Re: [Axiom-math] Solving symbolic vector equations: How do I do this in Axiom?

[Bill Page]

On Tue, May 13, 2008 at 3:51 PM, Zach wrote:
> Please excuse me, I am really quite new to Axiom and much of
> abstract algebra terminology.

Welcome.

> I hope this is the right place for these questions.  If not,
> could someone point me in the right direction.

Yes, it is a good place to ask such questions.

> I am currently reading the Axiom tome, er.. book, but it is good
> to have a human to bounce ideas off of.

Another useful place might be:

http://axiom-wiki.newsynthesis.org

>
> Here is a really simple example, we have real number `a' and
> vectors `v1' and `v2'.  Given:
>
> a*v1=v2
>
> solve(a*v1=v2)
> ==> [a = v2/v1]
>
> I would like to solve for a.  If I put this in, Axiom assumes v1 and v2 to
> be things that have a defined division (a field I guess).  But really we
> have no division by a vector (a ring, perhaps?),

You have the right general idea.

> so what I would like is for axiom to solve this by
>
> a * v1 . v1 = v2 . v1
> a = (v2 . v1) / (v1 . v1)
>

??? This is not a solution to the original equation!

> I assume that I do this by giving Axiom some type information,
> like specifying v1 and v2 as Vector (or Matrix) Fraction Integer
> or something.

Well no, not really. Currently Axiom has no domain for symbolic
computations with vectors. Specifying

  v1:Vector Fraction Integer

does not allow symbolic computations with the symbol 'v1', instead it
declares that the *type* of the variable 'v1' is 'Vector Fraction
Integer'. From this declaration Axiom expects 'v1' to eventually be
given a value, e.g.

  v1::= [1/2, 1/3, 1/4]

Axiom does have other domains like Polynomial which includes symbolic variables.

> What is the best way of tackling these types of problems?
>

It would be possible (and quite interesting) to create a new domain in
Axiom for symbolic vector calculations.

Regards,
Bill Page.