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>?
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". 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.
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.
-- 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
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