
Re: Failing Linear Algebra:
Posted:
Apr 30, 2004 9:51 PM


On 01 May 2004 00:34:06 GMT, Anonymous wrote:
>Marc: > >>The first time, I spent in a linear algebra course (Kaiserslautern, 1986) >>introduced sets, groups, rings and fields, modules and then vector spaces. >>(in that order) > >Odd. We did sets and then groups in algebraic structures. We touched upon >rings, I think, but I don't remember what they are. Have no idea about fields, >if they're not vector fields, which someone already said they aren't.
If you can't give the definition of a field, then why do you think you know what a vector space is? Look carefully again at the definition of v.s. and see if there are any words in that definition whose meaning you don't know.
(But probably for your purposes this semester it will suffice to know a couple examples of fields, two or three of them to be specific, instead of the definition. But no kidding, you *do* have to know at least what I'm talking about here. "A vector space over __"  fill in the blank.)
Here's an exercise. Is the set C of complex numbers with the usual complex addition a vector space over C? If so, find a basis. Is C a vector space over R? If so, find a basis. What dimension is it? Is the set of real numbers with normal addition a vector space over C? (Answer is no, why?) Can you ever have a vector space over Q (the set of rational numbers)?
Never >heard of modules. > >We did vectors, then vector spaces, then subspaces. Then spans, dependence, >bases, matrices, determinants, then eigenvalues and vectors. It's odd because >Schaum's Outlines does it in a different order. And Cliff's does an even >different order.
So? It's all stuff you have to learn. It all hangs together in a big network  the field isn't linear. (Puns intended.) Pick an organization that works for you; that's why there are different books.
> >>By the time we met vector spaces, the notion of image and kernel as well >>as the proof of the first homomorphism theorem > >"Homomorphism theorem"? Is this what's called "Dimension Theorem" now?
No, it's something in group theory. Not entirely unrelated, though.
> >>But nowadays nobody needs to be afraid of modules in a linear algebra course, >>unless it is only a MMC (Matrix Manipulation Course). >> > >What are modules? And why are they related to marix manipulation?
They aren't. Modules are a generalization of vector spaces, where "field" is generalized to "ring". Every vector space is a module, but not vice versa.
His point was that modules are not a difficult concept at all, and anybody who can grasp the subject matter of linear algebra (in a course like yours) should have no trouble understanding them. But in a strictly MMC course there would be many students who would find them scary. I don't know where that puts you personally; certainly right now (since you don't remember what a ring is) you would be scared. But fear not, looks like modules won't hit you this semester anyway.
BTW, I, for >one, would love to have taken a linear algebra course that was all MMC. I love >the stuff!
Yes, it can be fun. But once you know how to do it, it gets tedious and then thank heavens, you can leave it to the computers. Hopefully in time, you will find the abstractions even *more* fun, and anyhow there is plenty of MMC you have to do in manipulating those abstractions. Just read some of the postings here by the real mathematicians, and you will see matrices galore.

