conveniently stored in this array of numbers. The entries were respresented by counting rods; [...] the equations were then solved in a hands-on fashion, by manipulating the rods" I did some googling and found out that how Fang Chang is Chapter 8 in a book called Nine Chapters on the Mathematical Art which shows how almost 2000 years ago Chinese had a method similar to Guassian Elimination for solving linear equations even though they didn't call it Linear Alegbra. I thought it would be an interesting side note to add to the history section. the link about the book is here Nine Chapters on wiki and here Nine Chapters on google books —Preceding unsigned comment added by 128.61.43.160 ( talk) 18:05, 12 June 2008 (UTC)
I agree with people elsewhere on this page who feel that this entry is in need of major rewriting and changing. I have just made a dent in this. I began by tweaking the "history" section slightly (to treat the history of linear algebra as synonymous with the history of "abstract", ie, post-1850s linear algebra, is inaccurate) and removing a good chunk of stuff about quaternions. (Although it is certainly related to linear algebra, so are vectors--- which predate quaternions by centuries--- and it seems unusual to give so much attention to quaternions here.)
The statement that the use of Cramer's rule (which dates to the 1700s, mind you, not the 1850s) to solve partial differential equations led to the introduction of linear algebra to the math curriculum is, to this reader, laughable, and as it was not sourced in a way that makes sense to me I took it out. Copson's quote seems to have much more bearing on the difference in education between two universities, one English and one Scottish, in the beginning of the 19th century, than it does on anything specific to linear algebra. (In any case, the quoted portion of Copson does not say anything about the role of Cramer's rule in partial differential equations.)
Frankly, the article in its current form reads like a mish-mash of submissions by beginning students of linear algebra, well-intentioned people relating the observations of third parties (e.g. footnotes in general science works, or introductory paragraphs in introductory textbooks), and people too inclined to include abstract technical detail that is probably better left to more specific entries than "linear algebra". In my very humble opinion. 75.167.204.90 ( talk) 05:08, 7 July 2009 (UTC)
I've started a revamping of the article, which was long overdue. In the first installment, I've replaced a rambling and incoherent section Elementary introduction with a synopsis of the first few chapters in a standard linear algebra text. The section on history needs to go, too: there is no excuse to having such poor quality material, especially since a very good historical account is contained in the article on vector spaces (and, perhaps, elsewhere on wikipedia). It seems unwise to fork the content, especially from the maintenance point of view. I also feel that the list of results doesn't add much, but if anyone has ideas about how to incorporate some of them into the narrative, please, share them here or implement them yourself. Arcfrk ( talk) 05:07, 9 February 2010 (UTC)
This article must speak to the non-mathematical reader and to students. Mathematicians already know linear algebra and do not need to read about it on Wikipedia. Every line of the article should pass two tests. It should be mathematically accurate. And it should be readable by someone who is not a mathematician. The later parts of this article can address students who already know the material in the earlier parts of the article, but there should not be anything in the article that you need a Ph.D. in mathematics to read.
I've watched this article over the years swing back and forth between extremes. Sometimes it is oversimplified, sometimes too technical. I would like to see it at least move out of the start class, and I think there are currently some editors here working toward the same goal.
Rick Norwood ( talk) 16:48, 9 February 2010 (UTC)
Turning now to the picture, I find both the picture and the caption confusing. Were I beginning reader, I might think that all subspaces were lines through the origin. And the three dimensional effect is not clear: I'm not sure where the colored planes intersect. Also, vector subspaces are fundamental to the study of vector spaces, but more important in linear algebra, I think, is the use of a matrix to transform one vector to another. Does anyone have a picture showing this? Rick Norwood ( talk) 17:06, 9 February 2010 (UTC)
A "Matrix" isn't listed in the section on main structures, yet references to matricies are all over this page especially in the "most useful theorems" section. —Preceding unsigned comment added by 65.50.39.118 ( talk) 05:39, 7 September 2010 (UTC)
The statement about definite and semi-definite matrices is not correct as stated. Matrices should be assumed to be symmetric. Moreover, this is slightly off-topic: it is rather part of bilinear algebra rather than linear algebra.
The statement ``A non-zero matrix A with n rows and n columns is non-singular if there exists a matrix B that satisfies AB = BA = I where I is the identity matrix is much more a definition than a theorem
In my opinion, the main non-trivial result of linear algebra says that the Dimension of a vector space is well defined: Theorem: If a vector space has two bases, then they have the same cardinality.
A matrix is orthogonal diagonalizable if and only if it is normal (please check and edit!!!) —Preceding unsigned comment added by Niv.sarig ( talk • contribs) 23:06, 15 December 2009 (UTC)
Clearly, A matrix with only positive eigenvalues is not necessarily positive definite (also not semi for non-negative). These should be erased. (A counter example is M=[2 0;5 1] with eigenvalues 1,2>0 and vector v=[1;-1] but the inner product is v^tMv=-2<0.) —Preceding unsigned comment added by Niv.sarig ( talk • contribs) 20:46, 2 October 2010 (UTC)
The statement "any claim that the concepts of linear algebra were known to mathematicians prior to the end of the nineteenth century is inaccurate, an instance of the historical error of anachronism." seems strange. Herman Grassmann's The theory of Linear Extension (1834) seems to deal with linear algebra. 128.240.229.7 ( talk) 07:35, 21 January 2011 (UTC) Niko
Hi, I tried to re-write the introduction but someone reversed my changes. I think defining linear algebra as a branch of mathematics that studies vectors is not quite correct and also kind of circular. I'll re-write again if some people have suggestions where what I wrote wasn't clear, but I don't feel like writing everything out again only to have it deleted. Loadedsalt ( talk) 22:35, 18 February 2011 (UTC)
Clearly, I agree that linear algebra does not study vectors, and I have just changed it. Your re-write, however, was a bit too radical for a single change and, perhaps, hard for non-experts. Let's see if my revision survives longer :) 2andrewknyazev ( talk) 00:55, 19 February 2011 (UTC)
So what other stuff has the structure of a linear space but has elements that are not real or complex numbers?
You can have a space comprised of, say, all continuous functions or polynomials. In the polynomial case, however, Pn is isomorphic to Rn+1 (Pn being the space of all polynomials of degree at most n).
Veddan ( talk) 10:25, 24 March 2008 (UTC)
Consider the space C([0,1],R) of all continuous real-valued functions over the closed interval [0,1], this is a vector space, since linear combinations of continuous functions are continuous. The vectors in C([0,1],R) do not have "elements" in the same way n-tuples of real or complex numbers do. Also, the fact that polynomials of arbitrarily high degree exist in the space means that it is not finite-dimensional.
However, one way to unify the two ideas is to think about n-tuples as functions from the finite set {1,2,...,n} to R. Now finite-dimensional vector space Rn can be seen as a space of functions whose domain is finite, whereas the infinite-dimensional vector space C([0,1],R) is a space of functions whose domain is infinite.
tobilehman ( talk) 19:58, 21 December 2011 (UTC)
Hi all,
I have been editing this page lately, and here are some of my ideas on how to improve the page. Please feel free to act on these ideas, comment on them, reject them, and add your own thoughts!
1) In the scopes of study section, elaborate on determinants and inner product spaces, since they are important concepts. I am not sure if we should, but we could add some information on Hermitian and normal operators and the fact that they are diagonalizable and have orthonormal basis of eigenvectors.
2) In the applications section, elaborate on the solution of linear equations. I do not know if it is best to introduce this application through an example or through theory. I also don't know if we should use the augmented matrix notation or the equation notation that we are currently using; my inclination is that the augmented matrix notation might be cleaner and might connect better with the rest of the article.
3) Add the section on best-fit lines. Personally, I am not too familiar with this subject, so it might be better if someone else writes that section, but if no one will do it, I could relearn that material and write it up.
4) Add more applications? There are so many, so it is debatable how many we should include.
5) Expand the history section. The history of mathematics is really interesting, and when I read Wikipedia articles on math concepts, I like to read the history section.
6) Flesh out the generalization section. Maybe we should elaborate on what linear algebra theorems remains true in module theory and what becomes false, or put to symbols the concept of multilinear algebra. We should not write a whole expose of the subject, but link the subject to linear algebra.
7) Maybe mention its role in a mathematical education? That it is often used as a bridge to abstract math?
Best,
Majesty of Knowledge ( talk) 22:46, 26 January 2012 (UTC)
I think that that:
OrenBochman ( talk) 00:26, 19 March 2012 (UTC)
The following comment was removed from the eigenvectors and eigenvalues section today:
Perhaps true, but without reference, and inserted into introductory material, the comment is out of place. The topic requires a topology of matrices and an indication of density, beyond the scope of this article. Rgdboer ( talk) 22:16, 7 October 2012 (UTC)
This is inaccurate, for instance a matrix over the integers modulo 4 with a determinant of 2 would be uninvertible.
Was I being too pedantic for a Wikipedia article?
Maybe it should read "A matrix is invertible if and only if its determinant is nonzero (but see Invertible)" ?
I was taught this exact statement in school, and it cost me time and effort when I started trying to work with matrices over rings other than the integers or the reals. I'd rather not see anyone else misled by this implicit assumption. 91.84.221.238 ( talk) 02:24, 15 January 2008 (UTC)
Non-linear algebra redirects to this page! Really!? That's absurd..... someone, should write a new page on the general results of non-linear algebra. Here's a good text: http://arxiv.org/pdf/hep-th/0609022v4.pdf — Preceding unsigned comment added by 99.149.190.128 ( talk) 19:44, 6 May 2012 (UTC)
Can someone knowledgable in the social sciences further develop the sociology section under Applications? Right now it looks very underdeveloped and unprofessional. 134.126.68.80 ( talk) 18:15, 1 October 2014 (UTC)
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Please add some discussion of "lists" as used by Axler in Linear Algebra Done Right. 184.158.0.11 ( talk) 12:53, 10 October 2017 (UTC)
Currently there exists a popular ~2 hour long series of videos on Youtube known as Essence of Linear Algebra which briefly touches upon core topics in linear algebra, providing a visual depiction as well as intuition for why these concepts were created and what they are used for. I believe it is a valuable resource which presents information in a way that a text and image encyclopedia cannot. Upon adding this though it was removed with the person removing it clarifying on their talkpage that it was removed for not meeting the criteria of a reliable source. However, WP:RS specifically states that the reliability of a source is inextricably linked to the statement that it is being used to support. Since external links are not being used to support any statements, this would make consideration of whether or not an external link is a reliable source a moot point. To further support this, WP:EL makes no mention of a link's reliability, only requiring that it be accurate, and WP:ELPEREN (a non-official supplement) explicitly states that reliable sources and external links have different criteria. With this in mind I would like to reintroduce the link into the external links section. Thank you for your time. JustOneMore ( talk) 06:53, 26 July 2018 (UTC)
In computational number theory you sometimes get people doing linear algebra on matrices made out of integers modulo a prime. Often the prime is 2, but larger ones are also used.
My guess is the elements have to be from a ring or maybe a field. Anyway something with a group operation on the whole set, another group operation on the set except for identity of the first group, distributive law between the two group operations. — Preceding unsigned comment added by RoseParks ( talk • contribs) 10:45, 30 June 2001 (UTC)
You can do linear algebra over any field. If you're working with rings, they're called modules. Modules share many of the properties of vector spaces, but certain important basic facts are no longer true (the term dimension doesn't make much sense anymore, as bases may not have the same cardinality.) --Seb — Preceding unsigned comment added by 24.201.184.161 ( talk) 07:51, 25 February 2002 (UTC)
This is not an elegant section: it feels slightly like a dumping ground for a bunch of facts. Did anyone else have the same feeling? — Preceding unsigned comment added by Yiliu60 ( talk • contribs) 23:01, 05 July 2005 (UTC)
Here is my site with linear algebra example problems. Someone please put this link in the external links section if you think it's helpful and relevant. Tbsmith
http://www.exampleproblems.com/wiki/index.php/Linear_Algebra — Preceding unsigned comment added by Tbsmith ( talk • contribs) 19:13, 28 December 2005 (UTC)
Quoted from the main page:
This is truly a striking example :-)
Toby Bartels and I are going to correct this and I think we're also going to write about linear algebra over a rig (algebra) (this is not a typo!). -- Miguel — Preceding unsigned comment added by Miguel~enwiki ( talk • contribs) 01:41, 29 May 2002 (UTC)
@ Miguel @ Toby Bartels No rig? — Preceding unsigned comment added by Lbertolotti ( talk • contribs) 15:05, 06 September 2015 (UTC)
I think the reference to Hüseyin Tevfik Pasha's book should be removed. I looked at the English translation of his book on archive.org and it looks like it is a small textbook on very simple Linear Algebra. It doesn't look like it contains any new research nor does it look like a comprehensive textbook of the state of the art at (or near) the time of publication.
To be frank, it looks like it has only been included due to Turkish nationalism. The only reason why I haven't removed it myself is that I don't want to fight the Turkish nationalists. -- Peter Lund, 2018-08-06 — Preceding unsigned comment added by 80.62.117.202 ( talk) 07:38, 06 August 2018 (UTC)
Quote from the article: "In 1843, William Rowan Hamilton (from whom the term vector stems) discovered the quaternions."
Huh? I didn't find the answer on a quick perusal of the William Rowan Hamilton article either. I didn't see it in quaternions either. It sounds like an interesting story, but what (or where) is the story? Spalding ( talk) 18:25, 4 October 2004 (UTC)
I'm trying to learn something about linear algebra because I want to understand eigenvectors so I can read an article I have. The first part of the article was good; but then we get to the sideways "e" which means identical to? Then couldn't the section "Linear Transformations" be simpler? It's important to write for non-mathematicians. Edlong ( talk) 22:32, 14 August 2018 (UTC)
This section is does not respects
WP:NPOV, as considering only two countries (USA and France). Moreover, it is full of mistakes: The first sentence Linear algebra first appeared in American graduate textbooks in the 1940s
lets the user think that matrices, determinants, and systems of linear equations were not taught in American universities before 1940. The sentence "In France during the 1960s, educators attempted..." hides the fact that in France, the content of courses are fixed at central level (Minister of Education), and that "educators attempts" may only denote educational-research experiments.
This section is also a WP:Original synthesis, as not referring to any historical work, and referring only to WP:Primary sources.
Therefore, although educational history could be an interesting subject, the present content of the section is so far from an encyclopedic content that it is well suited for WP:TNT. Thus I'll remove this section. D.Lazard ( talk) 13:59, 19 August 2018 (UTC)
IMO, section "Applications" is a mess. It contains a list of applications, described in a way that make sense only for readers that are customized with the applied fields. Thus the content of the section is absolutely not useful for most readers. Moreover, most of the items of this list are narrow fields that are subfields of wider areas that use systematically linear algebra in the same way: robotics is a subfield of mechanics, where linear algebra is used for the geometry of the space, forces, ...; robotics has nothing specific from this point of view. Similarly, Fourier series expansion and quantum mechanics are applications of the theory of Hilbert spaces and functional analysis, that is are applications not of linear algebra directly, but of theories that use linear algebra. Therefore, my opinion is that we must apply WP:TNT to this section, for focusing onto the widest areas of applications. What follows is an essay for a section "Application" of encyclopedic nature.
Linear algebra is used in almost all areas of mathematics, and therefore in almost scientific domains that use mathematics. These applications may be divided in several wide categories
.
The modelisation of the ambient space passes through geometry and uses linear algebra for explicit models.(in the present list of applications, robotics and computer graphic are example of this)
Functional analysis studies function spaces that are vector space with additional structures, such as the Hilbert space structure.
Most physical phenomena are modeled by partial differential equations that are generally solved by linear approximations. This lead generally to compute with matrices of very large size. An example is weather forecasting.
More generally, almost all scientific computation involve linear algebra. For this reason, the algorithms of linear algebra are probably the algorithms whose implementation has been best optimized (see BLAS and LAPACK).
Clearly this essay deserve to be expanded, but, at is it, its content is certainly more useful for readers than the present state of the section. Further opinions? D.Lazard ( talk) 17:35, 10 September 2018 (UTC)