# Talk:Change of basis

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Field:  Algebra

Not to start an argument, but using " :-) " in an article is rather unprofessional, so I removed it. —Preceding unsigned comment added by 128.91.45.84 (talk) 19:02, 12 September 2007 (UTC)

It is too obstruse and esoteric to be useful. This is a simple topic, can be presented with very simple examples in low-dimensional cases with basic linear algebra, and then more abstract and less relevant (to the majority of users) bits can then be given to expand on the basic simple points. Really what a tragedy when a beautiful topic is been presented in such a way as to elicit a gag reflex in 99/100 readers of the site. Truly frustrating.

### LaTeX template

The template was removed, since I've done the job. :-) --NicApicella 19:18, 12 March 2006 (UTC)

Excessive use of PNG in inline text is to be discouraged, since it looks ugly. It is advised where when TeX renders as an image, to only to use it in displayed form. You should also aim to preserve the actual formatting that was there earlier -- the vectors should be in bold. Dysprosia 22:23, 12 March 2006 (UTC)

I've added an example of how a basis change can be used to solve a problem. I just realized how effective and powerful basis changes can be for an actual problem, and wanted to share it with the world. I've tried to make the example as simple as possible.

I do realize that it might be slightly un-encyclopedic, and a bit too specific, and the wikipedia-community will have my complete understanding if it is removed :-).--Avl 10:18, 17 April 2006 (UTC)

## Rewrite needed

It is great that this article has been created and it's off to an enthusiastic start. I've tagged it for rewrite, though, since in its present form it doesn't meet the style or content standards of most serious articles in mathematics. As a fundamental topic in linear algebra, the mathematics needs to be re-presented for theoretical accuracy, completeness and clarity, the section divisions redone, the writing style cleaned up, and the specific extended example that currently makes up most of the article is probably not appropriate. Your help and expert attention is appreciated!  — merge 00:22, 11 June 2006 (UTC)

I've had a go at a rewrite. We need a standard commutative diagram to insert in the appropriate place and the text is rather condensed; a few short strategically placed examples would be good. Cheers!  — merge 13:28, 26 June 2006 (UTC)
Clarity is still an issue, as it stands it seems a solid mathematical treatment, but lacks clear language describing exactly what is being proved at each step. 128.211.223.73 02:45, 12 April 2007 (UTC)
In the pursuit for 'solidness' they seem to build up the whole linear algebra. The subject is lost in there. I believe that it can be exressed very shorly, in few sencences. --Javalenok (talk) 10:42, 11 September 2010 (UTC)

<--insert change-of-basis diagram here--> ... ???

So where is the rewrite tag? I also think it should be rewritten, because it is written now like a page in a textbook. Readers on this page can refer to other articles for the notion of vector space, linear transformations, blah-blah-blah.--Netheril96 (talk) 15:47, 14 October 2010 (UTC)

Done. --Javalenok (talk) 17:01, 6 November 2010 (UTC)

I've found the original description for "Change of Basis for Vectors" to be poorly organized - I've redone it which should hopefully clear things up a bit. 6-16-2011 — Preceding unsigned comment added by 18.82.3.52 (talk) 20:41, 16 June 2011 (UTC)

This article is not didactic, basically I could understand that it is possible to change the basis of a vector, but not how (with exception of rotations for 45 degrees). The derivation of t_2 = q t_1 p^{-1} is confusing. This article does not meet its objective. — Preceding unsigned comment added by 143.107.180.128 (talk) 12:34, 28 April 2015 (UTC)

## From Euler to Linear Maps

1. I suppose that somebody attempts to turn the Wikipedia into his professional notebook. Why to bother about specific 3D cases and rotation operators? Just tell that rotation and Fourier transform are examples of basis change.

2. Added the definition in the beginning, Basis Matrix is a matrix that represents the basis, because its columns are the vectors of the basis this somebody also adds the warning in continuation that use vectors inside matrices is fast-and-dirty. Don't we have a contradiction?

3. He renames the Change of vector representation due to basis change section into covariant and contravariant considerations to add his, much longer (but not more informative), contravariant vector case once again. Nevertheless, the 'vectors inside matrix' persist after his edit.

4. His longer edit is actually less informative since it erased the definition of P = "change of basis matrix". The Variance and covariance considerations looks orphaned after that.

5. The meaning of [u]v was also lost. Shouldn't it be explained that these are vector coordinates with respect to a basis? The author also wiped out the word 'coordinates' from everywhere! But understanding that the components are the coordinates with respect to basis is fundamental to the topic!

6. Is the "tensor proof" needed after explained that "as basis scales up by P, W → WP, the coordinates of contravariant vector u are scaled down by the same factor, [u]w → P-1[u]w"?

7. The "Linear Map" seems to duplicate the "Linear transformation".

Since only useless things were added and useful removed, I propose to return to Old revision of Change of Basis . It has only two sections: the effect of change of basis on Vectors and Maps. The effect will be different depending on if object is co- or contra-variant, which was exposed right in the Vectors section. --Javalenok (talk) 10:31, 14 December 2010 (UTC)

## Textbooks style

Now, the garbage of that spoiler was mostly removed, however, it was replaced by the textbook. Somebody another attempts to turn this article into textbook by copypasting his textbook here. --Javalenok (talk) 14:06, 2 January 2013 (UTC)

## Tensor proof - expert for translation

I want to correct the English of tensor proof, but I'm not familiar enough with the procedure for a change of basis to be sure I'll supply a correct interpretation. ᛭ LokiClock (talk) 06:45, 15 May 2011 (UTC)

## propose

I think that:

"This means that given a matrix M whose columns are the vectors of the new basis of the space (new basis matrix), the new coordinates for a column vector v are given by the matrix product ${\displaystyle \scriptstyle M^{-1}v}$. For this reason, it is said that normal vectors are contravariant objects."

should be change it to:

"This means that given a matrix M whose columns are the vectors of the new basis of the space (new basis matrix), the new components for a column vector v are given by the matrix product ${\displaystyle \scriptstyle M^{-1}v}$ (M inverse times v). For this reason, it is said that components of vectors are contravariant objects."

... opinions.

Please--kmath (talk) 13:39, 4 August 2013 (UTC)

It seems to me that this is an abuse of terminology, and that to call vectors contravariant objects is perpetuating a confusion. I would suggest removing any such mention, though will defer to editors more familiar with notable usage. "Covariant" is applied to objects that vary using the same matrix transform as the basis vectors do when they are replaced, and "contravariant" is applied to objects that objects that vary using the inverse matrix. Thus, the scalar coordinates for multiplying with the basis to produce a vector are contravariant, the basis as a whole is covariant, the dual basis is contravariant and the scalar coordinates for multiplying with the dual basis to produce a covector are contravariant. Vectors and covectors themselves are neither contravariant nor covariant, only their scalar scalar coordinates are.
The entire article unfortunately does not consider vectors of any form other than an n-tuple of scalar components. The concept of a change of basis applies in the context of general vector spaces, in which no such equivalence is assumed. Understanding of a change of basis would be facilitated immensely by adopting this level of abstraction as it would remove the confusion. This is quite a substantial change, so I will not embark on this at this stage. —Quondum 03:43, 29 December 2013 (UTC)
Agree. But you should consider, by now, at this "low level of confusion" as a little challenge to be overcome by those person which embark in these kind of studies. kmath (talk) 19:03, 30 December 2013 (UTC)

Can you say the difference between the vectors and components? --Javalenok (talk) 17:25, 10 December 2013 (UTC)

To me the term "coordinates" are used for geometric purposes like the uses in manifolds terminology and since vectors represent geometrical objects then we have this little confusion. But the term "components" are used to emphasize the algebraic aspect, exclusive for the "flat" manifolds which are vector spaces. Consider that you don't say that a given position in a general manifolds have certain "components" instead of "coordinates", right? kmath (talk) 00:11, 27 December 2013 (UTC)
kmath, we have coordinates in the vector space. I do not see any geometrical purposes there. I cannot even find the word "geometry" in any form there. I also do not understand what topology is doing in the vector space. BTW, I like that that article on coordinates provides the idea spoiled here in much more concise way. --Javalenok (talk) 23:07, 5 January 2014 (UTC)
Linear spaces are special cases of manifolds (topological manifolds), so let us use "geometrical" language to allegorice algebraic matters to soften things for non-topologist kmath (talk) 23:27, 5 January 2014 (UTC)
Also, check https://en.wikipedia.org/w/index.php?title=Special%3ASearch&search=components+of+a+vector to see the times that "components of a vector" are used kmath (talk) 23:37, 5 January 2014 (UTC)
I can also query the times "rain" is used vs "electricity" but the relative frequency will not shed any light on the relevance of their usage. Do you say that "components" and "coordinates" can be used interchangeably? --Javalenok (talk) 11:40, 6 January 2014 (UTC)
My position is explained also in [1]

## coordinate tuple instead of Coordinate vector

The ugly copy-paste from some hardcore math textbook also replaces the notion of coordinate vector, known to the wikipedia, with coordinate tuple. Can we replace all those hardcore "preliminaries" garbage of advanced math, including ξ, morphisms, linear mappings, degradatory 2D/3D (Euler) angles and theorems with a simple reference to the coordinate vector? I cannot understand even the first sentence,

The standard basis for Rn is the ordered sequence (e1, …, en), where ej is the element of Rn with 1 in the jth place and 0s elsewhere.

What the fuck are they smoking? What is the jth position? Are they talking about ej? That is a vector! How can a vector be 0 or 1? What the fuck? Can anybody else see that shit? I guess that even copy-paster is blind, he does not realize what is he pasting. --Javalenok (talk) 09:26, 4 June 2014 (UTC)

Cool it. Are you perhaps misinterpreting something? ej is a vector, and may be considered as an n-tuple of coordinate values, one of which is 1, and the rest of which are 0. The ordered sequence is an indexed set of vectors. —Quondum 19:49, 4 June 2014 (UTC)
Ok. But what standard basis has to do with all of that? You have one basis and translate coordiantes into another basis. What standard basis has to do with all of that? --Javalenok (talk) 20:34, 8 June 2014 (UTC)
As you suggest in your first post, there is probably a good deal of confusion between the concepts of a vector space of tuples and the concept of coordinate vectors. In this case, the notation Rn suggests that n-tuples are meant as the definition of the vectors in the vector space, thus giving a canonical basis, which then is the standard basis. In a more general abstract (but nevertheless isomorphic) vector space that is defined in a basis-independent fashion, there would be no standard basis. I would personally prefer the more abstract approach that deals with vector spaces isomorphic to Rn, and not specifically the space Rn, especially in this article where a change of basis need make no assumption about anything but the vector space nature: what the base field/ring is and the dimension of the space over this field. So, the short answer is that a standard basis has nothing to do with it but for illustrative use. —Quondum 21:28, 8 June 2014 (UTC)
I fail to understand how entailing extra entity (Occam Razor), which leads to (mine) confusion, simplifies anythig? What do you need to illustrate? You need one basis (a set of vectors), let it ${\displaystyle B=[{\vec {b}}_{1}{\vec {b}}_{2}{\vec {b}}_{3}\ldots {\vec {b}}_{n}]}$, you need to point out that any vector can be decomposed into the basis vectors by linear combinatinon, as coordinates article suggests and refer to that article. Finally, you need to show that ${\displaystyle {\vec {v}}=E[{\vec {v}}]_{E}=B[{\vec {v}}]_{B}=BPP^{-1}[{\vec {v}}]_{B}}$ whereupon it follows that bases are changed according to (change of basis) matrix P, into another basis E = BP, and coordinates are changed contravariantly, ${\displaystyle [{\vec {v}}]_{E}=P^{-1}[{\vec {v}}]_{B}.}$ Voila! Why is that illustration hell, which removes all the cliarity? --Javalenok (talk) 14:44, 12 June 2014 (UTC)
For clarity, my reference to "a good deal of confusion" was intended to refer to what I consider an unfortunate preference amongst some editors to phrase linear algebra in terms of tuple vectors, which is understandable given that this is how we typically learn about vectors at school and college level, but does tend to hide some insights that become obvious in a more abstract approach. I'm in agreement with you: the chosen illustration does not help; give me some time to get to rewording the article and then see whether it is an improvement. Alternatively, you could do so yourself and others will review it. —Quondum 00:50, 13 June 2014 (UTC)

## Something missing?

Is there a phrase missing from the first sentence of https://en.wikipedia.org/wiki/Change_of_basis#Three_dimensions? --Ott0 (talk) 06:52, 2 November 2015 (UTC)

Yes, I think. I added the missing words "let R". Dexo568 (talk) 22:37, 4 November 2015 (UTC)

## Preliminary notions statement seems cyclical

In the second line of this section, namely: If T: RnRm is a linear transformation...

What then is the matrix of T(ej)?