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

 Messages: [ Previous | Next ]
 Marc Olschok Posts: 409 Registered: 12/6/04
Re: Failing Linear Algebra:
Posted: Apr 29, 2004 8:53 AM

Anonymous wrote:
> Olschok:
>

>> If I
>>> remember correctly, an abelian group is a group that has an invertible
>>table.
>>
>>What is an "invertible table" ? By the way, a group is abelian if the
>>group operation is commutative.

>
> Right. So, if you set up a table with all the set members on the horizontal
> and vertical, it'll form an invertible diagonal because xy = yx.

An "invertible diagonal" ? ( Are you inventing these on the fly? :-)

> So, a group has to be closed, ASSOCIATE, have an identity,
> and have an inverse?

The last one to mean, "each group element has an inverse".

> Whereas an abelian group has to have all the above, plus be commutative,
> right? It's been 6 months, at least.

Well, unless you burned all your notes from the last term, you can
look it up if you do not feel sure.

>
>>those axioms in (a) are those for an abelian group.
>>In most linear algebra courses, the definition of an (abelian) group
>>should at least be mentioned because
>>
>>- the axioms from (a) can be remembered easier.

>
> I agree. It's odd...we focused on abelian groups (and groups in general) in
> algebraic structures. We never mentioned abelian groups in linear,
> even though it's considered a continuing course from algebraic.

Bad course organization perhaps. The natural choice would be to

>[...]
>>- the definition of a field also includes them for the field addition.
>
> Vector field?

No. A field in this context is a ring such that every nonzero element
has a multiplicative inverse. That is where the scalars come from that
operate on the vectorspace via scalar multiplication.

For example the real numbers form a field in this sense.
Perhaps only vector spaces over the reals are considered in your
current course; in this case the above concept was probably never
introduced explicitely.

Marc

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