Hello all,
Perhaps you can field this question, Pat... one of the reasons I'm asking it here, rather than the Math Message Board, is because no one on the Math Message Board seems much interested, nor knowledgable, about Clifford, or Geometric Algebra.
I have a couple of David Hestenes books... "New Foundations...", and "Clifford Algebra to...", and a few other books about Clifford Algebra... and I've also seen sources on the Internet.
Perhaps it's a matter of understanding my background better. I have a BS in Computer Science. I wasn't a Math major, although many said I should have been. So, much of my mathematical education is self-taught,... and, for instance, I might know something traditionally taught in grad school, but not know something that every undergrad math major would know...
Basically, I'm hoping someone can show me how to form inner and outer products for arbitrary multi-vectors. Unfortunately, I don't seem to be able to take even David Hestenes explanations and apply them. I guess I need someone to show me specific examples with aribitrary multi-vectors.
Also... I know that some of the rules... like concerning commutivity (I believe?) with the wedge product... with bivectors have specifically spelt out rules... but, what happens if one's computing with trivectors? In other words, generalization,... in some easy to understand manner... is what I need.
If someone could show me how to do all the calculations... by hand. That's what I'm looking for. Specifically, by hand... such that I can apply it to arbitrary inner and outer products of multi-vectors.
Also... recently on the Math Message Board, http://www.math2.org/cgi-bin/mmb/server?action=read&msg=31392, Mark Tiefenbruck was helping me with understanding more about parametric equations. If you've read paragraph 3 above, perhaps it's easier to understand how I might have questions like this... along with questions about multi-vectors.
Anyway... The form of representing a nD-space as Mark pointed out seems like it could be very useful... except that the number of parameter variables might become quite large,... and then, trying to solve such systems of equations might be difficult. However, simply determining whether a particular point satisfies the equations still shouldn't be too hard.
What I'm wondering is here is... the multi-vector represents an arbitrarily shaped (seems to depend on who you read...) that doesn't seem necessarily to have any fixed position in nD-space. (If I understand correctly?) So, for instance, the wedge product of multi-vectors is the nD-hypervolume of an arbitrarily shaped nD-space... correct? But, a 1-vector determination of it is similar to the equation Mark provided... correct? Except Mark's formula would define the shape and the placement of the nD-space... correct?
Perhaps related to the preceding paragraph is a much more general issue... Sometimes I get a bit confused with vectors whether or not we're assuming that the tail of a vector is at the origin... I believe also know as a position vector. Again, maybe this is something a bit off... perhaps because I wasn't a Math major... but, can someone explain more about this? Is it a matter of loose convention, or are there hard-and-fast rules governing when a vector is assumed to have it's tail at the origin?
Also... the spinor is a different interpretation of the bivector... correct? Or, can spinors be of any dimension? That is, can a spinor be another interpretation of a trivector? As best as I can understand, a spinor is called a rotation operation... so, it's just considered to rotate geometric objects... is this correct?