While this is usually specified in terms of vectors nd vector spaces, I find these terms as alienating. Most of us will struggle with the concept of a reference frame, and with the word relativity. But in fact our natural reference frames are the perception we have that everything in our conscious experience, that is my conscious experience is relative to mr..
Any process or scheme I set up to measure in my reference frame is automatically or tautologically relative to me.
Algebraic notation sets up terminology to remind me hw I do things. It is this algebraic system of notation that in fact liberties me to adopt any other reference fram within , that is encoded within my natural reference frame. I do this by transforming the basis of my natural reference frame.
The basis or spanning set of " lines " is a fundamental nation in a natural reference frame. I tend to use Left, Right, Up, Down! Forward, Backward. .
My actual reference frame includes rotations but that is a quaternionic reference frame or a general Grassmann Algebra.
This is Norman brilliantly expounding these concepts.
Hyperbolic geometry, Wildlinearalg, all are good series of Norman to get a handle on this. This is the best investigation of Grassmann algebras I have found .
These considerations have revealed powerful shortcuts to computation and calculation , in practice. Joins and meets are introduced as projective " operations" . I prefer the concept of combinatorial process.
Taking these algebraic terms and methods Norman is able to compute simple results that are essential for analysis: where is a point, which direction is a line and where does it start.
Norman is going on to explore surfaces next.
These equations are algorithms that deliver a point object, a line object and soon a surface object. These are computer coding terms from object oriented programming. The power here is that we can programme these into a computer and automatically generate and display these solutions!
We can then look at how we can model surfaces and dynamic behaviours in our experiential continuum.
At last I have found an article which intuitively understands Grassmann's method and analysis. It is a bit technical( well quite a lot actually) but if you ignore the math speak you can get the drift of where Grassmann has been misunderstood with regard to his products.
This is my reply
This is the first article that I have found since beginning my research on Hermann that actually grasps his method and approach!
I have had to read in the german editions to pick this up, and I have had to divest myself of many subtly incongruent notions.
Grassmann also defines his inner product differently! The projection of the segmented lines( begränzte Linie or Strecke) is wertically onto each other in the parallelotopes . Referring directly to Grassmann's 1844 Vorrede, he defines the inner product as the product of these projected Strecken. In that case the area is given by abcos^2@ , not abcos@!
He actually indicates that he would use this product in a division or ratio with the exterior product, which is defined in the same paragraph!
He follows this with a derivation of Euler-Cotes identity from the hyperbolic trigonometric functions precisely by defining a division or ratio ( a rational form of Strecken, at the very most) which relate directly to his work on The Ebb and Flow of Tides.
The whole point of this last section was to show how far he had got on his own and to appeal for help in progressing his method, producing such amazing results!
His intention was therefore to use the ratio of the inner product with the outer product as some evaluation method related to trigonometric tables! I am not clear on the precise meaning of this crucial passage but it seemed to me that he defined the inner product as this ratio, not just the parallelogram that is projected internally by vertical projection of the Strecken..
If you want to know more google "jehovajah Grassmann" or email me on firstname.lastname@example.org