The Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Guest

Watch Watch this User
Re: Failing Linear Algebra:
Posted: Apr 27, 2004 4:28 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply


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
> 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,


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.
> 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 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
>
>
> -----= 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! =-----




Date Subject Author
4/22/04
Read Failing Linear Algebra:
Guest
4/22/04
Read Re: Failing Linear Algebra:
Michael N. Christoff
1/10/07
Read Re: Failing Linear Algebra:
Gerry Myerson
1/10/07
Read Re: Failing Linear Algebra:
Jonathan Miller
1/10/07
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
David C. Ullrich
1/10/07
Read Re: Failing Linear Algebra:
Acid Pooh
1/10/07
Read Re: Failing Linear Algebra:
Guest
4/23/04
Read Re: Failing Linear Algebra:
Russell Blackadar
4/23/04
Read Re: Failing Linear Algebra:
Brian Borchers
4/27/04
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
maky m.
4/26/04
Read Re: Failing Linear Algebra:
David Ames
1/10/07
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
Michael Stemper
1/10/07
Read Re: Failing Linear Algebra:
maky m.
4/23/04
Read Re: Failing Linear Algebra:
Porker899
4/27/04
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
Abraham Buckingham
1/10/07
Read Re: Failing Linear Algebra:
Mitch Harris
1/10/07
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
Grey Knight
1/10/07
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
Toni Lassila
1/10/07
Read Re: Failing Linear Algebra:
Thomas Nordhaus
1/10/07
Read Re: Failing Linear Algebra:
George Cox
4/28/04
Read Re: Failing Linear Algebra:
Dave Rusin
4/28/04
Read Re: Failing Linear Algebra:
George Cox
4/28/04
Read Re: Failing Linear Algebra:
George Cox
4/29/04
Read Re: Failing Linear Algebra:
Marc Olschok
4/29/04
Read Re: Failing Linear Algebra:
Mitch Harris
4/29/04
Read Re: Failing Linear Algebra:
Robert Israel
1/10/07
Read Re: Failing Linear Algebra:
Russell Blackadar
4/28/04
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
Russell Blackadar
4/29/04
Read Re: Failing Linear Algebra:
Russell Blackadar
4/29/04
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
Russell Blackadar
1/10/07
Read Re: Failing Linear Algebra:
Russell Blackadar
5/1/04
Read Re: Failing Linear Algebra:
Russell Blackadar
1/10/07
Read Re: Failing Linear Algebra:
Russell Blackadar
1/10/07
Read Re: Failing Linear Algebra:
Dave Rusin
1/10/07
Read Re: Failing Linear Algebra:
Russell Blackadar
4/30/04
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
David C. Ullrich
4/27/04
Read Re: Failing Linear Algebra:
Guest
4/27/04
Read Re: Failing Linear Algebra:
Guest
4/28/04
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
Law Hiu Chung
4/30/04
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
David C. Ullrich

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2017. All Rights Reserved.