
Re: Failing Linear Algebra:
Posted:
Apr 29, 2004 2:24 PM


In article <fc90f1e9.0404281935.394d563f@posting.google.com>, <gersh@bialer.com> wrote:
>>> the issue was whether there is any reason to >>> memorize definitions by rote >> >> 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. > >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.
I certainly agree that it is very useful to have particular models in mind when studying any axiomatic class of objects, but apart from that, I have to disagree with this last poster. If you think "vector space means R^n, generalized", then you will surely fall into certain traps which we instructors have seen time and time again. The problem is that there are _different_ generalizations of R^n possible. The facts that you know about R^n may or may not be relevant in a general case, and if all you have is your example, and not the correct definition, then you can't tell what will be true in the more general case. Worse, even if they are true, you may not be able to prove them.
Some examples:
"A vector is really a sequence of numbers."
"Every matrix is the sum of a symmetric matrix and a skewsymmetric matrix."
"Every linear transformation from V to V whose kernel is {0} is onto."
"Every vector is a scalar multiple of a unit vector"
etc. R^n is only the standard model (well, "models") for finitedimensional vector spaces over the real field, and these vector spaces have a natural basis and (therefore) a natural innerproduct structure, and thus a natural norm and a natural metric and a natural topology; these are refined statements that may or may not be true for general vector spaces.
dave

