[bmitc]

I will need to read through the rest of the article later, but the initial intuition building is a bit sloppy. None of those vectors drawn in the initial examples belong to the same vector space. Vectors need to emanate from the same origin to be considered as part of the same vector space.

[Tainnor]

> Vectors need to emanate from the same origin to be considered as part of the same vector space.

That's absolutely not the case.

[bmitc]

> That's absolutely not the case.

Can you elaborate? Which vector space(s) do these vectors belong to, as drawn?

If anything, this just highlights the sloppiness of these types of illustrations. Things aren't precise enough, in my opinion, for the illustrations to do anything except confuse.

[Tainnor]

In the mathematical sense, a vector is nothing but an object that can be added to other objects and scaled by some field (with some reasonable properties attached).

In physics, a vector is often more specifically something with magnitude and direction. This still doesn't mean that it needs to be anchored at the origin. Vectors that are anchored at the origin are IIRC called position vectors, but mathematically, if you translate them away from the origin they're still the same vector.

[bmitc]

The question is one of identification between the drawing and the vector space. See my comment here: https://news.ycombinator.com/item?id=44491474.

The graph of a function f: X -> Y is the set {(x, f(x)) | x in X}. It is much more clear and precise to associate elements of the graph with vectors such that the 0 vector is identified with the R^2 origin, and then points in R^2 are identified with vectors. Then there is a mapping between vectors in this vector space to the graph, i.e., to points (x, f(x)).

> In physics, a vector is often more specifically something with magnitude and direction.

Physics is sloppy. :) This is not a general description of a vector, where vector is an element of a general vector space. Not all vector spaces have a norm, which is required for magnitude to make any sense.

> but mathematically, if you translate them away from the origin they're still the same vector.

Right, and you cannot always translate vectors without more machinery, such as parallel transport.

[Tainnor]

> Physics is sloppy. :) This is not a general description of a vector, where vector is an element of a general vector space. Not all vector spaces have a norm, which is required for magnitude to make any sense.

Fine, then let's be precise. An element of the vector space R^n is nothing more than a function from {1,...,n} to R. And every n-dimensional R-vector space is isomorphic to it.

How you wanna draw this in a coordinate system is up to you. It is customary to identify vectors with points on that coordinate system and then equivalently with an arrow pointing from the origin to that point. In that case the zero vector is the origin, or an arrow from the origin to itself.

It is equally customary to use vectors as displacement, i.e. a directed difference between two points. In that case, the vector that you can't actually draw because it has zero length is your zero element. Now your arrows don't have to be anchored at the origin anymore.

Both of these spaces are of course isomorphic.

If your vector space is infinite-dimensional it's of course not gonna be really possible to draw anymore.

[MITSardine]

What do you call the origin of a vector? I don't think that's a notion in vector spaces.

[bmitc]

Vector spaces have an addition identity normally referred to as the 0 vector.

When drawing vectors, such as on a 2D space, the origin is identified with this 0 vector, and then the points in the 2D space can be associated with the vectors in the vector space.