biVector forum

Beyond Linear combination

#1

Hi. I would like to ask some questions.

[About my background]
First of all I am new to GA. I am shocked with the powerfulness of GA/Clifford algebra.
My research has heavy geometric contents, I guess GA may be well-suited for that.
My research is about matrix factorizations, and especially the factorization of nonnegative matrices, which has a lot of geometry behind.

[The question]
From the GAME2020 videos (which are very good), I understand that, if x is inside the span(a_1,…,a_n), then it is equivalent to the wedge product x ^ (a_1 ^ … ^ a_n) = 0. But this is only saying "x is a linear combination of {a_i} ", what about affine combination? conical combination? convex combination?

Thank you very much

#2

The wedge product is only half of the story. It makes higher dimensional elements out of lower dimensional elements by combining what unique parts each element contributes… i.e. the subspace spanned by the difference between them.

The full geometric product combines this with a grade lowering component… the subspace the elements share, the amount they have in common. This can look like a dot product for vectors but generalizes in more complex ways with elements of higher grade.

In addition for an N dimensional space you can map each D dimensional element to a corresponding N-D dimensional element; this operation is most commonly known as the Hodge Dual. Taking the dual, taking a wedge, and taking the dual again allows you to turn what was effectively a ‘union’ type operation into an ‘intersection’ type operation.

So yeah, try not to get stuck on the ‘grassman algebra’ subset that just deals with wedge products… there’s a lot more to it!

1 Like
#3

Thank you very much for the reply !
I will look deep into that !