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Topic: Failing Linear Algebra:
Replies: 91   Last Post: Jan 10, 2007 12:56 PM

 Messages: [ Previous | Next ]
 Dave Rusin Posts: 3,118 Registered: 12/6/04
Re: Failing Linear Algebra:
Posted: Apr 29, 2004 2:24 PM

<gersh@bialer.com> wrote:

>>> the issue was whether there is any reason to
>>> memorize definitions by rote

>>
>> If you cannot give
>> a definition that is logically equivalent to the one in the book, you
>> don't know the subject. You have to have such in order to show a structure
>> is a group, or a ring, or a module, etc. You certainly need one in order
>> to prove anything about them.

>
>You don't have to remember the defintion. You just have to remember a
>bunch of examples. For vector spaces, I just think of R^n, and then
>generalize. It relates to algebra to it has to have something to do
>with operations. So, vectors have to be additive, and scalars have to
>be nice. The definition is pretty obvious. Memorization is only really
>needed for geometry.

I certainly agree that it is very useful to have particular models in
mind when studying any axiomatic class of objects, but apart from that,
I have to disagree with this last poster. If you think "vector space
means R^n, generalized", then you will surely fall into certain traps
which we instructors have seen time and time again. The problem is
that there are _different_ generalizations of R^n possible. The facts
that you know about R^n may or may not be relevant in a general case,
and if all you have is your example, and not the correct definition,
then you can't tell what will be true in the more general case. Worse,
even if they are true, you may not be able to prove them.

Some examples:

"A vector is really a sequence of numbers."

"Every matrix is the sum of a symmetric matrix and a skew-symmetric matrix."

"Every linear transformation from V to V whose kernel is {0} is onto."

"Every vector is a scalar multiple of a unit vector"

etc. R^n is only the standard model (well, "models") for
finite-dimensional vector spaces over the real field, and these vector
spaces have a natural basis and (therefore) a natural inner-product
structure, and thus a natural norm and a natural metric and a natural
topology; these are refined statements that may or may not be true for
general vector spaces.

dave

Date Subject Author
4/24/04 Daniel Grubb
4/24/04 Marc Olschok
4/24/04 Daniel Grubb
4/24/04 Marc Olschok
4/24/04 Daniel Grubb
4/24/04 Thomas Nordhaus
4/24/04 Dave Rusin
4/25/04 Jonathan Miller
4/25/04 Felix Goldberg
4/24/04 Daniel Grubb
4/28/04 Tim Mellor
4/28/04 James Dolan
4/28/04 Daniel Grubb
4/28/04 James Dolan
4/28/04 Daniel Grubb
4/28/04 gersh@bialer.com
4/29/04 Daniel Grubb
4/29/04 Dave Rusin
4/28/04 Guest
4/29/04 Guest
4/28/04 Guest
1/10/07 David C. Ullrich
4/29/04 Dave Rusin
4/28/04 Guest
1/10/07 Law Hiu Chung
1/10/07 Dave Seaman
1/10/07 Marc Olschok
1/10/07 George Cox
4/28/04 Guest
1/10/07 Dave Rusin
4/28/04 Lee Rudolph
4/28/04 Guest
4/28/04 Guest
1/10/07 Marc Olschok
1/10/07 Toni Lassila
4/29/04 Guest
1/10/07 M L
1/10/07 Thomas Nordhaus
4/30/04 Guest
1/10/07 David C. Ullrich
1/10/07 Toni Lassila
4/30/04 Guest
1/10/07 George Cox
1/10/07 Marc Olschok
4/30/04 Guest
4/30/04 Guest
4/27/04 Guest
1/10/07 Thomas Nordhaus
1/10/07 David C. Ullrich
1/10/07 Dave Rusin
1/10/07 David C. Ullrich
5/9/04 James Dolan
5/10/04 David C. Ullrich
5/10/04 James Dolan
5/10/04 David C. Ullrich
5/10/04 Marc Olschok
5/10/04 David C. Ullrich
4/27/04 Guest
1/10/07 Thomas Nordhaus
4/27/04 Guest
1/10/07 magidin@math.berkeley.edu
1/10/07 David C. Ullrich
1/10/07 Marc Olschok
1/10/07 David C. Ullrich
1/10/07 Tim Mellor
4/28/04 Daniel Grubb
4/28/04 Daniel Grubb
4/27/04 Guest
1/10/07 David C. Ullrich
4/28/04 Dave Rusin
4/28/04 Daniel Grubb
4/27/04 Guest
1/10/07 Marc Olschok
4/24/04 Wayne Brown
4/24/04 Thomas Nordhaus
4/24/04 David Ames