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

 Messages: [ Previous | Next ]
 Dave Seaman Posts: 3,624 Registered: 12/8/04
Re: Failing Linear Algebra:
Posted: Apr 28, 2004 7:36 PM

On 28 Apr 2004 22:51:35 GMT, Anonymous wrote:
> George:

>>I was taught (33 years ago in the University of Birmingham, UK) that a
>>vector space is
>>(1) a set of elements (the vectors) which form an abelian group under
>>(2) etc....
>>The phrase "commutative group" or "additive group" might have been used
>>instead of "abelian group"--you'll forgive me if I can't quite remember!
>>

> I think you just made me remember what an abelian group is. It's closed under
> its operation, it's associative, commutative, it's got an inverse, and it's got
> the identity element in it. There may be some other condition, since aren't
> all the above needed for any group?

All except the "commutative" part. If the group operation happens to be
commutative, then the group is abelian.

>I remember something about an abelian
> group forming a table with a diagonal, or something....

Nope. "Abelian group" simply means "commutative group", and nothing
more.

> So, basically, "abelian group" is just an easier way of saying "vector space".
In approximately the same sense that "steering wheel" is just an easier
way of saying "automobile". You have omitted two of the three things
needed in order to have a vector space:

(1) A collection of vectors V, which form an abelian group with
respect to vector addition (this is the only part you got right),

(2) A scalar field F,

(3) An operation called scalar multiplication, which binds V and F
together. The operation is associative and satisfies the
distributive laws.

--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.

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
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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