Ok, things are more complicated than I thought they would be.
While
, it is not true, that
. Thus, we are in the weird situation, where
acts as some kind of non-commutative unity. That's quite like if
.
However, I can't actually find references in Garret Sobczyk's works, where he actually uses the nilpotents
in his GA basis. They don't appear to always occur or, more accurately, they often appear to just be
. So perhaps this actually is an instance of that.
In particular, if I require
, that requires me to set
, so
.
In that case I actually only have to deal with the idempotents
and the entire product-table becomes:
Interestingly, while
and
, it still is true that
.
This is an important property. It, for instance, means, that
(for some arbitrary Multivector
)
and through applying this idea to a Taylor series, that also means that
. This is what makes this basis so powerful. - Assuming you can easily find the value of a given function at
, you can easily calculate any function at some general M.
That's all assuming that I got that right and the
s are indeed 0 for this problem. If they actually are not, that means a general function would somehow use them.
A bit more explanation:
Since idempotents have the following propery:
, they act as projectors. i.e. if you multiply something with them muliple times, that thing will no longer change.
- that's what I used above when I said
. The very first product changes things around, but every further product doesn't change a thing.
Similarly, since
implies
and something of the form
will produce some form that has every combination
, we get, for all those terms, except for when
or
, simply the replacement term
. This term, however, by the properties of our special idempotents, is
. Thus only the two terms with the highest powers remain. Those two leftover terms reduce to
and
respectively.
If you put all this into a taylor series, you'll get the above result, that any analytic function can be expressed by
.
Lastly, there is one more weird thing about these idempotents:
would be the "reflection" of
in
. I put that in "quotation marks", because it only really would be a reflection if I used pure vectors, and I obviously am using vectors with a scalar part here.
This reflection completely knocks out the bivector part. Furthermore, no matter what side you multiply
onto that product, the result will no longer change:
So for this value, "reflection" and projection kind of coincide. I wonder about the exact geometric interpretation of this.