
Re: Failing Linear Algebra:
Posted:
Apr 28, 2004 11:35 PM


grubb@lola.math.niu.edu (Daniel Grubb) wrote in message news:<c6or0b$3tn$1@news.math.niu.edu>... > >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 antisymmetric > 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. > > Dan Grubb
You don't have to remember the defintion. You just have to remember a bunch of examples. For vector spaces, I just think of R^n, and then generalize. It relates to algebra to it has to have something to do with operations. So, vectors have to be additive, and scalars have to be nice. The definition is pretty obvious. Memorization is only really needed for geometry.

