A semi-decidable problem is still pretty bad news from a computational perspective, but I agree that it's not the best example of what I was trying to illustrate. I was aiming for something dramatic and (somewhat) approachable, but ended up emphasizing properties of vector spaces as free abelian groups, rather than as vector spaces per se (which undermines my emphasis of the specialness of vector spaces in comparison to other algebraic structures). That said, to the best of my knowledge, the algorithms for computing whether two finitely-generated* abelian groups are isomorphic take advantage of the close relationship between finitely-generated abelian groups and vector spaces to compute the Smith normal form of matrices associated with the groups and then compare the normal forms. This takes roughly O(nmsublinear factors) for n x m matrices[0]. So to revise my example, vector spaces with a finite basis (and any finitely-generated free abelian group) can be compared for isomorphism in constant time and finitely-generated non-free abelian groups take time roughly quadratic in the number of generators, so there is a huge win there still.
Do you have a favorite example that highlights the unique computational properties of vector spaces?
*I don't know how this changes in the finitely-presented case, but I assume the extra constraint can be used to improve the performance of the algorithms. It's a lot easier to find asymptotic analysis of the finitely-generated case though and I don't see a way around dealing with the fact that it's still not free.
Do you have a favorite example that highlights the unique computational properties of vector spaces?
*I don't know how this changes in the finitely-presented case, but I assume the extra constraint can be used to improve the performance of the algorithms. It's a lot easier to find asymptotic analysis of the finitely-generated case though and I don't see a way around dealing with the fact that it's still not free.
[0] - I'm basing this on Chapter 8 of https://cs.uwaterloo.ca/~astorjoh/diss2up.pdf, but this is a deep field in which I am not an expert, so if you are, I'd love to hear more.