Browne, Grassmann Algebra

I'm going through Browne's online book now. I think it's so far a
great reference, with lots of fantastic visual examples and
everything, but there are a few things that strike me as odd. For
instance, he talks about how "Grassmann algebra" (is this different
from exterior algebra?) allows us to distinguish between vectors and
points. I took this to mean that in some meaningful sense, Grassmann
algebra allows one to work with affine spaces and not just vector
spaces. But then, he defines the sum of two points as the point midway
between them, which is something I've never heard of before when
reading about affine spaces. He keeps using the term "linear space"
too, which I assume means affine space, since he's talking about
points...?

He also talks about something called the regressive product which I
thought was the same as the interior product, but apparently slightly
different.

Looks like a wonderful text, but is the terminology that's being used
in here standard? I'd like to know what the correct terminology for
these things is.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Looks like a wonderful text, but is the terminology that's being used
> in here standard? I'd like to know what the correct terminology for
> these things is.

I think it would be nice to get a better reference on the topic.

On Wed, Apr 4, 2012 at 7:38 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
>
> > Looks like a wonderful text, but is the terminology that's being used
> > in here standard? I'd like to know what the correct terminology for
> > these things is.
>
> I think it would be nice to get a better reference on the topic.

https://www.google.com/search?sourceid=chrome&ie=UTF-8&q=a+better+reference+on+exterior+algebra+than+browne

Well, I'm all out of ideas. Anything you could recommend?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> https://www.google.com/search?sourceid=chrome&ie=UTF-8&q=a+better+reference+on+exterior+algebra+than+browne
>
> Well, I'm all out of ideas. Anything you could recommend?

I put this in the Files section. What do you think of it?

http://f1.grp.yahoofs.com/v1/gOB8T5NXPXqlaoIQpd4-HJ_lOq67xxZpuT3phcO_lhbHAvczwMfgr1_BnwQG9rpH9975Kkmtr0WKjOmWy6HZ/books/Linear%20Algebra%20via%20Exterior%20Products.pdf

http://tinyurl.com/73sk4hb

On Wed, Apr 4, 2012 at 8:09 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
>
> >
> > https://www.google.com/search?sourceid=chrome&ie=UTF-8&q=a+better+reference+on+exterior+algebra+than+browne
> >
> > Well, I'm all out of ideas. Anything you could recommend?
>
> I put this in the Files section. What do you think of it?
>
>
> http://f1.grp.yahoofs.com/v1/gOB8T5NXPXqlaoIQpd4-HJ_lOq67xxZpuT3phcO_lhbHAvczwMfgr1_BnwQG9rpH9975Kkmtr0WKjOmWy6HZ/books/Linear%20Algebra%20via%20Exterior%20Products.pdf
>
> http://tinyurl.com/73sk4hb

Hey thanks, very nice! I'm about halfway through the first chapter and
will keep working my way through it. I think the Browne book was a bit
more intuitive in a few ways, but I'd just like to know that I'm using
the correct terms for things at this point.

I have one question which keeps coming up when I read these things.
What's the difference between "exterior algebra," "Clifford algebra,"
"geometric algebra," and "Grassmann algebra?"

Am I correct that exterior algebra is a specific type of Clifford
algebra, that geometric algebra is exterior algebra but equipped with
two additional products called the dot product and the geometric
product, and that Grassmann algebra is Browne's name for exterior
algebra?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Am I correct that exterior algebra is a specific type of Clifford
> algebra, that geometric algebra is exterior algebra but equipped with
> two additional products called the dot product and the geometric
> product, and that Grassmann algebra is Browne's name for exterior
> algebra?

Not really. Exterior algebra takes products of vectors to 2-vectors and so forth. You have a graded algebra, and you can't add vectors to 2-vectors or anything of that sort. Clifford algebras for trivial quadratic form are related, but they dump everything into a vector space of dimension 2^n instead of grading 1+n+n choose 2 + .... + 1. We are better off avoiding Clifford algebras and geometric algebra, which is a particular kind of Clifford algebra. Grassmann algebra is more or less synonymous with exterior algebra.