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From: Kaba on 20 Jun 2010 10:07
Hi,

I'd like construct matrix algebra as an algebra over a field. How would
you approach this?

--
http://kaba.hilvi.org
From: Axel Vogt on 20 Jun 2010 10:56
Kaba wrote:
> Hi,
>
> I'd like construct matrix algebra as an algebra over a field. How would
> you approach this?

multiplication = composition of morphisms ?

From: Kaba on 20 Jun 2010 11:49
Axel Vogt wrote:
> Kaba wrote:
> > Hi,
> >
> > I'd like construct matrix algebra as an algebra over a field. How would
> > you approach this?
>
> multiplication = composition of morphisms ?

Yes, but what confuses me is that there are all linear transformations
from R^n from R^m, for all n, m in N, where N is natural numbers. And
(R^n -> R^m) o (R^m -> R^p) gives (R^n -> R^p) etc. I would need some
common object to represent them all to keep composition closed.
R^{N x N} comes to mind.

--
http://kaba.hilvi.org
From: Axel Vogt on 20 Jun 2010 12:07
Kaba wrote:
> Axel Vogt wrote:
>> Kaba wrote:
>>> Hi,
>>>
>>> I'd like construct matrix algebra as an algebra over a field. How would
>>> you approach this?
>> multiplication = composition of morphisms ?
>
> Yes, but what confuses me is that there are all linear transformations
> from R^n from R^m, for all n, m in N, where N is natural numbers. And
> (R^n -> R^m) o (R^m -> R^p) gives (R^n -> R^p) etc. I would need some
> common object to represent them all to keep composition closed.
> R^{N x N} comes to mind.

Agreed on the problem.

I do no longer have Bourbaki at hand, but usually one only
takes the endomorphisms (not sure whether embedding in R^N
would give problems in functoriality).

From: Stephen Montgomery-Smith on 20 Jun 2010 15:12
Kaba wrote:
> Axel Vogt wrote:
>> Kaba wrote:
>>> Hi,
>>>
>>> I'd like construct matrix algebra as an algebra over a field. How would
>>> you approach this?
>>
>> multiplication = composition of morphisms ?
>
> Yes, but what confuses me is that there are all linear transformations
> from R^n from R^m, for all n, m in N, where N is natural numbers. And
> (R^n -> R^m) o (R^m -> R^p) gives (R^n -> R^p) etc. I would need some
> common object to represent them all to keep composition closed.
> R^{N x N} comes to mind.
>

How about using finite rank bounded linear operators on a Hilbert space
as the common object?
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