Points and vectors

Question

I've been working to understand the geometrical objects and intuition associated with the algebra which is used in the abstract theory which I've studied so far (topology, calculus, linear algebra) and I'm stumbling on the elementary distinction between points and vectors.

As far as I'm, points are just general elements of a set (maybe with some minimal structure such as a topology) whereas vectors can be added and scaled. So far so good. But then I don't see how we can talk about coordinates of points of a space. Take $\mathbb{R}^n$ for example. Seen as a general space (say a manifold), how does it make sense to represent the space in the standard way using coordinate axes etc.? Surely to write ${\bf{p}} = (2,3,5)$ means that I can get to ${\bf{p}}$ by starting from the origin and taking the prescribed step in the 3 directions, right?

If you reject this position vector interpretation of points of $\mathbb{R}^n$, as seems to be the norm, then shouldn't you be required to provide a definition of point that doesn't use coordinates? For instance, in the book A Visual Introduction to Differential Forms and Calculus on Manifolds, the author says:

'even though you are used to thinking of (3, 5) as a point, a point is actually more abstract thanthat. A point exists independent of its coordinates.'

All right, then to what point do the coordinates (2,3,4) correspond? How can you answer this question without reference to coordinates?

Have you already read this popular post: https://math.stackexchange.com/questions/645672/what-is-the-difference-between-a-point-and-a-vector — Joe, Jul 18 '21 at 19:11
@Joe yes I have, but I'm afraid it didn't help me much. I don't get how you can think about points free from any coordinate system. I mean, even by asserting that points are elements of $\mathbb{R}^n$, aren't you inherently defining points using coordinates? — Othman El Hammouchi, Jul 18 '21 at 19:23
You should not think of (physical) space as a vector space but as a manifold, as in differential geometry. There is a sharp distinction between points of space and the coordinates of these points. And coordinates are still not vectors the vectors exists in a tangent space of the manifold. — Physor, Jul 18 '21 at 19:28
When posting a question, if you have read other related posts, I recommend linking to those and explaining what the answers there did not address, just so no one posts a similar answer. — Joe, Jul 18 '21 at 19:28
Here's one possible approach: Define Euclidean space as a set that satisfies the Euclidean axioms. Open balls are naturally defined in such a space, which allowed you to define a topology on it. You can now also assume the space is complete. You can now choose two perpendicular lines and a point away from the origin on one of the lines. You can now define an isomorphism from this space to $\mathbb{R}^n$, which corresponds to the standard geometric structure on $\mathbb{R}^n$. — Deane, Jul 18 '21 at 20:29
@Joe I'll keep it in mind, thanks! I see from the dislike that my ignorance of protocol has annoyed someone. — Othman El Hammouchi, Jul 19 '21 at 11:55
@Deane that's actually a very satisfying way to think about it. Would you then say that structures defined in the manner of Euclid, with non-algebraic descriptions, would be the actual geometric objects like manifolds, topological spaces, vector spaces, etc., and that $\mathbb{R}^n$ is simply a 'model'? — Othman El Hammouchi, Jul 19 '21 at 12:00
Euclidean space with the complete topology is an example (is that what you mean by a model?) of a Riemannian manifold. — Deane, Jul 19 '21 at 19:07
@Deane Euclidean space is the collection of ideal points, lines and planes obeying Euclid's axioms, which can we described ('modelled') using $\mathbb{R}^n$ with the corresponding structures, right? But the geometric manifold would be the axiomatically defined structure, correct? — Othman El Hammouchi, Jul 19 '21 at 20:13
I don’t know what you mean by a “geometric manifold”. The word “manifold” is a modern term and has a precise definition using coordinate patches. The map from Euclidean space to $\mathbb{R}^n$ is a single coordinate patch that makes Euclidean space a manifold, whose topology is the same as the one defined on Euclidean space using open balls. Moreover, distances and angles in Euclidean space match those defined using the dot product on $\mathbb{R}^n$. This shows that there is a Riemannian metric on this manifold where distance and angle matches those of Euclidean space. — Deane, Jul 19 '21 at 21:27
@Deane I guess what I'm getting at is: do you still distinguish Euclidean space (consisting of lines, point, etc., i.e. geometric objects) from $\mathbb{R}^n$, which is an algebraic structure? — Othman El Hammouchi, Jul 20 '21 at 08:35
I think this is a matter of preference or perhaps philosophy. The way Riemannian geometry is taught, I would say that the standard approach is to start with $\mathbb{R}^n$ and observe that, using the standard inner product, one can define lines, distance, and angles, as well as its topology. This gives it a geometric structure. I do prefer to start more abstractly. I start by defining an abstract real vector space, use it to define first affine space and then Euclidean space. Then there are natural maps from affine or Euclidean space to $\math{R}^n$ . — Deane, Jul 20 '21 at 12:57

score 3 · Accepted Answer · answered Jul 18 '21 at 19:29

It sounds like the author is trying to drive at the difference between vector spaces and affine spaces. The idea here is to imagine the $3$D world we live in. Where is the origin? Obviously, there isn't one. So what does it mean to say that a point lives at the coordinate $(2,3,4)$? It's clear that there is still a notion of "point" in our space, even though there isn't a notion of coordinates!

However, for solving concrete problems, we first choose where we want the origin to be (oftentimes we can make a choice that renders our problem particularly simple) and then we have access to all of our vector space machinery. In fact, it can be really useful to work without coordinates for as long as possible. You only get one shot at choosing where the origin goes, so if you can work without making that choice, it might be a good idea to defer until you have a good idea where the best choice might be for your particular computation.

More generally, we can define manifolds without coordinates. Think of a circle. If you're only interested in the manifold structure, then a circle with radius $1$ centered at the origin, and a circle with radius $5$ centered at some other point are exactly the same, even though the points have different coordinates¹. Why should one parameterization be better than the other? Of course a circle (and the points on it) exist independent of our choice of coordinates. You might argue that, in this simple case, we should always use a unit circle centered at the origin, but for more complicated manifolds there isn't a good notion of "best" embedding. Already for a circle in $\mathbb{R}^3$, which plane should we put the circle in? $xy$? $yz$? These all give different coordinates to points on the circle, but obviously the geometric features of the circle don't care at all which one you choose.

So how, then, do we talk about points without talking about coordinates? Well, every manifold is (in particular) a topological space. So we have a bunch of points, which we can talk about abstractly as $x$, $y$, $z$, etc., but we can't necessarily give any individual point a name. This is analogous to working with a vector space before you choose a basis. The points still exist, but you'll get a different name for your point based on which basis you pick. This is still good enough for a lot of purposes, because we can say "Let $x$ be the point at which $f$ is minimized", etc.

Of course, eventually we have to work with a fixed coordinate system to really do computations. That's a really important part of the subject! What's important to remember is that, abstractly, we don't need this coordinate system for the points to exist. And sometimes, the geometry itself becomes more apparent if we work without a coordinate system for a while.

^1: I assume we're talking about smooth manifolds, not riemannian ones. If you want riemannian manifolds, then pretend I said both circles have radius $1$.

I hope this helps ^_^

Thanks a lot for your answer. I guess my confusion comes from the fact that text books in fact DO start with a coordinate description from scratch, e.g. 'consider the manifold $M = {(x,y): x^2+y^2 = 1} \subset \mathbb{R}^2$'. How would you give a description of $S^1$ without this? Using a description like in Euclid? — Othman El Hammouchi, Jul 19 '21 at 11:54
That's a fair question! It's hard because the machinery we have for working coordinate free is really best understood for an arbitrary manifold rather than a fixed one, like a circle. Because the way you name a circle (or any concrete manifold of interest) is almost always via coordinates. Since you asked, though, formally you would look at any circle, say $x^2 + y^2 = 1$ in the plane. This endows you with coordinate charts (say the charts $y = \sqrt{1-x^2}$, $y = - \sqrt{1-x^2}$, $x = \sqrt{1 - y^2}$, and $x = - \sqrt{1-y^2}$), which you can then extend to a maximal atlas. — HallaSurvivor, Jul 20 '21 at 02:56
It is this resulting (and unique!) maximal atlas structure which is how people formalize the "coordinate free" nature of a manifold. We take every possible (consistent) parameterization at once! It's not hard to show (and it's a good exercise to check) that the set of charts consistent with the ones we started with are consistent with each other, and any extra chart you try to add will render the set inconsistent. I would agree with you, though, that this feels a bit silly when we have perfectly good charts on hand. — HallaSurvivor, Jul 20 '21 at 02:59
Oh, I see, I'm familiar with those kinds of abstract constructions. That makes sense. Thanks a lot! — Othman El Hammouchi, Jul 20 '21 at 08:40
Happy to help ^_^. If this answered your question, you should mark it as such so other users know how to spend their time. Of course, you can also leave it open for a bit longer and see if someone else answers too. — HallaSurvivor, Jul 20 '21 at 17:40

Points and vectors

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