What I want is a text that starts with a blank slate, ignoring the past and building everything from scratch. If GA is superior then the result will be better, right? So forget about vectors, forget about complex numbers, forget quaternions. Show me your stuff direct. No one else is going to do that so if I want this I'm just going to have to do it myself.

The very terminology of GA is rooted in the past. Bivectors, trivectors. That's not what it's really about. Forget about vectors. Forget about the basis of the space, that just complicates things. The essence of geometric algebra is subspaces. Lines, planes, and volumes. Infinite closed subspaces. That's the place to start. You are better off not even choosing a space to work in. Don't fix a N dimensions for your space. Instead think of everything as a subspace. Instead of a vector you have a 1-space, a line. Instead of a bivector you have a 2-space that is a plane, and so forth. So what happens when you multiply two spaces? Not one to stand on the sidelines and throw stones I got a geometric algebra interpreter and let it give me the results. The geometric product of two subspaces is like the symmetric difference of those subspaces, or the exclusive or of logic. It can be explained as follows.

First an imprecise version.

------------

Let s

_{1}and s

_{2}be two subspaces.

Let S be the smallest space that contains both subspaces.

Let s be the intersection of the two subspaces.

The geometric product of the two subspaces, s

_{1}*s

_{2}, is the largest subspace of S that excludes s.

Next in the more exact jargon of vector algebra

-----------

Let S be the span of s

_{1}and s

_{2}. S = span(s

_{1},s

_{2})

Let s be the largest subspace that is a subspace of both s

_{1}and s

_{2}.

s

_{1}*s

_{2}is the dual of s in S.

OK, so then what use is the scalar attached to each subspace? Subspaces are infinite, so it can't be a size. There are two uses. The first is as a scaling factor when the subspace is used as an operator. For example, a 2-space is often used to rotate other subspaces. The scalar tells you how much to rotate whatever. The second use of the scalar is to specify a specific point in a 1-space. But quite often you don't care about a specific point, you just want the 1-space. So instead of a vector space it is better to think of GA as a space space. It's a ring, so one may call it a space ring. Each space has an inverse, one of the main motivations behind GA. But the sum of spaces usually doesn't have an inverse. So it is not a division ring. Some elements have inverses, others don't.

The scalars are also useful for specifying spaces by adding together basis spaces. But one of the advantages of GA is that you don't have to think about the basis. Just let a computer do it for you. That was a big gripe I had about the texts. They are preoccupied with basis vectors, heavily focused on them. That's shooting yourself in the foot, thinks I. I didn't get far until I got out of this frame of mind. The basis matters only when you are done and want to draw a figure on a 2D screen or otherwise relate to our real world. I think that's largely where the "dragging a boat anchor" feeling came from.

So in GA just what is a vector? There is a 1-space, a distance, and a sign. The 1-space is a line through the origin. The distance specifies an n-sphere centered at the origin with radius of that distance. The intersection of the n-sphere with the line is two points. The sign tells you which point to choose. That’s a vector.

A very nice feature of GA is that you don't need to worry about signs. There is no right hand rule or anything like that. Signs take care of themselves.

Now a 1-space together with a scalar, isn't that a two dimensional space? Indeed it is. But we are stuck with this.

One author mentioned that the scalars could be complex numbers. Why introduce that when complex numbers are a subset of GA? That inspired some silly speculation on my part. The ostensible scalars could be another full fledged GA. That "scalar" GA could then be have yet another GA as its scalars and so forth ad infinitum. As they say, "it's turtles all the way down."