I have skimmed it, and it looks very good. It is actually not solely about matrix calculus, but shows a practical approach to differentiation in different vector spaces with many examples and intuitions.
Decentralization and local governance aren't foreign things for China. It has a remarkable amount of authority on a local level (at least provincial). You can also google ratings based on share of subnational public expenditure in total public expenditure, China is the top.
Do you mean patricians? Because senatorial elite and senators in general were not become less relevant during the republic, as a roman became a senator after he took the office of a quaestor, the first position in the line of roman public service offices. And it wasn't limited to patricians only, for the most important offices there were no restrictions for plebeians, though there were some special positions for patricians only and for plebeians only. Examples - Cicero was a plebeian, even a novus homo (a politician without senators among his ancestors); Cato the Younger, one of the leaders of the optimates (traditionalist fraction in the senate), was a plebeian, though a member of plebeian aristicracy, descendant of Cato the Censor.
Edit: i think i misread your comment, so there is no arguing with you actually.
Hm, this book applies exterior algebra to linear algebra. I don't think there are a lot of applications of multilinear algebra outside of math, it is mostly needed to study differential geometry (forms and so on).