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

 Messages: [ Previous | Next ]
 David C. Ullrich Posts: 21,553 Registered: 12/6/04
Re: Failing Linear Algebra:
Posted: Apr 28, 2004 8:38 AM

On 27 Apr 2004 22:49:33 GMT, Anonymous wrote:

>Toni:
>

>>Define "vector". You can't really, since you haven't properly defined
>>a vector space. Hint: axioms.

>
>I don't understand. A vector is any collection: (x1, ...., x^n) of anything.
>In math, the vectors are numbers.

Absolutely not. (1,2,3) is an example of a vector in a certain vector
space. But what "vector" actually means is "element of a vector
space", and what "vector space" actually means is [insert
longish definition here].

An example I always give to emphasize that a vector space
is exactly what the definition says it is, not what we think
it is:

Let V = {the movie "Kill Bill"}. Let's say M is that movie, to
save typing; now V = {M}.

For x, y in V define x + y = M. For x in V and a real number
r, define fx = M.

Now it's easy to verify (_if_ you _know_ the definition of
"vector space" it's easy, anyway) that V, together with
the addition and scalar multiplication defined above,
is a vector space. So now the movie "Kill Bill" has
become a vector.

(Note it's the movie itself that's a vector, not (the movie)
or the sequence of frames of the movie or whatever.)

>>Is span({0,0},{0,2}) a plane?
>
>No, because a1 (0,0) + a2 (0,2) = 2a2. Multiplied by any scalar, this gives
>you a line through the origin and 2a2. So, it's just a line. I guess I should
>have said any vector space in R^2 is at MOST a plane?? Or, maybe, any basis
>for R^2 is a plane. Right? Since the (0,0) part of the span above is not
>needed. So, the above span is not independent and is therefore not a basis.
>That's why you just get a line.
>

>> Is span({0,0,0},{0,0,1},{0,0,2}) a solid
>>area?

>
>Again, this would be a line (????): 3a3. Only either (0,0,1) or (0,0,2) is
>needed to produce a basis.

************************

David C. Ullrich

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