>|It is certainly reasonable for a linear algebra exam to >|have a student show that a certain structure is a vector space!
>how do you manage to memorize mathematical definitions when you don't >even seem to be able to remember the issue under discussion in a >newsgroup thread? the issue was whether there is any reason to >memorize definitions by rote instead of just using a piece of paper or >computer to memorize them for you, so your comment is entirely >irrelevant.
No, I did remember the context. I *do* think it is reasonable for a student to show a certain structure is a vector space without having a machine or piece of paper with the definition on it. If they cannot show that, say, the collection of anti-symmetric matrices is a vector space without any computer or other pieces of paper, then they don't understand what a vector space is. I would say the same thing in an abstract algebra course with the definition of a group, or a ring, or a module over a ring, etc. If you cannot give a definition that is logically equivalent to the one in the book, you don't know the subject. You have to have such in order to show a structure is a group, or a ring, or a module, etc. You certainly need one in order to prove anything about them.