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Re: Arbitrary vector

  • To: mathgroup at smc.vnet.net
  • Subject: [mg121673] Re: Arbitrary vector
  • From: James Womack <james.c.womack at gmail.com>
  • Date: Sat, 24 Sep 2011 22:36:03 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <201109230742.DAA07655@smc.vnet.net> <4E7C3675.1040106@gmail.com>

Hi Jacopo

Thanks for your reply. I see your point. Having vectors without a 
clearly defined number of components would make it impossible to get 
definite answers from some vector operations.

The reason why I asked this question was because I wanted to derive an 
equation featuring vectors. I knew all the vectors had the same number 
of components, but did not know what those components were or how many 
of them there were (i.e. they were in the same, undefined, coordinate 
system). I was hoping there was some way that Mathematica could 
interpret this system without me needing to specify the components of 
the vectors explicitly.

So, rather than have a.b = a_x b_x + a_y b_y + a_z b_z, I'd like to see 
a.b = \sum_{i}^{N} a_i b_i where N is not explicitly defined (to clearly 
express my intent, I have used LaTeX-style notation).

Essentially, I'd like to tell Mathematica that a and b are vectors with 
N components and have Mathematica perform operations on vector 
operations a and b without explicitly defining N or the individual 
vector components.

Thanks for taking the time to consider my question.

James

On 23/09/11 08:34, Jacopo Bertolotti wrote:
> Hi,
> I have the feeling that your question is ill posed. A vector is (by 
> definition) an element of a vector space and a vector space is just a 
> set where certain properties are defined/respected (e.g. a commutative 
> addition between the elements is defined etc.). Some vectors can be 
> represented as a list of numbers (e.g. velocity) and some can not 
> (e.g. square integrable functions).
> If all you are interested in are those vectors that can be represented 
> as lists of numbers and you are happy with the canonical operations 
> (scalar product, multiplication by a matrix, outer product etc.) then 
> the problem is non-existent since Mathematica do it automatically.
> To be more clear: a.b is interpreted as the scalar product between the 
> two vectors/matrices/tensors a and b irrespectively from their number 
> of elements. If the operation is impossible (say a has three 
> components and b 4) you will get an error but nothing more.
> Of course do not expect any smart simplification or algebraic trick 
> from Mathematica unless you impose some assumptions.
>
> Jacopo
>
> p.s.
> If I misunderstood your question could you just try to reformulate it?
>
>
> On 09/23/2011 09:42 AM, James Womack wrote:
>> Hello,
>>
>> Does anyone know if it is possible to define an arbitrary vector in
>> Mathematica? What I mean by this, is can I tell Mathematica that a
>> particular variable is a vector, without having to define the components
>> of this vector?
>>
>> I'd like to be able to manipulate vectors with an arbitrary number of
>> components, but am not sure if this is possible in Mathematica.
>>
>> Many thanks,
>> James
>>
>>
>




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