Choosing a programming language as a competitive tool

[David C. Ullrich]

On Tue, 8 May 2001 10:00:38 -0400, "Paul Magwene"
<paul.magwene at yale.edu> wrote:

>
>"Isaac To Kar Keung" <kkto at csis.hku.hk> wrote in message
>news:7i4ruwigff.fsf at enark.csis.hku.hk...
>> >>>>> "Greg" == Greg Ewing <see at my.signature> writes:
>>
>>     Greg> But a collection of coordinates are just one way of representing
>a
>>     Greg> vector. The abstract concept of a vector doesn't depend on
>>     Greg> coordinates.
>>
>> That's not the mathematical meaning of vector.  It's the "physical" or
>> "engineering" meaning.  The mathematical meaning of vectors has to do with
>> linear algebra.  While a vectors can still have many "representation" when
>> different "basis" is used, it is consistent that multiple real numbers are
>> used to represent a vector.
>>
>
>That's not true!  

Of course it isn't true, but

>We can just as well talk about complex vector spaces as
>real vector spaces.  In fact, often times it's more convenient to do so
>(because the results are more general).

a much more important reason it's not true is that the elements of a 
vector space simply do not need to be "multiple real numbers" at all.
And the addition in the vector space need not have anything to do
with the actual addition of real numbers, etc. A vector space in math
is just an abstract structure with certain operations that satisfy
certain axioms. (Of course n-tuples of real numbers with the standard
operations give one of the most important examples.)

>See Sheldon Alexander's book "Linear Algebra Done Right" (1997, Springer)

Or if that's not available then Sheldon Axler et al's book of the same

title. (I know it's Sheldon Axler, not sure about the al.)

>for a good example of a text that derives many of the common results for
>complex vector spaces....
>
>--Paul
>

David C. Ullrich
***********************
"Sometimes you can have access violations all the
time and the program still works." (Michael Caracena,
comp.lang.pascal.delphi.misc 5/1/01)