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Topic: Failing Linear Algebra:
Replies: 54   Last Post: Jan 10, 2007 12:47 PM

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 Guest
Re: Failing Linear Algebra:
Posted: Apr 27, 2004 4:28 PM

borchers@nmt.edu (Brian Borchers) wrote in message news:<c6bea4\$t9q\$1@rainbow.nmt.edu>...
> I've also taught introductory linear algebra many times. David
> Ullrich's advice is correct as far as it goes. I have a few
>
> 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

Nope.

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.
> 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 borchers@nmt.edu
> 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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Date Subject Author
4/22/04 Guest
4/22/04 Michael N. Christoff
1/10/07 Gerry Myerson
1/10/07 Jonathan Miller
1/10/07 Guest
1/10/07 David C. Ullrich
1/10/07 Acid Pooh
1/10/07 Guest
4/23/04 Brian Borchers
4/27/04 Guest
1/10/07 maky m.
4/26/04 David Ames
1/10/07 Guest
1/10/07 Michael Stemper
1/10/07 maky m.
4/23/04 Porker899
4/27/04 Guest
1/10/07 Abraham Buckingham
1/10/07 Mitch Harris
1/10/07 Guest
1/10/07 Grey Knight
1/10/07 Guest
1/10/07 Toni Lassila
1/10/07 Thomas Nordhaus
1/10/07 George Cox
4/28/04 Dave Rusin
4/28/04 George Cox
4/28/04 George Cox
4/29/04 Marc Olschok
4/29/04 Mitch Harris
4/29/04 Robert Israel
4/28/04 Guest
4/29/04 Guest
1/10/07 Dave Rusin