email@example.com (Brian Borchers) wrote in message news:<firstname.lastname@example.org>... > I've also taught introductory linear algebra many times. David > Ullrich's advice is correct as far as it goes. I have a few > additional suggestions: > > Suppose that you have a definition like: > > A <frotz> is a <mumble> that has the properties > 1. <foo> > 2. <bar> > 3. <plugh> > > and theorems like: > > In any <frotz>, <x> happens. > > You'll find that lots of exercises are of the form: > > Consider <plover>. Is <plover> a <frotz>?
Exactly! My professor's big on these "true/false" type problems. 1/3 of every test is the "true/false" section. Another 1/3 is some type of long proof. Dimension Theorem, anyone (!!!)? That's where I blew it on the second exam. I understand that dim W = Im W + ker W, and it has something to do with that the Image represents every free variable and the kernel represents every basic variable (why?), so when added together they equal the overall dimension (ALL variables). But, how do you prove it? Explaining it like this got me like a 2 out of 25 on the exam.
> > When you solve such a problem it may be that your grader will accept a > "yes" or "no" answer,
and it's likely that the answer in the back of > the book will be a simple "yes" or "no".
Yes! Another reason why attempting book problems is irrelevant. I need more comprehensive solutions, including the reasoning that you're supposed to be following. For example, I know that if m > n and m is the number of equations and n is the number of variables, that there's at least one equation that isn't necessary (a linear dependence??), but I have trouble explaining WHY I know this is true and/or writing it out in a formal proof.
Do not be tempted by this. > If the answer is "yes", then your answer should be of the form "Yes, > because <plover> has properties <foo>, <bar>, and <plugh>", followed > by work that shows this. On the other hand, if the answer is "no", > then your answer should be something like, "No, because <plover> does not > satisfy <bar>", followed by work that shows this.
Right. If "yes", we need to do a proof. If "no", we need to show a counterexample. "No" is relatively easy, but "yes" confuses me.
> > The important point here is that mathematics is not simply a guessing game. > It's really about reading and understanding logical arguments, and then later > constructing your own logical arguments. > > You should also take time after memorizing the definition and the > theorem to come up with examples and counterexamples related to this > definition. Start by coming up with a <frotz>, verifying that is has > properties <foo>, <bar> and <plugh>, and that <x> happens. Then > construct something that is almost a <frotz> but doesn't satisfy > property <foo>. Does <x> happen? If it doesn't, then you can see one > reason why the property <foo> is part of the definition of a <frotz>. > > In general, you need to explore each of the parts of the definition, and > understand what "goes wrong" when one of the required properties is not > satisfied.
I've been doing more of this lately. I'm getting there.
> > > > -- > Brian Borchers email@example.com > Department of Mathematics http://www.nmt.edu/~borchers/ > New Mexico Tech Phone: 505-835-5813 > Socorro, NM 87801 FAX: 505-835-5366 > > > -----= Posted via Newsfeeds.Com, Uncensored Usenet News =----- > http://www.newsfeeds.com - The #1 Newsgroup Service in the World! > -----== Over 100,000 Newsgroups - 19 Different Servers! =-----