Yeah, ignore me there, you're right. I got confused between a linear coordinate transformation and a linear transformation on a vector space

. The former being a linear map between coordinates, and the latter a linear combination of basis elements the coefficients given by the elements of the Jacobian, so it doesn't matter whether the coordinate transformation is linear or not for the latter. Replace where I've said "linear coordinate transformation" with "linear transformation". This does of course render my saying a translation isn't a linear transformation void, but everything else I've said, with the replacement I just made, I believe is correct.

Gah, Dave, you're a fool. I'd cover my shame by editting my posts but it's too late to edit them :(