Talk:Basis (linear algebra)

Ordered Bases

More often than not bases are ordered. None of the examples uses set notation for the basis, all of them present an ordered basis. If nobody vetos, I will change the definition to "family of vectors". As any set X is naturally a family (x)_x∈X the current definition is included in the new one. Markus Schmaus 21:24, 14 Jun 2005 (UTC)

Of course, elements in a set may not be ordered, and since a basis is a set, the basis may not be ordered. So I think I am not sure why we have to say a basis is a family of vectors not a set. Probably putting some example of non-ordered bases to the article is helpful for both us and readers. -- Taku 22:11, Jun 14, 2005 (UTC)

Is ( (1,0), (1,0), (0,1) ) a basis of R²? Why? Markus Schmaus 15:09, 15 Jun 2005 (UTC)

No, a basis is a set and ( (1,0), (1,0), (0,1) ) is, at least in the notation that I use, not a set. I don't see where in the examples in the article the ordering is used. Do you have any references which define the basis as a family? All references I checked define it as a set. -- Jitse Niesen 16:14, 15 Jun 2005 (UTC)

You might (and should) start with Bourbaki, which is without doubt the most authorative reference for an axiomatic, comprehensive (nearly all-encompassing) presentation of mathematics one could conceive. — MFH:Talk 14:15, 30 October 2022 (UTC)

I am not sure what you mean by ( (1,0), (1,0), (0,1) ). Is it like, let a be a pair (1, 0) and b (0, 1). Then { (a, a, b) } forms a basis of R²? Isn't that a b forms such a basis? -- Taku 21:12, Jun 15, 2005 (UTC)

I guess you mean {a, a, b} and not {(a,a,b)} which is a singleton having just one element which is a triple of pairs. — MFH:Talk 14:08, 30 October 2022 (UTC)

By the way, strictly speaking $F^{n}$ does not have frame, just only a basis (cause is n-times the product of $F$ with no order at all) So strictly seaping, giving a frame on a vector space$V$ still needs to give an order in the canonical basis of $F^{n}$ so to have an isomporphisms defined :) --79.150.25.26 (talk) 16:32, 15 April 2011 (UTC)

I also agree that a basis should be defined as a family rather than a set, the order of the vectors is crucial. (Of course, any set is a family in a canonical way, indexing elements by themselves.) In 99.99% of the cases where you introduce a basis you want to introduce coordinates, and in 99.99% of the cases you index the coordinates with 1,2,... or 0,1,...; and when you make a change of basis (b1, b2, b3) to (b2, b1, b3) the first two coordinates must be exchanged (the change of basis matrix is the permutation matrix $P_{12}\in {\mathcal {M}}_{3,3}$), but with a set all that doesn't make sense. — MFH:Talk 14:06, 30 October 2022 (UTC)

@MFH: You have for some reason responded to a discussion that is more than a decade old. There was a much more recent discussion below in which a clear consensus was established for calling a basis a set; you might want to browse that discussion for the many good reasons not to define a basis as an ordered object. --JBL (talk) 19:39, 30 October 2022 (UTC)

Thanks for the link, I'll check it out. But whatever people will call it, a basis *is* a net/family and not a set. Otherwise permutation of basis vectors would not change the basis, but it does: The matrices of the maps will change. Oh, because they label rows/cols with indices instead of vectors? well these are the indices of the basis vectors...  ;-) Order of vectors definitely matters in a basis, but not in a set. — MFH:Talk 13:49, 24 January 2023 (UTC)

PS: that was not a consensus, all mathematicians in the discussion agreed that this is against mainstream terminology and should be forbidden. — MFH:Talk 13:51, 24 January 2023 (UTC)

Sequences or sets

Sequences are a more natural object with which to define the term "basis" than sets, and I'd like to edit the article accordingly.

Since sequences are ordered collections, it would obviate the need for phrases like "ordered basis", which frequently appear in the article. Using sequence notation also makes things much simpler notation-wise, and better lends itself to inductive arguments and other constructions on paper. For example, if you have two bases as sequences (a1,a2,a3) and (b1,b2,b3) of two orthogonal subspaces of R^6, then (a1,a2,a3,b1,b2,b3) is a basis for R^6, and the order is unambiguous despite not being implied by the subscripts alone. As another example, in concatenating (a1,a2,a3) and (a1), the obtained sequence (a1,a2,a3,a1) is not a basis and we can simply say "the sequence is not a basis". However, the union of the sets {a1,a2,a3}, {a1} would be a basis, and we'd have to say something like awkward like "the elements of {a1,a2,a3} and {a1} are not a linearly independent collection of vectors". Is there anything to be gained by calling them sets? The two books I have on linear algebra, Linear Algebra Done Right by Axler, and Linear Algebra by Shilov, use sequences or sequence-like notation. I'm having a similar conversation on the Change of basis talk page, but I see that they're called sets in this article too. AP295 (talk) 00:22, 16 January 2021 (UTC)

Addendum: Indexed family could be used as a "catch-all" structure, if we must include uncountable bases in the definition. For countable bases, these indexed families could be taken as sequences. Even those authors who do use the term "set" usually denote them as an indexed sequence or in matrix form. Without some order, expressed or implied, we cannot say that a basis (or bases) uniquely determines a vector of coefficients or the matrix of a linear map. It's also incongruous with the many sources (including some Wikipedia articles) that take for granted that each basis is ordered (and consequently that different orderings comprise unique bases) without always using the term "ordered basis", for example Triangular matrix. AP295 (talk) 14:10, 19 January 2021 (UTC)

You don't need an RfC for this, just discuss in the normal way. You may send a note to WT:MATHS if you like, but it's way too soon for a full-blown thirty-day formal RfC. --Redrose64 🌹 (talk) 08:04, 16 January 2021 (UTC)

I made it an RfC because I wanted to cast a wide net. If you look above, it's the same couple editors who get involved with each talk section. While I value their opinion, confusion about "ordered bases" and the question of what comprises a basis has come up a few times. The concept of a basis is a simple one which underpins basic linear algebra and I feel like it's somewhat obscured by the language and notation in this article, and this is reflected by many other sections in this talk page. AP295 (talk) 15:00, 16 January 2021 (UTC)

I re-added it. I think we can make room for this between the other very important RfCs about Pentagon UFOs, Jared Kushner, and covid FUD. AP295 (talk) 18:59, 16 January 2021 (UTC)

The common definition of a basis is that it is a set, and the definition does not imply any order on it. Moreover, infinite bases cannot always be indexed by integers, for example in the case of a Hamel basis of the reals over the rationals. So, if one does not change the common definition, it is an error to talk of a basis as a sequence.

However, for talking of the coefficients of a vector on a basis, one has to associate each coefficient to the corresponding basis element. This can be done through an indexing set. In the case of a finite basis, the most natural indexing set consists of the first positive integers. The choice of such an indexation does not change the basis, but is equivalent to the choice of a total order on the basis. So, The term ordered basis does not means that the basis is a sequence, but means that one talk of a basis (a set) on which an order has been chosen. In formal terms, a basis is a pair of a set and an order on this set. So, if one want to consider it as a sequence, this is a sequence of two elements, whichever is the size of the basis.

In summary, this request amounts to change the standard mathematical terminology, and this is strictly forbidded by policy WP:OR. D.Lazard (talk) 11:37, 16 January 2021 (UTC)

A countable basis can be indexed by the integers. I cited two well-known books that both use sequence notation (despite being very different approaches to the topic). I can see that an uncountable basis is not strictly amenable to representation via sequences, but the reader who is interested in such exotic things would not likely be stumped by a move from sequences to ordered sets. Also, I don't see a problem with generalizing the definition of a sequence so that it's indexed by real numbers or some other uncountable indexing set. Sequence operations like concatenation and taking a subsequence seem more useful for working with bases than union and intersection. For example, we have a nice correspondence between the columns of a matrix and a sequence of those vectors. However, the columns of infinitely many distinct matrices can correspond to a single set. If my earlier examples seem contrived I can cite more cases where using sequences make things easier notationally. For people like myself who look at things from a spatial point of view, sequences are much less abstract than sets and for people new to linear algebra it lessens the strain on one's working memory. AP295 (talk) 15:00, 16 January 2021 (UTC)

And would you consider all permutations of the standard basis as being the same basis? That's not much different from saying all invertible matrices (specifically their sets of columns) in a given vector space are the same basis. They are all bases for the same vector space but they are all distinct bases. Declaring a basis to be a set of vectors arbitrarily identifies matrices whose columns are permutations from the same subset of vectors. Is a permutation matrix a change of basis? Of course, but then we'd have to say "change of (ordered) basis". Considering how frequently we reference and take advantage of the correspondence between a basis and a matrix whose columns are those basis vectors, simply calling it a sequence removes this additional layer of complication and avoids ambiguities which might otherwise be a stumbling block for students and other novices. AP295 (talk) 17:36, 16 January 2021 (UTC)

As one way of resolving this, could the article be organized into sections about finite, countable bases and uncountable bases? Then a finite (or countable) basis can be called a sequence. In the article Schauder basis, they are defined using sequences. AP295 (talk) 20:25, 16 January 2021 (UTC)

Again, your suggestion is WP:Original research that goes against the main stream of mathematics. This is strictly forbidden by Wikipedia policy. This could be discussed further only if you provide WP:Reliable sources that support your views. D.Lazard (talk) 20:37, 16 January 2021 (UTC)

I have already provided a few sources that use the term sequence for countable-length bases, including another Wikipedia article. AP295 (talk) 20:48, 16 January 2021 (UTC)

A Schauder basis is not a basis as it involves a topology on the vector space, and infinite sums. Moreover, Wikipedia is not a source. Please, read WP:Reliable sources for learning what is a reliable source for Wikipedia. Moreover, you have not provided any textbook (for such a subject where there are plenty textbooks, a reliable source is necessarily a textbook) asserting that "a basis of a vector space is a sequence of elements such that ...". D.Lazard (talk) 21:28, 16 January 2021 (UTC)

I mentioned Axler's book earlier. He uses the word "lists" which means the same thing as "sequence", and for all intents and purposes treats them as sequences. We could say "list" if you want but "sequence" is probably more precise. In order for a choice of bases for the domain and codomain of a linear map to uniquely determine its matrix, the elements of each basis must have a concrete order. The connection between linear maps and their matrices is less obvious when we consider a basis as a set and not a sequence. The overwhelming majority of this article's readers are likely English-speaking students who may not understand or appreciate this concept yet, so do keep them in mind. AP295 (talk) 21:58, 16 January 2021 (UTC)

Also, while I do not have much experience with infinite vector spaces, the Schauder basis seems like a more natural generalization of the concept of a basis to infinite-dimensional normed vector spaces than avoiding countable sums and allowing uncountable bases, as in a Hamil basis. That seems bizarre to me. Can you explain how one might use a Hamil basis or where they come up? AP295 (talk) 22:20, 16 January 2021 (UTC)

Once again I've re-added the RfC tag. Many people have raised concerns in the span of several years about the clarity and general quality this article, along with the Change of basis article, and I feel it's entirely justified. AP295 (talk) 00:43, 17 January 2021 (UTC)

Many things in mathematics can be defined in several ways of equal validity in the sense that the objects created by those definitions behave the same. I'm sure that sequences are equally as valid a way to define bases as sets. That being said, Wikipedia needs to reflect how the majority of reliable sources in the field of linear algebra have defined bases, not our subjective idea of what's more "natural". A few textbooks aren't good enough; the textbook I learned from defined a basis as a set (A First Course in Linear Algebra by Kuttler) and I'm sure it's possible to find many more. I'd like to see some more sources that generally describe whether mathematicians tend to define bases in sets or using sequences. The only ones I could find pertaining to this were dictionary definitions; Merriam-Webster defines a basis in mathematics as being "a set of linearly independent vectors..." [1] If the article is to be changed to redefine the concept of a basis in terms of sequences there should be some sort of source (preferably better than a dictionary definition) that confirms that sequences are the most commonly used method of defining sets and unfortunately I was unable to find such a source at this time. I'm going to have to oppose this proposal for the above reasons, and for that matter, support that bases be defined as sets for the foreseeable future.Chess (talk) (please use {{ping|Chess}} on reply) 06:06, 17 January 2021 (UTC)

I sympathise with the request for comment, but my reaction must be read in the context of my limited knowledge of professional maths. This said, I favour the justification given by Chess. JonRichfield (talk) 12:32, 17 January 2021 (UTC)

I will try to find more sources, but keep in mind that I'm not "re-defining the concept of a basis" at all. Both ordered sets and sequences are adequate for developing the concept of a basis, but I propose that "sequence" is more convenient for the reader and the editor. Bases must be ordered to be of any use. This is purely a notational issue, and as far as I can tell the only concern would be the case of an uncountable basis. Perhaps you can comment on their significance. If nothing else, I hope we can include an explanation that they can be defined either way (at least for finite/countably-infinite bases), and that "basis" frequently means "ordered basis" or "sequence of basis vectors", just so that this article jibes with other material on the subject. For example, permutations of the same basis vectors are considered distinct bases in many contexts. If it's the case that we must distinguish the terms "basis" and "ordered basis" here, then we must not use "basis" to mean "ordered basis" in any other article, which would likely require a lot more work than simply getting it right in this article, and also seems rather idiosyncratic. Perhaps I did not explain this adequately in the RfC itself, but I hope it's self-evident that we need a precise and consistent definition of "basis" across Wikipedia. In another section of this talk page, an editor proposes "indexed family", which would probably be better as well. I would be fine with saying that a basis is an ordered/indexed set/family, and then disposing of the term "ordered basis". If we're to come up with a single definition that fits everything including uncountable bases, then "indexed family" would probably be ideal, and compatible with the various and less general definitions used in other material. Just look at all the other sections in this talk page. The concept of a basis is fundamental to linear algebra. The article must be improved and that starts with a precise definition of "basis". AP295 (talk) 13:52, 17 January 2021 (UTC)

One more note. Here's the current lead: In mathematics, a set B of elements (vectors) in a vector space V is called a basis, if every element of V may be written in a unique way as a (finite) linear combination of elements of B. The last part, if every element of V may be written in a unique way as a (finite) linear combination of elements of B. implies that we must have more than just an unstructured set. According to Merriam-Websters "usage notes" on "may vs. can", ""May" is the older word and has meanings that refer to the ability to do something, the possibility of something, as well as granting permission." If we have a set but no way to index individual elements, then we cannot write an element of V as a finite linear combination of B's elements. So at the very least, by our own definition, it must be something more than just a set. It must be an indexed family. We can be more specific, and therefore more clear and informative, without giving up anything. AP295 (talk) 15:22, 17 January 2021 (UTC)

In my experience, it is vastly more common in the sources I read (both research literature and textbooks) to define a basis as a set of vectors than as anything else. And then for computational purposes one assigns an order as a separate step (though sometimes this is implicit rather than explicit). Based on this sense of what sources do, I would not support rewriting this article to treat a basis as a sequence (or ordered set, or ...). I suspect that this article could be improved a lot, and I doubt that this question is a major stumbling block: when a basis need to be ordered, one can write "an ordered basis". --JBL (talk) 15:40, 17 January 2021 (UTC)

And in the sources that I have read, they use sequence. Even in those sources that do define "basis" as a set, they're almost always denoted using indices that imply order. Sure, they can make it work for their use. Just like Axler used sequences and made those work. If we're going to explain things in a general context, for all the world to read, I think we need to be as clear and informative as we can. Our definition of a basis implies it's more than just a set, so to separately distinguish "ordered sets" leaves something to be desired, and one might wonder when order does or does not matter. It needlessly complicates the notation and introduces ambiguity in our language. This issue was mentioned in 2005: https://en.wikipedia.org/wiki/Talk:Basis_(linear_algebra)#Ordered_Bases (I don't know how to link a specific section). It has been outstanding for over fifteen years. Is this article going to stay at C quality forever, or until enough literature is published to meet your arbitrary standard? How about using your goddamned head? If there's a real reason we shouldn't call them sequences or at least indexed families, please let me know. As far as I can tell, there isn't any. AP295 (talk) 15:54, 17 January 2021 (UTC)

You've started an RfC, I made a C; is it really necessary to repeat the same arguments in response to everyone who comments? Feels like WP:BLUDGEONing. (Incidentally, writing "a set {v_1, ..., v_n}" does not imply an order -- using some version of indices 1, ..., n is the only reasonable way to label the elements of a collection of size n, whether or not that collection is ordered.) --JBL (talk) 16:18, 17 January 2021 (UTC)

You've likewise repeated what others have stated in their comments. I'm not requesting votes, I'm requesting comments, and yours does not add anything. I feel as though I'm not being understood here, and it frustrates me when people ignore the point I am trying to make. Bases must be ordered or indexed to be of any use. The sentence in our current lead implies they must be at least indexed. Several sources use indexed/ordered structures like sequences. Without an order, a basis does not uniquely determine the coefficient vector of some element in a finite-dimensional vector space (or the matrix of a linear map). This property is important and most easily understood when a basis is considered as an ordered collection of elements, and many sources take for granted that a basis is ordered without using the phrase "ordered basis". Others have raised this concern over the years as well. Once again, if you know of a technical reason why this doesn't work, I'd like to hear about it. That is why I am asking, because I am not a professional mathematician but I do have some interest in this article and I feel it needs to be improved. Am I getting through to anyone here? AP295 (talk) 16:50, 17 January 2021 (UTC)

Your first sentence founders on the word "likewise". An RfC is a consensus-building process. It is constructive for new contributors to the discussion to indicate their views, especially when they agree with the views of other contributors to the discussion -- that helps establish consensus. It is not constructive for one editor to repeatedly restate their own views -- that does not help establish consensus, and is instead a form of WP:BLUDGEONing. (It is difficult to believe that you read the link and failed to understand this.) The rest of your comment is unnecessary repetition of your misconceptions, not supported by any sourcing. --JBL (talk) 17:03, 17 January 2021 (UTC)

Why do you call my points "misconceptions"? Can you refute them? Can you offer a counterargument? Something? Surely you can see that I am frustrated by this discussion. Improving Wikipedia always seems to be a grueling, uphill battle. It greatly strains my capacity to WP:AGF. If you disagree with me fine, but tell me why. Clearly there are several different and technically valid conventions we could use in this article. If our goal on this talk page is consensus, then we should evaluate the merits of each convention, and that involves more than saying "my book does it this way" or "my book does it that way". Clearly there are examples of both, we've established that. AP295 (talk) 17:18, 17 January 2021 (UTC)

Let me be more explicit: your behavior is aggressive, bordering on disruptive. You should read and reflect on WP:BLUDGEON#Dealing with being accused of bludgeoning the process before contributing further to this discussion. --JBL (talk) 17:29, 17 January 2021 (UTC)

It doesn't seem that I am able, much less welcome, to make contributions to Wikipedia without being persistent. I want this to be a cordial discussion just as much as you do, probably much more. For that to happen, you have to act reasonably too. I'm not sure how you expect me to react to the sort of replies that I get here, other than with frustration, as any human being who feels emotion probably would. AP295 (talk) 17:54, 17 January 2021 (UTC)

If this is true, it may be because you are opposing the consensus. Perhaps it is time to yield to the majority view instead of being persistent and bludgeoning.—Anita5192 (talk) 17:59, 17 January 2021 (UTC)

This sounds, like many suggestions to "improve" scientific or mathematical terminology, like Original Research. To the extent that sources supporting the change have been provided, the rationale is very thin: use of sequence-like notation or making the leap from the informal term list to the idea that an ordering is a necessary part of the formal definition. Doubtless the article is broken in many ways, but this is not the way to fix it. XOR'easter (talk) 00:07, 18 January 2021 (UTC)

• Oppose sequence. The Sequence article says "In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters." But in a set, ordered or otherwise, repetitions are not allowed. A basis is a set. Sure, an ordered set can also be represented as a sequence, but loosening the definition that way is not helpful. As for whether "set" or "ordered set" is better, I'm less sure. Typically we order the basis vectors, but we don't always need to, and it's not clear to me that it's always possible. Furthermore, if I understand the Sequence article, a sequence has a start (a first element), whereas an ordered set that is countable might be infinite in both directions, in which case it would not be representable as a sequence. So for example the Fourier series basis with N cycles per period, with N from negative infinity to infinity, is a countable ordered set of basis functions, but not a sequence (if the definition in our article is correct). Dicklyon (talk) 01:03, 18 January 2021 (UTC)

  Never mind, I see the lead there was overly restrictive, per Sequence#Finite_and_infinite about doubly-infinite sequences. Dicklyon (talk) 01:06, 18 January 2021 (UTC)

  @Dicklyon Their lead isn't overly restrictive. The set of integers can be mapped 1-1 with the set of non-negative integers to obtain a sequence (0,1,-1,2,-2,3,-3,4.....). The mapping itself is somewhat arbitrary but the element of any countable set can be put into a sequence through such a labeling with non-negative integers. That's actually the definition of "countable". AP295 (talk) 15:22, 18 January 2021 (UTC)

  Of course you can associate elements of a doubly-infinite countable ordered set with the natural numbers that way, but it doesn't preserve the order; so the lead definition of Sequence is too restrictive for that. Dicklyon (talk) 01:01, 19 January 2021 (UTC)

• It does if you want it to. If your original ordering is given by the relation R(x,y), and f maps each element 1-1 to the natural numbers, then there exists f^-1 (the inverse) and we can recover the original order: Define R'(i, j) iff R(f^-1(i), f^-1(j)) and so on. AP295 (talk) 01:31, 19 January 2021 (UTC)

• Here is a brief summary of some sources:
  • Friedberg--Insel--Spence, Linear algebra: unambiguously defines a basis as an unordered object, later explicitly introduces an ordering.
  • Axler, Linear algebra done right: unambiguously defines a basis as an ordered object (a "list"); it is not clear what this is supposed to mean in the infinite-dimensional case.
  • Artin, Algebra: It will be convenient to work with ordered sets of vectors here. The ordering will be unimportant much of the time, but it will enter in an essential way when we make explicit computations. Bases are defined as ordered sets, but the definitions and theorems are only in the finite-dimensional case, and many results have an extra finiteness condition (e.g., in a fin dim space, every finite [ordered] spanning set of vectors is shown to contain a basis), avoiding the question of what an ordered infinite set is.
  • Weintraub, Linear algebra for the young mathematician: bases are definitely unordered sets. Sets are often indexed in a way that suggests all sets are countable, and in the fin dim case, the ordering is introduced implicitly via the indexing.
  • Beezer, A first course in linear algebra: bases are definitely unordered sets. When choosing matrix/vector representations in the fin dim case, the ordering is introduced implicitly via indexing.
• I will admit surprise both at the number of texts I found that use the sequence definition (I was expecting a more lopsided ratio than 3:2) and also surprise at the failure of the authors using the sequence definition to contemplate what it means in the case of infinite (especially uncountable) dimension. In my opinion, this is good support for the current approach of mentioning both possibilities prominently, and definitely does not support jettisoning the unordered definition from the article. --JBL (talk) 02:55, 18 January 2021 (UTC).

  I note that Ordered basis (a redirect to section "Coordinates" of this article) is not sourced but could be sourced by most of these sources. Moreover, this appears after the statement of the numerous properties that are ordering independent, and it is explicitly explained that this ordering is needed for manipulating coordinates (Artin's "explicit computations"). So I agree with JBL's conclusion. D.Lazard (talk) 09:36, 18 January 2021 (UTC)

  In hindsight, Indexed family might have been better. If you're also taking into account the uncountable case, then "ordered set" isn't quite right either. You need to be able to reference specific elements of the basis to "write every element as a linear combination of basis vectors". This definition implies that a "basis" is partly a notational tool and its elements must either be known or able to be referenced via an index. How do you use (that is, "write with") an uncountable Hamil basis, some of whose elements (if I understand) might be arbitrary irrational numbers? AP295 (talk) 15:34, 18 January 2021 (UTC)

  The statement "every element of V can be written as a linear combination of basis vectors" is not about what a human being can do; it has exactly the same meaning as "for each v in V, there exists a finite set of vectors, each of which is a scalar multiple of one of the basis vectors, whose sum is v". Hamel bases are not "useful" for explicit computations in the way that finite bases are, and they cannot be conveniently listed, ordered, or indexed; nonetheless they are bases. --JBL (talk) 17:31, 18 January 2021 (UTC)

  My point was that they have to be indexed in some manner, even if using the reals. And in more convenient cases where a basis is finite (and I'm sure also countable case), the ordering of those basis vectors is what allows us to determine a unique coefficient vector (and unique matrices corresponding to linear maps). Vectors (and column/row matrices) are inherently ordered, and if a basis is to induce a unique and specific coefficient vectors, its elements must be considered ordered as well. So often an order is taken for granted. Here's a random example from Triangular matrix Here, whether or not a matrix is triangular w.r.t. a given basis depends on the order, but they don't say "ordered basis". They just say basis. Is the reader supposed to miraculously understand that the order is relevant here? AP295 (talk) 17:40, 18 January 2021 (UTC)

  Your point is unsourced and also wrong. If one accepts the appropriate set-theoretic axioms, then every vector space has a basis; indexing is utterly irrelevant. To choose a finite example, the space of matrices of a given shape has a set that I think many people would be happy to call its "standard basis", but it definitely does not have a standard ordered basis. The article on triangularity is instructive: you need only read the beginning of the section Triangular_matrix#Triangularisability (of which you linked a sub-section) to see this issue spelled out completely explicitly and clearly (if in excessively technical language). So you've illustrated only that it's a bad idea to start reading a technical article in the middle of a section without having read what comes before it.
  It is a shame that you have not used this experience to reconsider your strongly held but erroneous or confused beliefs. It's quite clear that the particular proposal you've made is going nowhere; you should drop the issue and consider other ways of improving the article (which you obviously are capable of). Given the unpleasantness of interacting with you above, I am also serving notice that if you write something in response to this, I will read it, but I will not write more. --JBL (talk) 18:12, 18 January 2021 (UTC)

So let me get this straight. If I ask you to write the matrix of the identity map I(v) = v with respect to the standard basis (yes I know, I was trying to say "the matrix of the standard basis", my earlier comment was deleted twice by another editor), you believe that something other than the identity matrix would be a reasonable answer? AP295 (talk) 21:17, 18 January 2021 (UTC)

Ok, I guess I will make an exception for direct answers to questions. The matrix of the identity map on any fin dim vector space with respect to any ordered basis is the identity matrix; this has nothing to do with the issues under discussion here. Now let me ask you a direct question: what would it take to convince you to reconsider your approach to this discussion? --JBL (talk) 22:10, 18 January 2021 (UTC)

I can't argue with you all week so I guess I'll just leave. You've worn me down, I'm starting to make typos, and I have other things I need to do. Goodbye. AP295 (talk) 23:01, 18 January 2021 (UTC)

For the record though, you are still wrong. Nobody calls a permutation matrix "the standard basis". I've pointed out several very subtle but important issues and you've deflected every time. You can pretend to be civil but part of a civil dialectic is acknowledging reason and common sense. Acting like a pig-headed fool until the other party tears out their hair in frustration is hardly a civil way of talking with someone. So I'd probably re-consider my approach if you'd stop doing that. You are not editing in good faith and it shows, and so I cannot assume good faith. If you will not even speak with me on equal terms then I have no reason to speak with you. AP295 (talk) 02:58, 19 January 2021 (UTC)

And I guess every set can be indexed by itself, with the identity function. If the set exists, there exists a function that can index the set, "if one accepts the appropriate set theoretic axioms". And the order/indexing becomes very relevant should you want to use that basis for any practical purpose. God. AP295 (talk) 03:15, 19 January 2021 (UTC)

I'm not sure why I even bother at this point, but https://mathworld.wolfram.com/HamelBasis.html implies that they are indexed. For all intents and purposes, we must be capable of indexing or ordering a basis. Unusual bases whose existences are only implied non-constructively are likely of little concern to the vast majority of readers. If they are, I'd be interested in reading any relevant material that you can cite. AP295 (talk) 18:46, 18 January 2021 (UTC)

You should read Monomial basis#Several indeterminates. You will find there an example of a basis that not only does not has a natural order, but of which many different orders (of the same basis) are commonly considered (see Monomial order). Your suggestion for changing the common terminology would imply that changing of monomial order would change the basis. This is definitively against the common practice that, if the indeterminates are fixed, the monomial basis is uniquely and explicitely defined. D.Lazard (talk) 18:55, 18 January 2021 (UTC)

What is meant by "natural order"? In R^n or C^n, the order of basis elements is (rather, may be) arbitrary but must be fixed in order for that basis to uniquely determine coefficient vectors/matrices. Basis vectors do not determine their own order in a basis per se. AP295 (talk) 19:04, 18 January 2021 (UTC)

I read the section. In that case, a monomial order is chosen to be compatible with the product, since having a unique, easy representation via coefficients is entirely the point, as far as I can see. AP295 (talk) 19:47, 18 January 2021 (UTC)

And the current lead of this article even states (as do many texts on the subject) that it pertains mostly to finite-dimensional vector spaces, in which case "sequence" is not only adequate, but likely more useful and less ambiguous. Even if we want to cover infinite-dimensional vector spaces, these differences could be resolved by calling them indexed families having separate sections in this article for finite, countable, and uncountable vector spaces, and then going into greater detail in each respective section, which is something I proposed earlier. AP295 (talk) 20:23, 18 January 2021 (UTC)

One more thing, according to Monomial basis#Several indeterminates, the basis elements are the set of all monomials: "...which has the set of all monomials as a basis". A total order on the set of all monomials (rather than just their indeterminates) does give an order on the coefficients of a polynomial comprised of those monomials. So yes, ordering the indeterminates does not fully specify an order on the coefficients, but that's not at all the same as ordering the basis elements, and not analogous to my point. It's also pretty far removed from the scope of this article. AP295 (talk) 20:38, 18 January 2021 (UTC)

• Set. It is usually defined as a set. Maybe it works to define it as a sequence (though would that still work for uncountable-dimension vector spaces?), but most sources define it as a set in my experience. —Granger (talk · contribs) 13:25, 19 January 2021 (UTC)

Since the question of uncountable bases has come up a few times (and is the only real concern I had in the first place), and to avoid being accused of "bludgeoning", I'll add the answer at the top instead of replying each time. AP295 (talk) 14:10, 19 January 2021 (UTC)

@JayBeeEll, @D.Lazard This book, "A Basis Theory Primer", specifically calls Hamel bases sequences: https://link.springer.com/chapter/10.1007%2F978-0-8176-4687-5_4 According to this source, a Hamil basis is a sequence, but the indexing set need not be countable. If you want to take a narrow view of the term "sequence" to mean sets indexed by the natural numbers then "indexed family" would be appropriate as well. Either is fine by me. AP295 (talk) 22:52, 19 January 2021 (UTC)

Any more comments? "A Basis Theory Primer" seems to settle the concern about uncountable Hamel bases. I'm new to the RfC process, but I'm disappointed that most RfCs on Wikipedia appear to be settled pollice verso, and not through measured comments and dialectic. WP:NOR applies to the articles, not to talk pages. I'm not interested in your upvotes/downvotes, I am interested in your two cents. Surely any democratic process is debased by participants who only see fit to cast their vote and not, once in a while, take a critical look at things and ask how they might be improved. AP295 (talk) 15:00, 20 January 2021 (UTC)

I've tried to think about how the article might present both definitions, but it is too awkward. The sequence definition is largely compatible with sources that use the set definition, since basis elements are unique anyway and you simply drop the indexing function. The set definition is not compatible with sources that use the sequence definition because they frequently rely on on the fact that a basis uniquely determines a coefficient vector, and two bases uniquely determine the matrix of a linear map. For example, chapter 5 of "Linear algebra done right". Many sources that discuss structured matrices frequently rely on a basis being ordered, and consider permutations of the same basis vectors as unique bases. The theoretical concern of whether "sequence" is appropriate to describe an uncountable basis is a good point, but my earlier reference, a book specifically about basis theory, calls them sequences and allows them to be indexed by an uncountable set. Likely the author found it convenient to call them sequences (implying a well-order in the countable cases) and simply relax the definition to include uncountable indexing sets when the sequence elements are uncountable. We can call them sequences and generalize to indexed families to discuss uncountable Hamel bases. I believe this article should maintain a clear distinction between countable bases and uncountable bases rather than just say "infinite basis", because they're quite different concepts. Finite/countable bases are likely to be of more interest to a vast majority of readers (even including advanced readers), and I feel that the article should not give undue weight to uncountable Hamel bases, although we can still reconcile them with a sequence-based definition of "basis". I will probably start editing the article at some point. This article is too important to remain at "C" quality so I must insist on these changes unless someone has any other technical/semantic concerns about the word "sequence". AP295 (talk) 15:26, 21 January 2021 (UTC)

You requested comments, and you got comments, mostly against your proposed change. I don't see how your proposal relates to the "C" quality rating. Why not work on that in a way that is less controversial? For the current a problem, it might fly to say "a set (often described as an ordered set or a sequence)". Dicklyon (talk) 03:19, 22 January 2021 (UTC)

@Dicklyon That would be even more confusing. There should be nothing controversial about this. The precedent for using "sequence" exists and it makes more sense for reasons I've already explained. I'd rather have it here as a sequence and explain that some authors define them with sets than the other way around. It's an important topic and this article needs to be sorted out sooner or later, and the definition is where we should start. Sure, the "set-based" definition can be made to work, but is it as easy to use or understand? There are more featured articles about different species of birds than there are about computing, mathematics, psychology and philosophy, health and medicine all combined. That is a travesty. Utterly shameful. AP295 (talk) 17:50, 22 January 2021 (UTC)

Yes, and your contribution is 0. (Negative, if we include all the time of other editors that you've wasted.) Go troll elsewhere. --JBL (talk) 18:05, 22 January 2021 (UTC)

Certainly not for lack of trying. AP295 (talk) 18:23, 22 January 2021 (UTC)

• I'd advise AP295 to focus a bit less on this conversation and focus on editing other maths articles to improve unambiguous error or lack of clarity (which shouldn't be hard to find). "Winning" this RfC wouldn't achieve all that much in comparison to how much you could do with that time in other areas, or even the same area. (For instance, you could write a lead with the set definition accessible to high school students, a challenging but attainable goal, per WP:ONEDOWN. It wouldn't even be that hard to change it to the sequence formulation if consensus did later arise at that.) I won't reply further because I have read this discussion in full and can't imagine new points being raised. Consensus is arising against a sequence definition, and for the current set definition. I support that—use sets, not sequences or families.
  From my perspective, I learned it with sets and "ordered basis" (as well as even the use of the term "basis" informally for "ordered basis") is undoubtedly well-accepted in mainstream mathematics, so it is not our place to object to it. I don't like "indexed family" because that just begs the question, "what is a family? (Is it a set? A class?)" I think uncountable sequences are a lot less natural than uncountable sets. It's slightly more natural to me to think of a basis without order, even though orders are often used "in practice". I guess it depends on your field—if you want to show that every vector space has a basis then you want (for simplicity) Zorn's lemma and (unordered) sets as bases. The simple fact is that none of the common types of collections (set, sequence, multiset etc.) fit the exact properties AP295 wants here: arbitrary cardinality; ordered; no repetition. Every definition will be undesirable in some way and with the set way, it's that "change of basis" really means "change of ordered basis" (so that we allow permutation matrices). — Bilorv (talk) 00:16, 26 January 2021 (UTC)

@Bilorv Looks like I'm outnumbered. The points you make are not unreasonable, but I hope you and everyone else in this RfC can recognize the extra layer of obscurity/ambiguity that comes with defining "basis" as a set without order. If our goal is to communicate mathematics rather than simply document mathematics, then we should think very seriously about it. What is the point of introducing the idea of a "basis" if not to speak about coefficient vectors and matrices of linear maps with respect to a basis or bases? For that purpose, an order must be fixed. Most people encounter the concept of a basis in an undergraduate linear algebra class, and undergraduate students likely comprise a large majority of the article's readers. Something like this "A sequence B of vectors in a vector space V is called a basis of V if every element of V may be written uniquely as a linear combination of the elements of B. Given a vector v (in the standard basis) and another basis B, there exist unique coefficients c1,...,cn such that v can be written as c1B1 + ... + cnBn. The vector [c1,...,cn] is the representation of v in the basis B.... etc." immediately motivates the concept of a "basis" and makes it clear what we mean when we talk about a vector in some other basis. The minor semantic issue of whether "sequence" is appropriate for uncountable bases is easily reconciled (as in the textbook I cited about basis theory) and not at all worth complicating the presentation of a simple and important idea. I hope you'll reconsider. AP295 (talk) 15:54, 8 February 2021 (UTC)

A basis is a set. The current definition is perfect. Awoma (talk) 22:28, 4 February 2021 (UTC)

No. A basis is a family (or net or sequence). What qualification do you have to make such a categorical statement as "A basis is a set."? I'm professor of mathematics at university, I have PhD, Habilitation, and many papers published in renowned international journals, and I'm telling you a basis is not a set, order of vectors matters (obviously - for defining co-ordinates of vectors and matrices of linear maps etc...).

I have taught linear algebra and definition of linearly dependent and independent families of vectors for over 25 years. Such families can have duplicates, in which case they are not linearly independent, but they can also not have duplicates and still be not linearly independent and therefore NOT be a basis. So, it's not because a basis can't have duplicates that it must be a set, it's even confusing and leading to misunderstanding and error to think that way. How can you make a change of basis where just some basis vectors are permuted (which affects the matrix of vectors and linear maps) if there is no order? How can you even define matrices of linear maps? You will not want to use the basis vectors as indices, that would be a little too much of original research....

Summarizing, as the O.P., A.P. above and other knowledgable persons in theis thread, I hope ("for you") that you'll reconsider this. — MFH:Talk 23:30, 24 January 2023 (UTC)

The easiest way to support this change would be to go look at a dozen or so popular linear algebra books, do a literature search of recent research papers, etc., and see how many of them use the term “basis” to refer to an ordered vs. unordered set of vectors (either implicitly or explicitly). If there is a clearly dominant definition in the literature, Wikipedia should probably stick to that; if both turn out to be about equally common, the article can mention the disagreement. –jacobolus (t) 04:03, 25 January 2023 (UTC)

For example, Greub (1963) Linear Algebra defines a basis as a "system of n linearly independent vectors ...", and immediately starts indexing them by numbers. Halmos (1947) Finite Dimensional Vector Spaces defines a basis as a "set of linearly independent vectors ...", and then immediately starts giving finite examples indexed by numbers. –jacobolus (t) 04:21, 25 January 2023 (UTC)

I've already done this above (but of course MFH hasn't actually read any of the discussion they're commenting on). --JBL (talk) 18:09, 25 January 2023 (UTC)

Why "of course"? 🤔 Of course I did... — MFH:Talk 04:19, 14 January 2024 (UTC)

I have read the article again to see what should be changed and what would be improved or disimproved if one would follow MFH suggestion.

First of all, everything in the article is presently mathematically correct, and changing the definition would induce many changes, which would be difficult to do without introducing logical errors.

In particular, in the infinite-dimensional case, a definition as a family or as a net would need to introduce (at least implicitly) an auxiliary set; I do not see any way to do this without some circular reasoning.

Also, the first bulleted property of § Properties would be much less clear without using set inclusion, and the proof would be more complicated.

Even in the finite dimensional case, this would be problematic to consider that bases must be ordered by definition. For example, the polynomials of degree d in n variables form a finite dimensional vector space, which has the monomials as a basis. There is no natural order on this basis, and so, the definition of a basis as a sequence could not be used easily in this case.

In summary, MFH's suggestion is philosophically interesting, but has not practical value. D.Lazard (talk) 10:28, 25 January 2023 (UTC)

It's not a suggestion but a plain stubborn fact, that the order of vectors of a basis does matter, whereas the elements of a set do not have a well defined order. In particular, for sets you have {u, v, w} = {w, u, v}, so "these" two "bases" would be the same, but there is a nontrivial permutation matrix associated to the change of a basis (u, v, w) to the basis (w, u, v). Once again, if a basis was a set these two would be the same and one could not consider a change of basis from one to the other, but one does and they aren't. — MFH:Talk 04:28, 14 January 2024 (UTC)

"in mathematics" vs "in linear algebra"

Given that the article is entitled "Basis (linear algebra)", the first sentence should read "in linear algebra" rather than "in mathematics" to establish an appropriate context for the article. "in mathematics" seems unnecessarily vague and, lacking a clear scope, many articles seem to become very fragmented and disjointed. AP295 (talk) 16:33, 21 January 2021 (UTC)

This has been discussed many times in Wikipedia; see MOS:MATH#Article introduction. This is aimed to help readers with little knowledge to decide whether this article may be useful for them. As "linear algebra" is in the article title, repeating it in the first sentence is totally useless. Moreover, readers who have encountered the topic in another area of mathematics or physics may think that the article is not for them if the first sentence is too restrictive. This is why it is standard in Wikipedia to put "in mathematics" even for very specialized subjects, as soon as it is used in many areas of mathematics. So your opinion goes against a long standing consensus. D.Lazard (talk) 16:52, 21 January 2021 (UTC)

From variance, In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean. The concept of "variance" is used in many many different fields. Sociology, biology, economics, etc. and in such instances it is commonly understood as an application of probability/statistics. Similarly, linear algebra can be applied in many areas. From the article on linear algebra, Vector spaces are the subject of linear algebra. Is the word "basis" used for anything other than the basis of a vector space? And if so, would such cases belong in this article? I don't think so. This change isn't that important so fine, we can keep it this way. I don't understand you though. There are only nine featured articles in mathematics. This is a low quality article and it needs to improve. AP295 (talk) 17:14, 21 January 2021 (UTC)

And bizarrely, the page Linear algebra isn't even linked in this page until below "external links". AP295 (talk) 17:17, 21 January 2021 (UTC)

Actually, I think my edit agrees with MOS:MATH#Article introduction. Any "basis" not pertaining to vector spaces (the subject of linear algebra) is probably best covered in a separate article. AP295 (talk) 17:52, 21 January 2021 (UTC)

The article title specifies the subfield, so repeating it does not help anyone, whereas if someone doesn't know what is linear algebra then it is more helpful to let them know it is part of mathematics. --JBL (talk) 18:10, 21 January 2021 (UTC)

Happens all the time. I thought the article was about baseball before I read that it was actually about math. Imagine my shock when I learned the cardinal numbers weren't a baseball team. AP295 (talk) 19:17, 21 January 2021 (UTC)

A high-schooler doesn't know what "linear algebra" is, but does know what mathematics is, and the lead should be accessible to them. Just knowing the two words "linear" or "algebra" could reasonably given a non-mathematician the thought that it might be a part of economics or statistics, or it might be a nonsense phrase they can't make sense of. "In mathematics" is better. — Bilorv (talk) 01:46, 3 February 2021 (UTC)

I think mathematics is the best choice here, as anyone working in mathematics who says "basis" is referring to this concept, regardless of whether they are working in linear algebra or not. Similarly, the article on square number does not say "in number theory". Awoma (talk) 17:44, 16 February 2021 (UTC)

Bold or italic is required

@D.Lazard: Hi, the object of the verb "called" need bold or italic font. But here as you say "target of a redirect" we should use italic instead of bold. Is that Ok? Hooman Mallahzadeh (talk) 13:44, 17 November 2021 (UTC)

In this case, bold is forbidden per MOS:NOBOLD. If the definition would be here, emphasizing with italic would be recommended. Here, the definition is in the linked article, and a sufficient emphasis is provided by the blue of the link. D.Lazard (talk) 14:06, 17 November 2021 (UTC)

@D.Lazard: So, please review my last edit. Thanks. Hooman Mallahzadeh (talk) 14:13, 17 November 2021 (UTC)

"Basis(mathematics)" listed at Redirects for discussion

An editor has identified a potential problem with the redirect Basis(mathematics) and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 April 22#Basis(mathematics) until a consensus is reached, and readers of this page are welcome to contribute to the discussion. Steel1943 (talk) 06:00, 22 April 2022 (UTC)

Shorter proof of basis theorem

Every well-ordered vector space has the basis of vectors that aren't generated by earlier ones. 2A02:3032:207:881D:53D9:1E88:A103:AE94 (talk) 20:05, 11 May 2024 (UTC)