On Fri, Oct 15, 2010 at 8:11 PM, Douglas Theobald
<dtheob...@brandeis.edu> wrote:
> Vectors are not only three-dimensional, nor only Euclidean -- vectors can be
> defined for any number of arbitrary dimensions. Your initial comment
> referred to complex numbers, for instance, which are 2D vectors (not 1-D).
> Obviously scalars are not 3-vectors, they are 1-vectors. And contrary to
> your earlier assertion, you can always represent complex numbers as vectors
> (in fortran, C, on paper, or whatever), and it is possible to define many
> different valid types of multiplication, exponentiation, logarithms, powers,
> etc. for vectors (and matrices as well).
I didn't say that vectors are only 3D or only Euclidean (note my
qualification 'Euclidean *or otherwise*'). I was stating that we're
not talking here specifically about 1D (or 2D) vectors; my use of 3D
is only an example, since my original example referred to structure
factors which are usually defined in 3D reciprocal space. Quite
obviously vectors can be generalised to any number of dimensions (as
in the example below).
Let's take a simple example of an operation that's trivial to express
using complex numbers as scalars. Suppose we have 2 vectors of
complex structure factors of equal dimension (n) and we want to form
the (complex) scalar product. This kind of equation arises in, for
example, the theory of direct methods.
Let F = (F1, F2, F3, ... Fn)
and G = (G1, G2, G3, ... Gn)
Then the scalar product F.G = F1*G1 + F2*G2 + F3*G3 + ... Fn*Gn (or
SUM [j=1 to n] Fj*Gj),
where '+' and '*' here are the normal addition and multiplication
operators on scalars (here complex of course).
Note that the result of a scalar product of 2 vectors is by definition
a scalar and here it's complex!
Also note that the RHS can be programmed exactly as written, in fact
something like:
COMPLEX F, G, S
S = 0
1 READ (*,*,END=2) F, G
S = S + F*G
GOTO 1
2 PRINT *,S
END
Now, how would you express the scalar product F.G in a way that could
be programmed, using vector notation for all the complex numbers, and
obviously you can only use operators that are defined for vectors,
namely addition, subtraction, scalar multiplication, scalar and
exterior product?
Then when you've done that, how would you express a ratio of complex
numbers (say F1/G1), again using only vector notation?
-- Ian
>
> On Oct 15, 2010, at 12:40 PM, Ian Tickle wrote:
>
>> Any vector, whether in the 'mathematical' or 'physical' sense as
>> defined in Wikipedia, and which is defined on a 3D vector space
>> (Euclidean or otherwise - which I hope is what were talking about),
>> has by definition 3 elements (real or complex). This clearly excludes
>> all scalars (real or complex) which have only 1 whatever the dimension
>> of the space. Therefore it's plainly impossible for an entity in 3D
>> space to be both a scalar and a vector at the same time. Your
>> conclusion that scalars and complex numbers fulfil the axioms of a
>> vector space is applicable only in the case of a 1D vector space, and
>> therefore is not relevant. My original observation which started this
>> thread was intended to be general one, not for a particular special
>> case.
>>
>> -- Ian
>>
>> On Fri, Oct 15, 2010 at 5:17 PM, Douglas Theobald
>> <dtheob...@brandeis.edu> wrote:
>>> On Oct 15, 2010, at 11:37 AM, Ganesh Natrajan wrote:
>>>
>>>> Douglas,
>>>>
>>>> The elements of a 'vector space' are not 'vectors' in the physical
>>>> sense.
>>>
>>> And there you make Ed's point -- some people are using the general vector
>>> definition, others are using the more restricted Euclidean definition.
>>>
>>> The elements of a general vector space certainly can be physical, by any
>>> normal sense of the term. And note that physical 3D space is not
>>> Euclidean, in any case.
>>>
>>>> The correct Wikipedia page is this one
>>>>
>>>> http://en.wikipedia.org/wiki/Euclidean_vector
>>>>
>>>>
>>>> Ganesh
>>>>
>>>>
>>>>
>>>> On Fri, 15 Oct 2010 11:20:04 -0400, Douglas Theobald
>>>> <dtheob...@brandeis.edu> wrote:
>>>>> As usual, the Omniscient Wikipedia does a pretty good job of giving
>>>>> the standard mathematical definition of a "vector":
>>>>>
>>>>> http://en.wikipedia.org/wiki/Vector_space#Definition
>>>>>
>>>>> If the thing fulfills the axioms, it's a vector. Complex numbers do,
>>>>> as well as scalars.
>>>>>
>>>>> On Oct 15, 2010, at 8:56 AM, David Schuller wrote:
>>>>>
>>>>>> On 10/14/10 11:22, Ed Pozharski wrote:
>>>>>>> Again, definitions are a matter of choice....
>>>>>>> There is no "correct" definition of anything.
>>>>>>
>>>>>> Definitions are a matter of community choice, not personal choice; i.e.
>>>>>> a matter of convention. If you come across a short squat animal with
>>>>>> split hooves rooting through the mud and choose to define it as a
>>>>>> "giraffe," you will find yourself ignored and cut off from the larger
>>>>>> community which chooses to define it as a "pig."
>>>>>>
>>>>>> --
>>>>>> =======================================================================
>>>>>> All Things Serve the Beam
>>>>>> =======================================================================
>>>>>> David J. Schuller
>>>>>> modern man in a post-modern world
>>>>>> MacCHESS, Cornell University
>>>>>> schul...@cornell.edu
>>>>>
>>>>>
>>>>>
>>>>>
>>>
>>>
>
>
