# Talk:Vector space/Archive 1

(Redirected from Talk:Vector space/Archive)
 Archive 1 Archive 2

## "Associativity" of scalar multiplication

I am concerned that axiom 7 of the current axioms is misnamed. It says that "scalar mutliplication is associative". There are, however, two multiplication operations in question, one from Field x Field -> Field and the other Field x Space -> Field. In Associativity, one finds a definition of "associative" that applies to only one binary operation on a set.

I would like to rename the axiom in question as the "consistency axiom", because it asserts the consistency of the two multiplications. Ben 10:12, 22 July 2006 (UTC)

I'm going to edit the page to change this, since I think it's an error. If you don't agree, please tell me why. Ben 09:19, 28 July 2006 (UTC)

## Notation for operators

For the sake of correctness. The * was used in the vector space axioms both as a map * : F x F -> F and as a map * : F x V -> V. I changed a*b to ab. What about the double used + : F x F -> F and + : V x V -> V? -- Georg Muntingh

I wouldn't worry about it. Usually, each multiplication is written without a symbol, and each addition with "+". The reader has to infer from context what operation it is, and this is possible by looking at the elements operated on and asking where they came from. If this is too much for the reader to handle, they probably won't understand it anyway. (Sorry if I sound dismissive.) Otherwise, we would have 4 or 5 different symbols, (+, _, #, *, ...??) and this is just incredibly cluttered. Revolver 21:40, 9 Feb 2004 (UTC)

To say the same things in more technical language: we have a case here of operator overloading (see also overloading); which is not necessarily a bad thing when types can be inferred. Charles Matthews 19:13, 11 Feb 2004 (UTC)

How about using + for vector addition and + for field addition? —Daniel Brockman 07:16, Mar 8, 2004 (UTC)

The section on 'vectors in physics' really belongs at vector (spatial), rather than here.

Charles Matthews 09:24, 23 Feb 2004 (UTC)

## -x + x = 0

I agree about the 'vectors in physics'... Is it really true what is atated "Note that property 5 (commutativity) actually follows from the other 9"? I am almost certain that we need to modify the 4:th axiom (exist -x: x + -x = 0) so that the invers of a vector is commutative in the following way: "exist -x: x + -x = -x + x = 0" for the statement to hold. So I will now modify the 4:th axiom myself.. I would be glad to see a proof of the statement if it was true in its original version. Dj, 14 Mar 2004

The extra axiom you have added (-x + x = 0) is not needed, as it follows from the first four axioms. Any introductory book on group theory should have a proof, but here's one anyway. Suppose s is an idempotent (that is, s + s = s). Then s = s + 0 = s + (s + -s) = (s + s) + -s = s + -s = 0. So 0 is the only idempotent. For any x we have (-x + x) + (-x + x) = -x + (x + -x) + x = -x + 0 + x = -x + x, that is, -x + x is an idempotent, so -x + x = 0. --Zundark 09:09, 14 Mar 2004 (UTC)

## Typo

Sirs:

You have a mathematical typesetting error in statement 4 of your formal definition of a vector space over a field F.

In the last part of that sentence, when you type v + w = 0, the 0 should be in BOLD FACE, because it is a vector 0, and not a scaler 0.

Regards, Harold

You are correct; I've fixed it now. (You could have fixed it yourself.) --Zundark 19:07, 26 Jun 2004 (UTC)

## Property 5

How does property 5 follow from the other nine? I've added this as an open mathematical question, since it seems to have gone unanswered for several months. Prumpf 18:01, 15 Aug 2004 (UTC)

(x+x)+(y+y) = (1+1)(x+y) = (x+y)+(x+y), so x+y = y+x. The same proof shows more generally that you can't have a "non-abelian module". --Zundark 20:01, 15 Aug 2004 (UTC)

## F is an abelian group

It should be noted that F is an abelian group. Stefan Udrea 12:01, 29 October 2005 (UTC)

F is a field, so by definition, it is an abelian group under both addition and multiplication. --203.206.183.160 07:45, 30 October 2006 (UTC)

## Connection

Using the vector space R and a ultrafilter U on the natural numbers we can creat a hyperreal field!

## removal of closure axioms

I've added them back in since most literature I've seen lists them as part of the formal axioms. It would seem to me that they are also fairly important. Why were they removed? Does some other part of the definition imply the closure axioms? --Antiduh 02:26, 29 July 2005 (UTC)

Yes, see the text

A vector space over a field F (such as the field of real or of complex numbers) is a set V together with two operations:

• vector addition: V × VV denoted v + w, where v, wV, and
• scalar multiplication: F × VV denoted a v, where aF and vV.

which satisfy following axioms (for all a, bF and u, v, and wV):

in the article. Oleg Alexandrov 02:31, 29 July 2005 (UTC)

I think this is obviscates an important point and steepens the learning curve. Yes, the codomain specification does imply closure. However, it's easy to overlook, particularly if your a student just learning about vector spaces. No harm is done by including closure explicitly, and significant good is done removing obviscation. -- T, September 27, 2008 —Preceding unsigned comment added by 64.178.105.25 (talk) 15:17, 27 September 2008 (UTC)

## etymology

i have always wondered the origins of the term "linear spaces". what is linear about spaces or vectors for that matter?

you can take linear combination and remain in the same space. --MarSch 12:14, 31 October 2005 (UTC)
linear functions are called linear because their graphs are lines. -lethe talk 04:03, 24 January 2006 (UTC)

## someone defend this paragraph

"The concept of a vector space is entirely abstract, like the concept of a field itself, and analogous to the concept of a module over a ring, of which it is a specialization. To determine if a set V is a vector space, one only has to specify the set V, a field F, and define vector addition and scalar multiplication in V. Then, if V satisfies the above eight axioms, it is a vector space over the field F."

I changed the word "concept" to "formal definition", since the concept of a vector space (as well as a group or field, but perhaps not module) is older than the 20th centuries reliance on set theory for formal definitions. But it seems to me that the whole paragraph is spurious; all it says is that a vector space is a set which satisfies the axioms. Entirely redundant, it would seem.

OK, so I know that beginning students might choke on the fact that polynomials can form a vector space, having had it drummed into their heads that vectors are arrows with magnitude and direction. So it does need to be emphasized that the modern formal definition is more abstract, but I don't think this paragraph does it. -Lethe | Talk 15:24, 13 November 2005 (UTC)

I notice that the introduction contains a paragraph which attempts to address this concern. I'm inclined to simply delete the paragraph causing offense. -Lethe | Talk 15:28, 13 November 2005 (UTC)

I have a question for you Lethe: If we define a vector space of the complex numbers with scalars being reals, and then define addition to (x,y) + (a,b) = (cube root of (x+a, y+b)), and the scalar multiplication being the normal operation with complex numbers. Then the sum of two objects would equal three different ones, since we are taking the cube root of a complex number. Can we say that v+w = w+v?. I'm a bit confused since v+w can equal three numbers, and so can w+v, so how can we say its equal? As you said I'm only a beginner at this, so you would expect a bit of simple questions. —Preceding unsigned comment added by 211.31.37.88 (talkcontribs)

Well, you're right. Since there are three distinct complex cube roots for all nonzero complex numbers, then defining any operation of complex numbers as the cube root of something is ambiguous. To have a well-defined operation, you must specify which cube root you want. If you did so, you would have complications with continuity, since the cube root function has branch cuts. Even if you take care of the choice of cube root, and don't worry about continuity, your proposal suffers from a more immediate problem: it doesn't define a vector space because this choice of addition does not satisfy distributivity. You don't have k(a+b)=ka+kb. This choice of addition is also not associative, which means that you don't have a+(b+c)=(a+b)+c. So this isn't a vector space. Addition in vector spaces is always "linear", which basically just means abelian, associative, and distributive. In such a case, you won't run into the problems that multi-valued functions have; those functions are not linear. -lethe talk 03:47, 24 January 2006 (UTC)

## New formatting

I must say I liked the old formatting better than the new one. The article now looks like its shouting at me. Can we go back? -- Fropuff 06:51, 26 January 2006 (UTC)

I thought it’d be good to use standard definition lists, but maybe you’re right about the shouting. I’ve now changed it to a lighter style that seems to me like a nice compromise. What do you think? — Daniel Brockman 08:41, 27 January 2006 (UTC)

## Commutativity

I think it should be interesting to note that commutativity does not necessarily need to be an axiom as it follows from the rest. This is what has been taught in my algebra class, and shown online in some places. Generally, though, most web pages seem to include it (although those are not definitive sources of proof.) Proof as soon as I find it again. WingZero 09:23, 5 February 2006 (UTC)

## Help w/ disambiguation

I'm trying to fix links to three-dimensional. When an article is referring to dimensions of a mathematical space, such as: Ambient space, is it appropriate to say: "three-dimensional space"; linking to this article? Is an n-dimensional space always a vector space? EricR 02:53, 17 February 2006 (UTC)

No, an n-dimensional space could be a topological space (e.g., a manifold), for example. --Zundark 08:37, 17 February 2006 (UTC)

## Merging from Complex Vector Spaces

See talk:Complex Vector Spaces. Oleg Alexandrov (talk) 02:14, 18 February 2006 (UTC)

## How many axioms

The section on the formal definition ends "Then V is a vector space over the field F if and only if it satisfies the eight axioms listed above." However, there are 10 axioms above. I guess this is just a mistake, but thought I would ask here before changing it in case two of the axioms are redundant or something. —This unsigned comment was added by 139.184.30.16 (talkcontribs) .

I'd bet money that the reason for the discrepancy is the two closure axioms. It's not usual to list closure as axioms, products on sets are closed by their very definition, in the formal set-theoretic formulation. Since this section advertises itelf as being formal, I would advocate removing the closure axioms, rather than adopting 10 axioms. Note that we can still mention the closure axioms, we just should mention that it's not necessary to assume them. This is what is currently done at group (mathematics), and I'm in favor of doing the same thing here. What do you think? -lethe talk + 15:58, 20 March 2006 (UTC)
I agree. In fact this is somehow a "pedagogical" issue : if one defines "a vector subspace is a subset which is a vector space (for the induced laws)" then usually the closure properties are the most important ones to verify ; but if you define a vector space from scratch, you usually say : two maps + : E x E -> E and * : K x E -> E such that... (with only the 8 final axioms) - since the closure prop. is "automatic" if +,* is defined as a map from there to there. I suggest explaining it like this, more explicitely, but maybe in the (a?) section on subspaces and not right at the beginning. I'm rather in favour of defining +,* correctly as maps and to delete the closure axioms here. — MFH:Talk 22:41, 20 March 2006 (UTC)
PS: just noticed higher on this page : Talk:Vector_space#removal of closure axiomsMFH:Talk 22:45, 20 March 2006 (UTC)

## the axioms

Per the above discussion, I have moved the closure axioms to the bottom and de-emphasized them. Also, I find a long unorganized list of 10 axioms a bit to hard to absorb, I have to understand the vector space axioms in terms of what structures they make, so I've rephrased the axioms that way: vectors form an abelian group, each scalar multiplication is a group homomorphism, and the map to group homomorphisms is a ring homomorphism. Unfortunately, this makes the axioms more abstract than strictly necessary, and I wonder if a student in his first linear algebra course would be put off by this. The section did promise it was going to be formal though. What you guys think? -lethe talk + 03:58, 21 March 2006 (UTC)

I like the change, but I also think that some of the group titles are too complicated: While I think that "abelian group" is OK for the +, "Map to scalar action is a homomorphism of rings" is really confusing (what exactly is the second ring?). Also, maybe one should better write "multiplication by a fixed scalar is a group homomorphism". (and this is also a somehow complicated expression for a distributive law.)
Personally I prefer grouping the 2 "action"/operation properties (involving only scalar mult.: 1x=x and a(bx)=(ab)x - which is b.t.w. also an associativity property in a generalized sense), and the 2 distributive laws. (Rather for mnemotechnic / pedagogical reasons than for reasons of mathematical logic, maybe, I admit.) — MFH:Talk 22:26, 21 March 2006 (UTC)
Yeah, you're right. I've removed those headings. Furhermore, I'm starting to think that talking about the scalar action as homomorphisms and the association of scalars to homomorphisms as a ring homomorphism is something that really belongs in the module article, not here. I'm surprised it's not there already, actually. I'm going to add it there, and then maybe we'll decide whether we want to keep that description here (looking for input here). Anyway, in the mean time, while I was removing the headings, I reworked and removed a lot of stuff, so have a look at the diff and bring your comments. -lethe talk + 13:15, 24 March 2006 (UTC)

## Definition

Um, it would be very helpful if there was a definition at top. The only definition i can see on this page is the long-winded formal definition. There should be a concise definition at the top. Saying that its a "basic object of study" doesn't cut it. Its like definin chocolate chip cookie as "a type of food that many people eat as a dessert in north america". Fresheneesz 20:52, 27 March 2006 (UTC)

The third sentence of the intro is "In general, a vector space is any abstract mathematical structure on which these operations [scaling and addition], satisfying their natural axioms, are defined." Can you mention why this does not meet your needs? -lethe talk + 21:02, 27 March 2006 (UTC)
Sure. Its very vauge. "Any object" seems like it could be just that - any object. However, a mathematical object is left undefined. Where and apple is an object in the physical sense, the information that an apple comprises must be converted or interpreted to represent a different type of object: a mathematical object. Now, obviously I know that a scalar and a vector are objects (and sets and variables, and matricies, etc), but the meaning of "object" isn't bound by anything. In short, that sentence could mean practically *anything* because it doesn't define "object". And I don't know how to go about seeing if some axioms are defined on the objects. Actually, by saying that, doesn't it imply that one must simply define the axioms on the objects, which doesn't quite limit the scope of an "object".
Sorry if thats a little long winded, but since I don't understand exactly what its trying to say, I can't try to make it better. Fresheneesz 09:16, 28 March 2006 (UTC)
Now you seem to be complaining about the use of the word "object" in the fourth sentence, whereas I asked you about the definition in the third sentence. But OK, I'll try to keep up with your changing whims. So you don't like the word object? Let's see if we can find a better word then. How do you like it now? -lethe talk + 09:45, 28 March 2006 (UTC)

I disagree with putting the definition on top. Fresheneesz, Wikipedia is a general-purpose encyclopedia. Treat the Wikipedia articles like essays. I would very strongly disagree with making things more concise and more techical than they already are. Oleg Alexandrov (talk) 19:54, 28 March 2006 (UTC)

Oh no i definately agree with you Oleg, putting that formal definition would be very confusing. I'm looking for a definition that is more understandable, less vauge, but not technical.
Sorry, i guess I read structure and thought object (I sorta see them as meaning the same thing in this context). I suppose its slightly better, but its still quite vauge. So vector space is a set, I would add that to the definition (probably to oleg's shagrin), but I'm not quite sure what its a set of. And must a vector space be defined "over" a "field"? Fresheneesz 11:41, 29 March 2006 (UTC)
Vector spaces have little to do with set theory, and I think connecting linear algebra to formal set theory is dubious. I oppose such an addition (it's there in the formal def anyway). But let's hear what Oleg says. As for what a vector space is a set of, well it's a set of vectors. Not really sure what more you're looking for there. Finally, yes, a vector space is always defined over a field. -lethe talk + 13:09, 29 March 2006 (UTC)
Agree with Lethe. Fresh, of course vector spaces are over fields, and that for very many, very good reasons. :) Oleg Alexandrov (talk) 19:38, 29 March 2006 (UTC)
One of these reasons is that if it is not over a field but over a ring, then it's not called a vector space but a module. -- Jitse Niesen (talk) 03:07, 30 March 2006 (UTC)
Hmm, i guess i'll just leave these topics alone. Theres too much I don't understand about them. Fresheneesz 05:50, 31 March 2006 (UTC)

I see that the first sentence has now been changed to

"Vector spaces (or linear spaces) are algebraic sets of mathematical objects satisfying a number of basic relationships. The most common instance of a vector space is the n-dimensional coordinate space in geometry, whose vectors are ordered n-tuples of numbers."

I'm afraid I don't like this very much: what are algebraic sets? The vectors themselves do not satisfy "basic relationships", but the vectors plus the operations define on them do. It is badly integrated with the next sentence ("The most familiar vector spaces are spaces of geometrical vectors, usually depicted as arrows with magnitude and direction."), which steps back and takes a more concrete approach.

However, the previous version was

"Vector spaces (or linear spaces) are the basic object of study in linear algebra, and are used extensively in almost every area of mathematics and many branches of the sciences."

I don't like that very much either, because it does not actually define the term, so I did not revert.

Unfortunately, I don't have a satisfactory proposal right now, but I intend to change it as soon as inspiration strikes. -- Jitse Niesen (talk) 06:34, 7 April 2006 (UTC)

I had also noticed that the newest edit is not an improvement, rather the other way around, for the reason you noticed. I reverted, for now. Looking forward to Jitse being struck by inspiration. :) Oleg Alexandrov (talk) 07:14, 7 April 2006 (UTC)
I am also not so pleased. I find the new edits confusing (or perhaps confused). -lethe talk + 07:51, 7 April 2006 (UTC)

## new intro

I've reverted Markan's new intro. It had some things I liked, but also enough things that I didn't like that I decided to revert. Let me list those things I didn't like: this article should be as accessible as possible to as many people as possible in its first sentence. Things like "mathematical object", "set", and "operations defined with respect to it" are needlessly formal for the first sentence and will scare people off. The old intro was much more gentle; it first gave the idea, but didn't give the formal notion until the second paragraph. There are some sloppinesses. Scalar multiplication and addition are not properties, they are operations. A large number of words is spent detailing closure in the third sentence, which I don't like. Let me recommend also in the future not to capitalize stuff in mid-sentence (unless you've a proper noun). -lethe talk + 13:42, 28 June 2006 (UTC)

I see your point about formality, but I still think it's necessary (see below). Regarding "property," I use that word informally, because I'm trying to get across the intuition that what's special about a vector space is the fact that you can meaningfully take sums/products (hence the implicit emphasis on closure). I'm open to a better phrasing there. The capital letter is of course a mistake but hardly relevant. Markan 14:07, 28 June 2006 (UTC)

The use of the word "object" in the first sentence is not good (in my opinion). First, the word "object" is often used in category theory and could be confusing. Secondly, a vector space is just a set. This is simple. If the article is to be made more accessible, it might be better to emphasise the examples of R^2 or R^3. In lower division classes on linear algebra, the notion of a vector space is often omitted, and attention is restricted to these types of examples. 19 July 2006.

## proposed new intro

A Vector space (or linear space) is a mathematical object consisting of a set and two operations defined with respect to it: vector addition and scalar multiplication. The elements of a vector space are known as vectors. Formally, the two operations are defined to satisfy certain axioms, detailed below. But informally, a vector space is just a mathematical space distinguished by two special properties: vectors may be added, yielding some other vector in the space, and a vector may be multiplied by a number, again yielding a vector in the space. Vector spaces are the basic object of study in linear algebra, and are used throughout engineering and the sciences to represent physical quantities.

The most familiar vector spaces are the two- and three-dimensional Euclidean spaces, in which the vectors correspond to geometric vectors--quantities with a magnitude and a direction, often depicted as arrows. These vectors may be added together with the triangle rule (vector addition) or scaled by real numbers (scalar multiplication). The behavior of geometric vectors under these operations provides a good informal model for the behavior of vectors in more abstract vector spaces, which need not have a geometric interpretation. (For example, the polynomials with real coefficients form a vector space.)

I tried this and got reverted, so I figured I'd put it up for discussion here. A few comments:

• Regarding the level of formality, the notion of a "vector space" is inherently abstract; I think it's only logical to give a precise definition first, leaving technicality for later, and then follow up with some intuition, as the proposed first paragraph does. I'm not convinced it's meaningful to describe a VS as a space in which things can be "added" and "scaled" without defining those terms and making it clear that those two operations are whatever one defines them to be.
• The second sentence in the current version seems a bit imprecise to me; I'd say that it's linear algebra that provides the framework for studying linear phenomena.
• I tried to preserve the essence of the second paragraph while moving the central statement about what a VS is to the first paragraph. I think the main point is to relate the idea of a vector space to the notion of a geometric vector. But the current version also makes passing reference to n-tuples and the correspondence between them and geometric vectors, all of which is good and should get mentioned somewhere, but which dilutes the paragraph. We should realize that if you don't know what a vector space is, the second paragraph as it stands will be confusing.
• In the current version, the phrases "abstract structure," "natural axioms," and "the axioms" are all used imprecisely.

Thoughts?Markan 14:00, 28 June 2006 (UTC)

I feel that it's not true that vector spaces can only be understood abstractly. Many high school students learn the rules of addition and scaling of arrows in a very intuitive way. I Have been through several wars between people who want widespread accessibility and people who want technical accuracy. I'm usually in the latter camp, but I think there is wide agreement that the first couple of sentence or so should be readable by really everyone. The first couple of sentences should describe what a vector space is in a way that anyone can understand who learned that vectors are arrows with magnitude and direction with the triangle rule for adding. Abstract ideas like sets and operations and axioms should wait until later. We can haggle over other stylistic issues, but let's decide on this big issue first. -lethe talk + 17:52, 28 June 2006 (UTC)
Ok, I'm willing to start off with an informal definition as long as we mark it as such--how about this?
Informally, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. More formally, a vector space is a set together with two operations defined with respect to it: vector addition and scalar multiplication. These two operations are defined to satisfy certain axioms, detailed below. Vector spaces are the basic object of study in linear algebra, and are used throughout engineering and the sciences to represent physical quantities.
The most familiar vector spaces are the two- and three-dimensional Euclidean spaces, in which the vectors correspond to geometric vectors--quantities with a magnitude and a direction, often depicted as arrows. These vectors may be added together with the triangle rule (vector addition) or scaled by real numbers (scalar multiplication). The behavior of geometric vectors under these operations provides a good informal model for the behavior of vectors in more abstract vector spaces, which need not have a geometric interpretation. (For example, the polynomials with real coefficients form a vector space.)
Markan 07:24, 29 June 2006 (UTC)
Alright, I think this is better. Put it into the article. I have some minor alterations in mind for this intro, which I'll just do and we can discuss them after you see them. -lethe talk + 15:22, 29 June 2006 (UTC)
I think your new intro contains several improvements over the old intro. I want to thank you for your forbearance in the face of reverts of your work. That is often my first reaction to to largish undiscussed edits to important articles. Anyway, as promised, I have made several changes to your new intro. Your comments are welcome. -lethe talk + 19:57, 29 June 2006 (UTC)

Triangle law is not a good term. The standard term is parallelogram law:

http://mathworld.wolfram.com/ParallelogramLaw.html

## polyglut's ideas

It seems that Polyglut is insistent that a vector be defined as an n-tuple. Unfortunately this definition is insufficient for many reasons. Nevertheless, that is a very common notion of vector spaces, and deserves coverage in this article. It actually already has coverage; in the section labeled subspaces and bases, we find the sentence

"A basis makes it possible to express every vector of the space as a unique combination of the field elements. Sometimes, vector spaces are introduced from this coordinatised viewpoint."

which seems to address the issue. Does this mention not meet our needs for some reason? -lethe talk + 21:36, 29 June 2006 (UTC)

## Linear transformation

I don't think that this article should discuss Linear Transformations as there is a separate entry for this topic. The French version of this entry does not and, as far as I can tell, neither does the German version. --anon

Well, it is just a small section at the bottom of the article, while referring to linear transformation for more details. I think things are fine the way they are.  :) Oleg Alexandrov (talk) 08:34, 22 July 2006 (UTC)
Linear transformations are the natural maps to consider when studying vector spaces, because they preserve the structure. That seems to me to be an excellent reason for discussing them here. I don't see why their having a separate entry means that we shouldn't discuss linear transformations briefly in this article. -- Jitse Niesen (talk) 14:04, 22 July 2006 (UTC)

## Quotient Space

It might be useful to have a discussion of quotient spaces.-Englebert, July 25, 2006.

## Improvements

What are the main improvements that are needed to move this article to the next class? --—The preceding unsigned comment was added by Englebert (talkcontribs) .

Quite a bit could be done. There is no history - who developed the concept, what where the major developments, and by whom? It lacks any illustrations, which could be useful for a visual topic. I'd like to see a brief mention of some of the examples, and it also needs better inline referencing.
The next step up is Wikipedia:Good articles status which is quite tough and involves a review by a third party. I've been nominating a few maths articles for GA status recently. Looking at these may help you get a feel for whats required
There is some coordination of improving key maths articles at Wikipedia:WikiProject Mathematics/Wikipedia 1.0. --Salix alba (talk) 08:42, 25 July 2006 (UTC)

## Subspace and bases

I would like to change section 4 to a discussion of subspaces, quotient spaces, direct products etc, renaming it to something appropriate, and move bases to a newly created section that discusses them and related ideas (linear independence, spanning sets etc). These latter concepts are, of course, very important and should, perhaps, be discussed more fully. Any support for this idea? Englebert 09:34, 28 July 2006 (UTC)

What about sums of vector spaces, like ${\displaystyle \oplus }$ and ${\displaystyle +}$?

## we & one

Hi, Would it not be better to remove the occurences of 'we' and 'one' in the article? On a related note, in the list of axioms in the formal defintion, there seems to be a lot of words. Given that this is a formal defintion, it is not reasonable to be more concise? Englebert 05:16, 14 August 2006 (UTC)

I assume the reasoning is to make sure it doesn't end up like Godel's Incompleteness Theorem's mathspeak. Fephisto 03:55, 13 December 2006 (UTC)

## Diagram

Some diagrams would be nice. For example, diagrams of adding vectors and of scalar multiplication--enough to show the basic properties. Beyond that it would be nice to have a diagram for each of the rules. —Ben FrantzDale 21:12, 6 May 2007 (UTC)