2 ms·
Right, but the individual vector spaces you speak of there are the tangent spaces of a manifold M. There's a vector space TpM attached to M at a point p. And th
by gspr 5y ago
Right, but the individual vector spaces you speak of there are the tangent spaces of a manifold M. There's a vector space TpM attached to M at a point p. And the point you make is that there isn't necessarily a canonical way to relate vectors in TpM to vectors in TqM for p≠q. In this sense, we may well say that "a vector in TpM is anchored at p, and a vector in TqM is anchored at Q". I agree with that, obviously. However, within a fixed vector space, of which such a TpM is merely one example, there is no notion of vectors being "anchored".
- kmill 5y agoGiven the definitions in a textbook on smooth manifolds, I agree with you -- the modern notion of a vector space has no notion of root or anchor since these words have no associated definitions. It's interesting to think about how we came to the modern viewpoint. Some of the underlying context behind my comment is that long ago manifolds were concretely subsets of R^n that could be locally parameterized. The tangent space at a point was the affine subspace of R^n that was tangent to the manifold there, and you could regard a tangent vector as an arrow rooted at that point inside that subspace. Even though we have our clean modern notion of an abstract vector space, somehow these old intuitions linger on in other disciplines In math we have a precise definition for a vector (an element of a vector space, full stop), but I don't think experts have the authority to be prescriptive about jargon outside their discipline. Of course, a student should be willing to drop preconceptions and to absorb correct usage.