Examples of vectors
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Examples of vectors
Hello. Can you give me a few examples of vectors, so that I can fully grasp this concept? It's not that I do not understand any certain part of this, and I've read the wikipedia page on vector spaces. So vector spaces are groups of vectors. And what are vectors? They are items onto which we can define addition and multiplication, but can they be for example water droplets? I don't understand how a vector space with such vectors would work.
Besides a series of numbers (a,b,c,...), and functions, what other can vectors be? And what is the difference between geometric vectors in a line, plane, space... etc, and (a), (a,b) (a,b,c) ... (that is, vectors as a series of numbers)?
Besides a series of numbers (a,b,c,...), and functions, what other can vectors be? And what is the difference between geometric vectors in a line, plane, space... etc, and (a), (a,b) (a,b,c) ... (that is, vectors as a series of numbers)?
 phlip
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Re: Examples of vectors
Barcode wrote:Hello. Can you give me a few examples of vectors
(0,0)
(1,5,2)
(4,7.3,[imath]\sqrt{\pi}[/imath])
... Sorry.
Barcode wrote:And what is the difference between geometric vectors in a line, plane, space... etc, and (a), (a,b) (a,b,c) ... (that is, vectors as a series of numbers)?
Basically the same as the difference between a bag of a dozen apples, and the number 12.
R^{3}, say, as a vector space is just a bunch of abstract numbers... but they happen to be useful in they can represent positions and velocities and such in 3D space... and when viewed in this way, things like vector addition, scalar multiplication, norms and dot products all have physical analogues. In much the same way as when you're using the integers to represent bags of apples, things like addition/subtraction/multiplication/division on those numbers are all analagous to physical operations on the apples themselves.
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Re: Examples of vectors
It's good to keep in mind that mathematically speaking it doesn't make very much sense to say that something by itself is a vector. It makes more sense to say that some collection of objects is a vector space.
One common example of a vector space is the space of possible forces on an object. It makes sense to add them, because two forces applied to an object is somehow equivalent to one force, their sum. It makes sense to multiply them by a number, because you can pull twice as hard, or half as hard in the opposite direction. These two operations interact in the way you'd expect.
However, it turns out that vector spaces are a much more general thing, and they pop up all over the place (in some sense, that's because we understand them pretty well so we try to use them as much as we can).
For example, the set of all functions from the real numbers to the real numbers is a vector space.^{[1]} You can add two of them by taking (f+g)(x) = f(x)+g(x) and you can multiply one by a real number by taking (t*f)(x) = t*f(x). It's infinitedimensional, of course. Thinking of this (and subspaces of it) as a vector space is sometimes conceptually a very useful thing to do.
[1]: More generally, the set of all functions from any set to the real numbers is a vector space. The usual vector spaces written as R^{n} are an example of this, since strictly speaking an ordered ntuple (x_{1}, x_{2}, ..., x_{n}) is just a function from the set {1, 2, ..., n} to the real numbers. Specifically, it's the one which takes k to x_{k}.
One common example of a vector space is the space of possible forces on an object. It makes sense to add them, because two forces applied to an object is somehow equivalent to one force, their sum. It makes sense to multiply them by a number, because you can pull twice as hard, or half as hard in the opposite direction. These two operations interact in the way you'd expect.
However, it turns out that vector spaces are a much more general thing, and they pop up all over the place (in some sense, that's because we understand them pretty well so we try to use them as much as we can).
For example, the set of all functions from the real numbers to the real numbers is a vector space.^{[1]} You can add two of them by taking (f+g)(x) = f(x)+g(x) and you can multiply one by a real number by taking (t*f)(x) = t*f(x). It's infinitedimensional, of course. Thinking of this (and subspaces of it) as a vector space is sometimes conceptually a very useful thing to do.
[1]: More generally, the set of all functions from any set to the real numbers is a vector space. The usual vector spaces written as R^{n} are an example of this, since strictly speaking an ordered ntuple (x_{1}, x_{2}, ..., x_{n}) is just a function from the set {1, 2, ..., n} to the real numbers. Specifically, it's the one which takes k to x_{k}.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?
Re: Examples of vectors
Don't know how helpful this is, but am explaining this as best as I can. If anyone can add something to clarify what I'm saying or flatout jinx it as bad pedagogical juju, go ahead.
Anyway, Half of the intuituve high school concept of vectors I actually find to be pretty solid. Namely, vectors are objects that have two components: a magnitude and a direction. Geometrically, you can think of vectors as encoding all the places you can go to from the origin if you move in some direction a certain amount(the magnitude). Hence, we can uniquely represent (ndimensional) vectors as points in ndimensional space, where the origin has 0 for all of its coordinates. Think about a "ball" of arrows, all of which eminate from a single point, which we call the origin, and the arrows going in a bunch of directions, and as far out sa you please: these arrows would be the vectors.
The questions that rise then are what does it mean for two vectors to be in the same direction, and what distinguishes one magnitude from another. Well, naively, we say that two vectors go in the same direction if one's coordinates are a multiple of the other's. For example, in 2D space ([imath]\mathcal{R}^2[\imath]) the vector (think arrows going around in a circle centered around (0,0)) (3,4) and the vector (1/4,1/3) are in the same direction because (3,4)=12(1/4,1/3).
So the concept of direction, vectorwise, is closely related to the concept of multiplication by numbers, which we call scalars. But we want to be able to multiply by numbers in order to talk about directions. However, we also know that (1/4,1/3)=1/12(3,4)., i.e. (1/4,1/3) is in the same direction as (3,4). So, intuitively, our scalars need to have multiplicative inverses, which makes them into a field (the scalars inherit additive properties from the idea of how vector addition should work; just think about adding arrows whose heads are given by coordinates, and think about what happens when you scale the arrows by numbers and then add them together). So the point is that the idea of a vector inherently holds in itself the idea of a vector space hat is an abelian group of vectors and field of scalars so that scalar multiplication on the vectors works properly.
The magnitude question I will not answer, because it doesn't really make sense in a general setting, whereas the idea of a direction works well enough. The high school idea of magnitude though is more precisely expressed by the fact that we can tell if vectors along the same direction are different. The ordered part of it comes from the field of scalars being ordered, which is generally not the case.
Anyway, Half of the intuituve high school concept of vectors I actually find to be pretty solid. Namely, vectors are objects that have two components: a magnitude and a direction. Geometrically, you can think of vectors as encoding all the places you can go to from the origin if you move in some direction a certain amount(the magnitude). Hence, we can uniquely represent (ndimensional) vectors as points in ndimensional space, where the origin has 0 for all of its coordinates. Think about a "ball" of arrows, all of which eminate from a single point, which we call the origin, and the arrows going in a bunch of directions, and as far out sa you please: these arrows would be the vectors.
The questions that rise then are what does it mean for two vectors to be in the same direction, and what distinguishes one magnitude from another. Well, naively, we say that two vectors go in the same direction if one's coordinates are a multiple of the other's. For example, in 2D space ([imath]\mathcal{R}^2[\imath]) the vector (think arrows going around in a circle centered around (0,0)) (3,4) and the vector (1/4,1/3) are in the same direction because (3,4)=12(1/4,1/3).
So the concept of direction, vectorwise, is closely related to the concept of multiplication by numbers, which we call scalars. But we want to be able to multiply by numbers in order to talk about directions. However, we also know that (1/4,1/3)=1/12(3,4)., i.e. (1/4,1/3) is in the same direction as (3,4). So, intuitively, our scalars need to have multiplicative inverses, which makes them into a field (the scalars inherit additive properties from the idea of how vector addition should work; just think about adding arrows whose heads are given by coordinates, and think about what happens when you scale the arrows by numbers and then add them together). So the point is that the idea of a vector inherently holds in itself the idea of a vector space hat is an abelian group of vectors and field of scalars so that scalar multiplication on the vectors works properly.
The magnitude question I will not answer, because it doesn't really make sense in a general setting, whereas the idea of a direction works well enough. The high school idea of magnitude though is more precisely expressed by the fact that we can tell if vectors along the same direction are different. The ordered part of it comes from the field of scalars being ordered, which is generally not the case.

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Re: Examples of vectors
I guess this is a good thread to ask a similar question: What isn't a vector, or vector space?
I took an (intro, freshman) class in linear algebra this winter and we didn't go over the vector space chapter, which was only 10 or so pages in the manual anyway. I glossed over it, and it basically said that vector spaces satisfied addition and scalar multiplication and gave a few basic exercises. What I can't seem to get is what on earth would satisfy the property k(a,b) = (ka, b)?
I guess it wasn't taught yet for a reason, but then I'm curious. What made the distinction necessary?
I took an (intro, freshman) class in linear algebra this winter and we didn't go over the vector space chapter, which was only 10 or so pages in the manual anyway. I glossed over it, and it basically said that vector spaces satisfied addition and scalar multiplication and gave a few basic exercises. What I can't seem to get is what on earth would satisfy the property k(a,b) = (ka, b)?
I guess it wasn't taught yet for a reason, but then I'm curious. What made the distinction necessary?
Re: Examples of vectors
The natural numbers aren't a vector space, for one.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

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Re: Examples of vectors
antonfire wrote:The natural numbers aren't a vector space, for one.
That's actually only partly true. What you meant to say was: The natural numbers don't form a vector space under the usual addition and multiplication. We can however define these operations in a way to make the natural numbers a vector space: Identify the naturals with the rational numbers (which habe the same cardinality, and don't forget to identify 0 if your natural numbers don't include it) and then define addition and multiplication like on the identified rationals. Since the rationals are a field they are trivially also a vector space.
Re: Examples of vectors
Yeah, it's fun to be pedantic, so I'll go with that: I meant to say exactly what I said. "The natural numbers" refers to the set together with the additive structure on it (among other things). With that structure, it's not a vector space.
To add some actual content, the sets which have no vector space structure on them are precisely the sets whose cardinality is a finite number with more than one prime factor.
To add some actual content, the sets which have no vector space structure on them are precisely the sets whose cardinality is a finite number with more than one prime factor.
If you take (a,b) to denote [imath]a\otimes b[/imath] in some tensor product then that equality is satisfied. The tensor product of two vector spaces is still a vector space, though.Quenouille wrote: What I can't seem to get is what on earth would satisfy the property k(a,b) = (ka, b)?
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?
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Re: Examples of vectors
I'll try to intuitively explain why number sequences and functions can be viewed as vectors.
The concept of a vector space of geometric vectors in space ("arrows" pointing from the origin to some point) is the intuitive picture you should keep in your head.
If we represent those vectors by their cartesian coordinates (such as (1,2,3)), we get a vector space of (finite) sequences of numbers. This should still be intuitive.
The vector (1,2,3) can be given another geometrical interpretation. We can picture it as three arrows extending up from the xaxis, the first going pointing up and having 1 as length; the second pointing up and having 2 as length; and the third pointing down and having 3 as length. Adding and scaling these vectors just amounts to adding and scaling all these little arrows.
With this interpretation, it becomes obvious that functions can be viewed as vectors: you can view a function as a bunch of arrows pointing up from (x,0) to (x,f(x)), and adding and scaling such vectors just amounts to adding and scaling all these little arrows. By the same token, infinite sequences of numbers can also be viewed as a vector space: just view each vector (such as (1,2,3,...)) as a bunch of arrows pointing up from the xaxis in sequence.
The concept of a vector space of geometric vectors in space ("arrows" pointing from the origin to some point) is the intuitive picture you should keep in your head.
If we represent those vectors by their cartesian coordinates (such as (1,2,3)), we get a vector space of (finite) sequences of numbers. This should still be intuitive.
The vector (1,2,3) can be given another geometrical interpretation. We can picture it as three arrows extending up from the xaxis, the first going pointing up and having 1 as length; the second pointing up and having 2 as length; and the third pointing down and having 3 as length. Adding and scaling these vectors just amounts to adding and scaling all these little arrows.
With this interpretation, it becomes obvious that functions can be viewed as vectors: you can view a function as a bunch of arrows pointing up from (x,0) to (x,f(x)), and adding and scaling such vectors just amounts to adding and scaling all these little arrows. By the same token, infinite sequences of numbers can also be viewed as a vector space: just view each vector (such as (1,2,3,...)) as a bunch of arrows pointing up from the xaxis in sequence.
Re: Examples of vectors
Barcode wrote:And what is the difference between geometric vectors in a line, plane, space... etc, and (a), (a,b) (a,b,c) ... (that is, vectors as a series of numbers)?
This is actually a subtle question, and I'm glad you asked it. The slogan here is that vectors are not points, even though we write them the same way.
Why? When we talk about points on a plane, we don't need to specify either an origin or coordinate axes. We often do, to make it easier to talk about particular points, but we don't need to. When we talk about vectors, however, we have to specify an origin  even though we still don't have to specify coordinate axes  because the origin tells us how to add two vectors. (This is sometimes explained with the informal definition "an affine space is a vector space that has forgotten its origin.")
Another way of thinking about this distinction is that vectors are "formal differences" of two points. If this is too abstract, vectors are displacements, but points are positions. Displacements measure changes of position. We can specify positions using displacements by computing the displacement from an arbitrary point we call the origin, but again, we don't have to.
Re: Examples of vectors
OK, I think I'm going to tell you what I have in my mind about this subject and then we can discuss it further if needed.
1) I think phlip answered straight to the point, at least concerning one of my questions. So in fact geometric (or eucleidian) vectors are by definition a different thing than a ntuple of numbers, since the former is a geometrical object with magnitude and direction, and the latter just a list of numbers, so they cannot be the same thing. However, they are so closely related, that there is a onetoone correspondence for each object of any group to the other, and every operation on geometrical vectors can be represented by another operation on the list of numbers. So it is easier to add pure numbers than to add vectors with the parallelogram rule.
Did I get this correctly?
2) Why aren't the natural numbers a vector space? Because of the non existing inverse elements of addition? If yes, the integers should form a vector space, right?
3) One more point that I would like to make is that I am trying to get rid of the intuitive idea of vectors as arrows, since this is only a very specific example or a vector space. And also, repeating my crude example above, how would a vector space with water droplets work? We can certainly add them, getting a droplet with the sum of their masses. Multiplication is obvious. Assuming there is some form of antimatter droplet (excuse me for this), so that we also get our additive inverse elements, does that form a vector space?
4) If the notion of arrows (or even angles, and magnitude) does not hold in a generalised notion of a vector space, and it doesn't according to wikipedia, why should we consider functions (or other objects) as vectors forming a vector space? We can add/multiply them anyway, so what are the benefits?
1) I think phlip answered straight to the point, at least concerning one of my questions. So in fact geometric (or eucleidian) vectors are by definition a different thing than a ntuple of numbers, since the former is a geometrical object with magnitude and direction, and the latter just a list of numbers, so they cannot be the same thing. However, they are so closely related, that there is a onetoone correspondence for each object of any group to the other, and every operation on geometrical vectors can be represented by another operation on the list of numbers. So it is easier to add pure numbers than to add vectors with the parallelogram rule.
Did I get this correctly?
2) Why aren't the natural numbers a vector space? Because of the non existing inverse elements of addition? If yes, the integers should form a vector space, right?
3) One more point that I would like to make is that I am trying to get rid of the intuitive idea of vectors as arrows, since this is only a very specific example or a vector space. And also, repeating my crude example above, how would a vector space with water droplets work? We can certainly add them, getting a droplet with the sum of their masses. Multiplication is obvious. Assuming there is some form of antimatter droplet (excuse me for this), so that we also get our additive inverse elements, does that form a vector space?
4) If the notion of arrows (or even angles, and magnitude) does not hold in a generalised notion of a vector space, and it doesn't according to wikipedia, why should we consider functions (or other objects) as vectors forming a vector space? We can add/multiply them anyway, so what are the benefits?

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Re: Examples of vectors
Barcode wrote:OK, I think I'm going to tell you what I have in my mind about this subject and then we can discuss it further if needed.
1) I think phlip answered straight to the point, at least concerning one of my questions. So in fact geometric (or eucleidian) vectors are by definition a different thing than a ntuple of numbers, since the former is a geometrical object with magnitude and direction, and the latter just a list of numbers, so they cannot be the same thing. However, they are so closely related, that there is a onetoone correspondence for each object of any group to the other, and every operation on geometrical vectors can be represented by another operation on the list of numbers. So it is easier to add pure numbers than to add vectors with the parallelogram rule.
Did I get this correctly?
Yes and no. The magnitude and direction of your geometric arrow are just as present in the ntuple of numbers, they are just slightly less obvious. There really is a point at which it doesn't make much sense to think of two things as being different even if they may, at the core, not really seem identical. Obviously, the ntuples and the geometric arrows behave in exactly the same way, one is just a different way to represent the other. You really do not gain anything by considering them to be something different, it's much more useful to think of BOTH as a different REPRESENTATION of the same exact thing. If you take a really close look at it, the numbers 1, 2 and 3 are not exactly the same in the natural numbers and the real numbers, depending on the way you construct them. In fact, there have been over half a dozen different approaches to construct what we collectively know as the real numbers, in ways where the respective representation of the number pi, for example, may really not look like the representation in other constructions. However, all these structures behave in the exact same way (as they were certainly constructed to do just that), it really doesn't make much sense to consider the real numbers as constructed by dedekind cuts to be something different than the real numbers as constructed by adding in the limits of cauchy sequences or any of the other approaches. All of them are essentially just ways to represent the deeper structure of what we call the real numbers. I hope that, to a certain degree, clears things up a bit.
2) Why aren't the natural numbers a vector space? Because of the non existing inverse elements of addition? If yes, the integers should form a vector space, right?
There are two aspects to a vector space. For one, you have to have the Set you want to turn into a vector space by defining the addition and scalar multiplication on it. The other thing you need is what mathematicians call a field, a structure that is, to a certain degree, a generalized version of something that behaves like the rational numbers. This is what you define the scalar multiplication with: You take an element out of the field and multiply it onto a vector from the vector space and you get another vector. For many purposes the Fields in question are the real or the complex numbers, but there are other, potentially more confusing fields as well that you could define your vector space over. Something you might not have quite realized yet is that the multiplication in vector spaces isn't multiplying two vectors together, thinking about it geometrically, what is an arrow times and arrow supposed to be? You might have come accross the Vector or Cross Product in three dimensions, but that is more or less a quirk: No construction like it exists in other dimensional vector spaces. Generally, the question of Vector times Vector doesn't make much sense. There may be ways to define it reasonably in some Vector Spaces, but it is not generally expected to give it a sensible meaning.
So not only do the natural numbers lack additive inverses, you would also need a field to define the scalar multiplication on. The Integers do have additive inverses, so all you need to turn them into a vector space is some field for the multiplication. I am not aware of any canonical way to do this while still reasonably identifying the resulting structure as closely related to the Integers, maybe somebody else knows more about it. If you use the ordinary multiplication on the Integers, the smallest structure with this multiplication that satisfies the field axioms are the rational numbers. At that point you run into the issue that 1/5 * 3 is not an Integer.
3) One more point that I would like to make is that I am trying to get rid of the intuitive idea of vectors as arrows, since this is only a very specific example or a vector space. And also, repeating my crude example above, how would a vector space with water droplets work? We can certainly add them, getting a droplet with the sum of their masses. Multiplication is obvious. Assuming there is some form of antimatter droplet (excuse me for this), so that we also get our additive inverse elements, does that form a vector space?
Technically, yes, you could create a vector space of water droplets and antiwater droplets that way. However, all your operations work solely on their mass, a one dimensional unit, so that vector space is pretty much the same as the field you use for the multiplication of the droplets (Most likely the rational or real numbers). It is not a particularly interesting creation, though it may make sense in special context to use it. Kind of how all finite dimensional (say: Dimension n) vector spaces over a field are not really all that different from the vector space of ntuples over that field. It may still make sense to think of them as something different depending on the context you are working in. Things get more interesting once you start talking about infinite dimensional vector spaces, like previously mentioned function spaces and the likes.
4) If the notion of arrows (or even angles, and magnitude) does not hold in a generalised notion of a vector space, and it doesn't according to wikipedia, why should we consider functions (or other objects) as vectors forming a vector space? We can add/multiply them anyway, so what are the benefits?
Simply put: We spend quite some time studying the behaviour of vector spaces, developing a pretty deep understanding of them and developed a lot of tools to deal with and manipulate them in certain ways. If you can find a way to reasonably describe your structure, for example a set of functions with a certain properties, as a vector space, you all of a sudden have that huge piece of knowledge and instruments at your disposal without having to verify that it works with the specific structure you have. That's one of the reasons we try to do things as abstractly as possible: If you know something holds for a general structure, you are free to use it on all instances of such a structure without much more thought. It's not really all that much about the addition and multiplication, it's more about the benefits you get from knowing these behave in a certain way.
The notion of arrows and the likes you are used to is pretty much just a visualization of a vector space. It is certainly less abstract than other ways to express them, but it doesn't really fully capture what the structure of a vector space is really about. To a certain degree, vector spaces are some of the simplest and best known structures we have, which is why we try to find them in as many things as we can.
As a side note, if your vector space has an inner product (which is a good thing to have, though not all vectorspaces neccesairly do) you can define something like that magnitude and the angle in a canonic way with it. It is possible to define the angle between two matrices, for example, in a not really arbitrary way. How useful that number is really very much depends on what you need it for in the first place.
Re: Examples of vectors
LLCoolDave wrote:So not only do the natural numbers lack additive inverses, you would also need a field to define the scalar multiplication on. The Integers do have additive inverses, so all you need to turn them into a vector space is some field for the multiplication. I am not aware of any canonical way to do this while still reasonably identifying the resulting structure as closely related to the Integers, maybe somebody else knows more about it. If you use the ordinary multiplication on the Integers, the smallest structure with this multiplication that satisfies the field axioms are the rational numbers. At that point you run into the issue that 1/5 * 3 is not an Integer.
I just had an idea about that... someone tell me if my reasoning is faulty somewhere.
Let us investigate what the necessary properties are of the underlying field that makes the integers into a vector space.
Let us call the field F, and the integers: Z. Now for any scalar a in F, and any nonzero integer x in Z we have that ax is an element of Z, meaning that so is ax+ax+ax+...+ax (n times)=(a+a+a+...+a)x n times. But ax+ax+ax+...+ax is nonzero for any n times you add them together, which means that (a+a+a+...+a)x should also be nonzero vector, which means that a+a+a+...+a should not a nonzero scalar. (the reason we know that 0x=0 (where the first 0 is the scalar) is that 0x=(0+0)x=0x+0x, and subtracting the vector 0x from both sides we get that 0=0x).
So this means for example that the sum of scalars 1+1+1+1...+1 is never the scalar 0, which means the field has to have characteristic 0. But then if we look at what the multiplicative identity generates under addition, that is precisely the integers. Taking their inverses (we're in a field), it means that we have a subfield isomorphic to Q. But then, if x is some integer, we have that n*1/n x=1x for any scalar n. But this is equivalent to having 1x=1/n x + 1/n x + 1/n x + ... + 1/n x (addition n times if n is an integer scalar.) i.e. the vector 1x has to be the sum of n identical vectors for any n.
But the abelian group of the integers doesn't work that way, because 1x being the sum of n identical integers means that the integer 1x is a multiple of the integer n (here the multiplication of the abelain group of the integers follows naturally from the group structure. Specifically, being a multiple of n, means that you're an nfold sum). But 1x=x(by axioms of vector space), and the only integer with the property that it is the sum of n identical integers for any n is 0.
Hence, we can't have the integers be a vector space in any way.
Re: Examples of vectors
Sad, isn't it. They are a free module though, so maybe, one day when they grow up...
Re: Examples of vectors
phr34k wrote:Hence, we can't have the integers be a vector space in any way.
There's a very concise way to phrase the argument you're making. Giving an abelian group [imath]G[/imath] the structure of a vector space over [imath]F[/imath] is equivalent to giving a bilinear map [imath]F \times G \to G[/imath], and any such map factors through the tensor product [imath]F \otimes G[/imath]. The observation you made is equivalent to the fact that [imath]\mathbb{Q} \otimes \mathbb{Z} = 0[/imath].
Re: Examples of vectors
Hm? It's true that there are no nonzero bilinear maps [imath]\mathbb{Q} \times \mathbb{Z} \rightarrow \mathbb{Z}[/imath], but there is one [imath]\mathbb{Q} \times \mathbb{Z} \rightarrow \mathbb{Q}[/imath]: the one that takes (q,z) to qz. In fact, as abelian groups (which is the only reasonable way I can think to interpret that tensor product), [imath]\mathbb{Q} \otimes \mathbb{Z} = \mathbb{Q}[/imath].
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?
Re: Examples of vectors
Whoops. I think I remembered something incorrectly. Anyway, the first thing you said.
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