This question seems really controversial for some reason, but I don't understand why (most of the time every upvote has been balanced out by some determined downvoter, now it's in the negatives).
The question has been a genuine source of confusion of mine for a really long time, and a major part of that confusion was the effort of integrating all of the sub-answers together into one consistent answer. (Basically everything is multilinear algebra, the study of tensor algebras, and in the case of objects meant to capture geometric intuition or used most frequently in mathematical physics, they are over the field of real numbers.)
I.e. sure I could have asked twenty questions and then tried to put the pieces together myself, but my greatest confusion was in putting the pieces together, and it's generous to assume that someone could ask 20 questions about similar topics and expect them all to be answered to the same level of detail or accuracy.
Is it controversial because people feel it is a "soft question" (I would actually disagree with that characterization to some extent) or because it is a "big list"? (again, it seems like most of these are closely related and should be possible to fold into each other easily, plus I specifically state I'm not concerned with extensive justification of the answers, just what the answer is).
The reason why I haven't made the question community wiki yet is (1) I don't know how to, (2) I don't know when it is appropriate to do so.
To the extent that I felt it necessary/appropriate to answer my own question, this does seem to suggest that community wiki would be appropriate, but then again, I don't quite understand the purpose of community wiki.