Affine Spaces and Vector Spaces

[vanhees71]

[Moderator's Note: Spun off from previous thread due to increase in discussion level to "A" and going well beyond the original thread's topic.]

An affine space is basically a vector space without an origin.

A vector space has no origin to begin with ;-)).

An affine space is a set of points and a vector space $(M,V)$. Then you have a set of axioms which boils down to what you know from Euclidean geometry, i.e., to a pair of points $A,B \in M$ there's a vector $\overrightarrow{AB}$ (an arrow connecting $A$ with $B$). If you have two other points such that the arrow $\overrightarrow{CD}$ is parallel to $\overrightarrow{AB}$ these to vectors are by definition the same.

If you have an arbitrary vector $\vec{v} \in V$ and a point $A \in M$ there's a unique point $B$ such that $\overrightarrow{AB}=\vec{v}$.

Then you can add vectors $\vec{v}_1,\vec{v}_2$ and the meaning in connection with the arrows connecting points is $\vec{v}_1+\vec{v}_2=\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}$. Here we have chosen $\overrightarrow{AB}=\vec{v}_1$ and $\overrightarrow{BC}=\vec{v}_2$.

The dimension of the vector space is by definition the dimension of the affine space.

If your vector space has in addition a scalar product, you can measure distances and angles between points in the usual way using this scalar product. That's then an Euclidean affine space. E.g., space as described by Euclidean geometry is simply a 3D (or if you are restricted to a plane 2D) Euclidean affine.

In special relativity you have a 4D pseudo-Euclidean affine space to describe spacetime. Here the Minkowski product is not positive definite and thus not a proper scalar product, but it has the signature (1,3) (or (3,1) depending on the convention of your book). Otherwise it's pretty much analoguous to a Euclidean affine space.

 

[Dale]

A vector space has no origin to begin with ;-)).

it has a zero vector. It is perfectly reasonable to describe the zero vector as the origin.

 

[vanhees71]

No! If you have an affine space the origin of a reference frame is an arbitrary point $O \in M$. For any (!) point $A \in M$ the zero-vector is reprensented by $\overrightarrow{AA}$. Only if you arbitrarily fix $O \in M$ you can uniquely map any point $P$ by a vector $\vec{r}$ (which then is called "position vector") via $\vec{r}=\overrightarrow{OP}$, and of course then $\vec{r}=0$ uniquely refers to the origin, because $\overrightarrow{OO}$ is the unique zero-vector with $O$ as one end of the corresponding "arrow".

There's a lot of confusion in the physics literature, because the authors don't carefully distinguish a vector space and an affine space which consists of a pair $(M,V)$ ($M$: set of points, $V$: vector space)!

 

[Dale]

No! If you have an affine space the origin of a reference frame is an arbitrary point $O \in M$. For any (!) point $A \in M$ the zero-vector is reprensented by $\overrightarrow{AA}$. Only if you arbitrarily fix $O \in M$ you can uniquely map any point $P$ by a vector $\vec{r}$ (which then is called "position vector") via $\vec{r}=\overrightarrow{OP}$, and of course then $\vec{r}=0$ uniquely refers to the origin, because $\overrightarrow{OO}$ is the unique zero-vector with $O$ as one end of the corresponding "arrow".

There's a lot of confusion in the physics literature, because the authors don't carefully distinguish a vector space and an affine space which consists of a pair $(M,V)$ ($M$: set of points, $V$: vector space)!

Which is exactly why an affine space doesn’t have an origin but a vector space does. The zero vector in a vector space is unique and can reasonably be called the origin. In an affine space any point can equally serve as the subtrahend to produce vectors.

I think you are the one not carefully distinguishing between a vector space and an affine space since I made a comment about a vector space and you responded with an emphatic “No!” and a discussion about an affine space

 

[PeterDonis]

A vector space has no origin to begin with ;-)).

What is your definition of "origin"?

 

[PeterDonis]

The zero vector in a vector space is unique and can reasonably be called the origin.

What is your definition of "origin"?

 

[vanhees71]

What is your definition of "origin"?

There's simply no origin in a vector space. At least I've never ever seen any book, where one defines something as "the origin" of a vector space.

In an affine space it's what you usually introduce to map each point of the affine manifold one-to-one to a vector (see also #16, where a textbook is quoted). Of course, then using a basis you can map this "position vector" to its components wrt. this basis.

Wikipedia is not too bad a start:

https://en.wikipedia.org/wiki/Affine_space#Definition

 

[weirdoguy]

There's simply no origin in a vector space.

I think it's simply too strong of a statement since I guess quite a lot of mathematicians say that "an affine space is like a vector space without origin" meaning exactly that vector spaces have a unique zero vector. At least three of my math lecturers said that, but of course this is not an argument. But if you try to visualise a vector space of "arrows" those arrows have to have a common origin. Also this origin is a zero vector. Vectors in an affine spaces (as pairs of points) are free vectors, so that they don't have to have a common origin. That is the basis of the statement discussed.

 

[PeterDonis]

There's simply no origin in a vector space

This doesn't answer my question. I didn't ask you whether or not an affine space or a vector space has an origin. I asked you what is your definition of the term "origin"?

The reason I am asking you that, and why I asked @Dale that, is that I strongly suspect the two of you have different definitions of the term "origin", and so you are talking past each other. But the only way to find out for sure is for each of you to explicitly state the definition you are using. That doesn't seem like a difficult thing to do.

 

[vanhees71]

I gave it: In an affine space $(M,V)$ you introduce an arbitrary point $O$ as the origin of a reference frame. Then any point $P \in M$ is uniquely mapped to a vector $\vec{r} \in V$ via $\vec{r}=\overrightarrow{OP}$ and vice versa, i.e., for each $\vec{r}$ there's a unique point $P$ such that $\vec{r}=\overrightarrow{OP}$.

Maybe I'm not familiar enough with the English literature concerning math, but I've never ever seen, how "origin" is defined in a vector space. How do you define it in a pure vector space (not an affine space, where it is commonly defined as given above).

 

[weirdoguy]

Every vector space is an affine space, so if you can talk about origin in the latter, you can also in the former.

 

[vanhees71]

Maybe, there's another convention in the literature, but I learned that an affine space is something different than a vector space.

Here's the proper formulation of the definition, I referred to above:

https://www.encyclopediaofmath.org/index.php/Affine_space

Another definition (equivalent to the one I've given above) of an affine space is given here:

https://mathworld.wolfram.com/AffineSpace.html

Again: An affine space consists of a set $M$ (with elements called "points") and a vector space $V$ (usually a finite-dimensional vector space over $\mathbb{R}$).

Here indeed I find this (for me absurd) idea of an affine space as a "vector space if you forget about the origin". Maybe there's a difference in the mathematical conventions in the English and the German math literature:

https://en.wikipedia.org/wiki/Affine_space

For those, who understand German. Here's the explanation that this point of view is not as absurd as I thought, because you can define an affine space also starting from a vector space only:

https://de.wikipedia.org/wiki/Affiner_Raum#Der_affine_Punktraum_und_der_ihm_zugeordnete_Vektorraum

 

[weirdoguy]

but I learned that an affine space is something different than a vector space.

Of course, but a vector space can be considered an affine space by the very definition you cite. The set $M$ is then $V$ and the addition of a vector to a point is the standard addition in $V$.

Besides, I think that this whole "affine space is like a vector space without an origin" should be treated mainly as a heuristic. I've heard something similar about fibers of principal bundles, that they are "like Lie group but we are forgetting where the identity is".

 

[PeterDonis]

I gave it: In an affine space $(M,V)$ you introduce an arbitrary point $O$ as the origin of a reference frame.

Ok, so you are using "origin" to mean "origin of a chosen reference frame".

How do you define it in a pure vector space

The definition @Dale is using seems obvious to me: the origin of a vector space is the zero vector.

@Dale even gave a way of constructing a simple correspondence between his definition and yours. By your definition, an "origin" is a point you pick in an affine space, which you designate as the origin of a reference frame. Once you have done this, the set of displacement vectors based at that origin is a vector space, which has an obvious one-to-one correspondence with points in the affine space given your chosen origin point. And the zero vector of that vector space corresponds to your chosen origin point in that obvious one-to-one correspondence.

this (for me absurd) idea of an affine space as a "vector space if you forget about the origin"

It's not absurd given the obvious one-to-one correspondence I just described above: if you reverse the process described, i.e., view the points in the affine space, given the chosen origin point, as a vector space via the one-to-one correspondence, and then "undo" the choice of origin, you are left with just the affine space itself.

I would agree that this viewpoint is heuristic and not very rigorous, but I don't think it's absurd.

 

[vanhees71]

Well, yes. I understand this now, but it seems to me more confusing than helpful.

 

[robphy]

Some mathematical structures are gotten (or motivated) by building up from more primitive constructions,
and some are gotten by relaxing or weakening more complicated constructions.

 

[Dale]

What is your definition of "origin"?

I don’t have a rigorous definition. I just think that it is perfectly reasonable to describe the 0 vector as “the origin“.

In R2 (0,0) is known as “the origin”, and since R2 is the prototypical vector space and the point (0,0) is the zero vector the zero vector is in fact the origin for R2. Similarly for RN. So calling the zero vector “the origin” is exact for the prototypical vector spaces and therefore a clearly reasonable general statement.

 

[robphy]

Possibly enlightening:
https://en.wikipedia.org/wiki/Forgetful_functor
https://ncatlab.org/nlab/show/affine+space

 

[sysprog]

So this only applies to flat spaces, since there isn't in general a well-defined difference vector between points in a curved space? Is there a name for whatever a curved manifold is? Other than "curved manifold", I mean.

As you presumably will recall, a curved manifold can be be described as a tensor system.

 

[PeterDonis]

In R2 (0,0) is known as “the origin”, and since R2 is the prototypical vector space and the point (0,0) is the zero vector the zero vector is in fact the origin for R2.

Yes, but note that R2 as a vector space is not the same thing as "the points in a plane", which is the affine space in question in this particular example. So saying that (0, 0) is "the origin" is not, strictly speaking, justified if all you have is R2 as a vector space--at least not if you take "origin" to mean a point, which I think is what most people intuitively expect. You need to first explain how to get from (0, 0) as a vector, the zero vector, to (0, 0) as the coordinates of a particular point in the plane.

More precisely, R2 as a vector space interprets the 2-tuple (x, y) as the components of a vector, not the coordinates of a point; but the term "origin" is normally thought of as a point, not a vector. You can, of course, construct a one-to-one correspondence between points and vectors, in the way I described in an earlier post. Once you have done that, the 2-tuple (0, 0) does indeed correspond to both the zero vector and the point in the plane that you picked as the origin. But you need the correspondence to make that identification, and so it seems to me that you need the correspondence to justify calling the zero vector "the origin".

It's also worth noting that it's a little odd, if you are calling the zero vector "the origin" of a vector space, to then say an affine space is a vector space "without the origin", since the zero vector is still there in the vector space; you can't just remove it. There is some unpacking that needs to be done to justify that statement.

 

[Orodruin]

I would say that the concept of "origin" is not inherent to an affine space. There is nothing in the axioms of an affine space that needs you to define an origin. Instead, they only talk about difference vectors and translations between different points. The concept origin becomes relevant once you pick a fixed point to refer to other points by vectors in the tangent space, i.e., it is more related to the concept of coordinate systems on affine spaces, not to the affine spaces themselves.

 

[SiennaTheGr8]

I thought I didn't know what an affine space is. Turns out it's what I thought a vector space was. Thanks everyone.

 

[Dale]

A vector space has no origin to begin with ;-)).

So saying that (0, 0) is "the origin" is not, strictly speaking, justified if all you have is R2 as a vector space

Sorry, but I think that both of you are wrong to harass me over this. Calling the zero vector the origin is well established, especially in the context of distinguishing them from affine spaces, and particularly given the one-to-one mapping between vectors of any real vector space and points in RN (so you can always consider vectors to be points).

http://math.ucr.edu/home/baez/torsors.html (John Baez) "An affine space is like a vector space that has forgotten its origin. "
https://en.wikipedia.org/wiki/Affine_space "When considered as a point, the zero vector is called the origin "
https://www.cis.upenn.edu/~cis610/geombchap2.pdf "the point corresponding to the zero vector (0), called the origin"
https://ncatlab.org/nlab/show/affine+space "An affine space or affine linear space is a vector space that has forgotten its origin."
https://www.math.utah.edu/~tan/Convex Analysis/CA - Chapter1.pdf "the point corresponds to the zero vector 0, called the origin"

Saying that a vector space has an origin and that said origin is the zero vector and that an affine space is a vector space without an origin is perfectly well justified. Can either of you produce any references that explicitly say that a vector space does not have an origin or that the zero vector is not an origin?

 

[vanhees71]

Yes, but note that R2 as a vector space is not the same thing as "the points in a plane", which is the affine space in question in this particular example. So saying that (0, 0) is "the origin" is not, strictly speaking, justified if all you have is R2 as a vector space--at least not if you take "origin" to mean a point, which I think is what most people intuitively expect. You need to first explain how to get from (0, 0) as a vector, the zero vector, to (0, 0) as the coordinates of a particular point in the plane.

More precisely, R2 as a vector space interprets the 2-tuple (x, y) as the components of a vector, not the coordinates of a point; but the term "origin" is normally thought of as a point, not a vector. You can, of course, construct a one-to-one correspondence between points and vectors, in the way I described in an earlier post. Once you have done that, the 2-tuple (0, 0) does indeed correspond to both the zero vector and the point in the plane that you picked as the origin. But you need the correspondence to make that identification, and so it seems to me that you need the correspondence to justify calling the zero vector "the origin".

It's also worth noting that it's a little odd, if you are calling the zero vector "the origin" of a vector space, to then say an affine space is a vector space "without the origin", since the zero vector is still there in the vector space; you can't just remove it. There is some unpacking that needs to be done to justify that statement.

Exactly that's my point!

 

[vanhees71]

Saying that a vector space has an origin and that said origin is the zero vector and that an affine space is a vector space without an origin is perfectly well justified. Can either of you produce any references that explicitly say that a vector space does not have an origin or that the zero vector is not an origin?

It seem to be well-established in the anglo-saxxon tradition. In Germany our math professor in the 1st semester in fact emphasized the importance of this very distinction between affine spaces and vector spaces.

A point in an affine space is not a vector, but you can make a one-to-one map between the points of the affine space after defining an arbitrary point $O$ as the origin of a reference frame. Then according to the axioms of an affine space each point is uniquely determined by a vector $\vec{r}$ (the "position vector") and vice versa via $\vec{r}=\overrightarrow{OP}$, and then of course you are right the origin is uniquely given by the zero-vector $\vec{r}=0$, but it's not the same as the point. It's just a one-to-one mapping between a point and the zero-vector. Also it's not a "canonical mapping" from the point of view of an affine manifold since it depends on the arbitrary choice of one point $O$ as the origin of the reference frame.

It's a bit similar in a vector space, when it comes to a basis. If you have a $d$ dimensional vector space $V$ (say over $\mathbb{R}$) and a basis $\vec{b}_j$ then you have a one-to-one map between vectors in $V$ and vectors in the vector space $\mathbb{R}^d$ via
$$\vec{v} = \sum_{j=1}^d v^j \vec{b}_j \mapsto (v^1,v^2,\ldots,v^d)^{\text{T}} \in \mathbb{R}^d.$$
Also here $\vec{v}$ is not the same as a column vector in $\mathbb{R}^d$, but there's a one-to-one mapping between these two vector space, which again is not "canonical", because it depends on the arbitrary choice of a basis.

 

[weirdoguy]

in fact emphasized the importance of this very distinction between affine spaces and vector spaces.

But no one here is saying anything about them being the same!

 

[vanhees71]

Then I don't understand the whole debate to begin with, and then there's no "origin" in vector spaces.

But as I learned from the German Wikipedia article there's a realization of an affine space with a vector space only (not an additional set of "points"), and this construction seems to be what's taught in the anglo-saxxon tradition and that's why this misunderstanding came up. According to the German Wikipedia this interpretation of a vector space as an affine space is as follows.

Start with a vector space $V$ and define an affine space with vectors in $V$ as the points of the affine space. Then the map $V \times V \rightarrow V:(\vec{v}_1,\vec{v}_2) \mapsto \vec{v}_2-\vec{v}_1)$ defines the vector connecting the "points" $\vec{v}_1$ and $\vec{v}_2$. In this construction of course the zero-vector is the natural choice of the "origin" of this special kind of affine space.

On the other hand, as we are discussing spacetime models, it's clear that you rather are after the usual definition with a set $M$ of points and a vector space $V$ making up together an affine space, and there points are not the same as vectors, but you have a one-to-one-mapping of the points after choosing arbitrarily one point as an "origin".

Minkowski space of special relativity is an affine pseudo-Euclidean space, i.e., on the vector space $V$, which is a four-dimensional real vector space, there's defined a non-degenerate fundamental form with signature (1,3) or (3,1). The corresponding symmetry group is the ISO(1,3), aka the Poincare group aka the inhomogeneous Lorentz group.

 

[Dale]

it's not the same as the point. It's just a one-to-one mapping between a point and the zero-vector.

I disagree. First, affine spaces don’t “own” the concept of a point nor do affine spaces have exclusive rights to the term.

Second, even if you grant affine spaces exclusive rights to the term “point” and insist that the origin is a point so that origins must always be elements of an affine space, the zero vector is still the origin. This is because vector spaces themselves are also affine spaces (the reverse is not true, affine spaces are not vector spaces, but vector spaces are affine spaces).

Vector spaces satisfy all of the axioms of affine spaces so a vector space is an affine space (not just mapped to an affine space). So any element of a vector space (a vector) is also an element of an affine space (a point) and therefore both a vector and a point, with the special vector known as the zero vector being the special point known as the origin. This is an identity, not just a mapping, all vectors themselves are in fact also points (again the reverse is not true, not all points are vectors).

A point in an affine space is not a vector

But a vector is a point in an affine space.

 

[vanhees71]

Yes, I just wrote that in #40.

As I also wrote there, I find this special construction of an affine space not useful for spacetime models. There a point (an event) is not a vector, before you have introduced an origin (see also #33 by @PeterDonis for the example of the Euclidean plane) and defined the one-to-one mapping between points in spacetime and vectors.

 

[Dale]

I find this special construction of an affine space not useful for spacetime models.

That is fine, I don’t disagree. But your opening comments of posts 12 and 40 are wrong. A vector space does in fact have an origin even if you restrict the origin to being a point and points to being elements of affine spaces.

 

[vanhees71]

You have to make the qulification that you interpret the vector space as an affine space in the above explained way. Otherwise it's not understandable, at least not for one trained in the German mathematical tradition.

 

[PeterDonis]

"An affine space is like a vector space that has forgotten its origin. "

This just shows that this claim is fairly common. It doesn't show why it is true (if it is true).

"When considered as a point, the zero vector is called the origin "

Do I have to point out that Wikipedia is not a valid source?

"the point corresponding to the zero vector (0), called the origin"

You took this quote out of context. Here is the context:

"One could model the space of points as a vector space, but this is not very satisfactory for a number of reasons. One reason is that the point corresponding to the zero vector(0), called the origin, plays a special role, when there is really no reason to have a privileged origin."

In other words, this source is arguing against interpreting a vector space as a space of points, not for it.

"An affine space or affine linear space is a vector space that has forgotten its origin."

Same comment as for the Baez quote above.

"the point corresponds to the zero vector 0, called the origin"

This source is quoting the same source above that you quoted out of context, and which when taken in context, is arguing against the view you are promoting.

I'm sorry, but I don't accept any of these references as supporting your position.

A vector space does in fact have an origin even if you restrict the origin to being a point and points to being elements of affine spaces.

And, as I said, this amounts to saying that an affine space has an origin, when it doesn't, which is, to say the least, odd. That's why the source you took out of context, when the context is put back in, is arguing against this interpretation of a vector space.

 

[dextercioby]

@fresh_42 Can you read this with your (Germany-based) mathematical education and express your opinion to Dale, Peter and Hendrik?

 

[fresh_42]

@fresh_42 Can you read this with your (Germany-based) mathematical education and express your opinion to Dale, Peter and Hendrik?

Do you have something specific in mind? Hendrik should have the same qualifications.

I like Dale's nLab quotation:
"An affine space or affine linear space is a vector space that has forgotten its origin."
although it is a bit misleading. An affine space is $\mathbb{A}=\vec{v_0}+\mathbb{V}$, so it's a vector space kicked off its origin, not forgotten. There had been some strange statements about the word origin. I think it should be clear that the zero vector is meant.

I think the formal definition is: An affine space $\mathbb{A}$ with an underlying vector space $\mathbb{V}$ is the set of all points, which can be written as a difference of vectors from $\mathbb{V}$.

Things can become unclear if differential geometry is involved. A tangent space can be viewed as both, a vector space $T_0M$ and as an affine space $p+T_pM$. However, everybody writes $T_pM$ and calls it vector space, ignoring the fact, that $p$ has to be identified with the zero vector first. Makes sense, because the tangent space $T_0M = (p,T_0M)$ at $p$ is even more confusing.

 

[PeterDonis]

I think the formal definition is: An affine space $\mathbb{A}$ with an underlying vector space $\mathbb{V}$ is the set of all points, which can be written as a difference of vectors from $\mathbb{V}$.

This doesn't make sense. The difference of two vectors is another vector, not a point. If you have a set of points, the difference between two points is a vector, but that's going from the affine space to the vector space, not the other way around.