I love to see more discussion of geometric algebra, but I have a few disagreements with this introduction:

– Starting with standard linear algebra (vector spaces, matrices, etc.) leads to confusion. These ideas are best developed within the framework of geometric algebra, where their features can be easily proved / reasoned about, but only after the basic features of geometric algebra have been presented first. After a bit of exposition, the relationship to more commonly used mathematical formalisms can be explained. Same story for trigonometry, barycentric coordinates, complex numbers, quaternions, projective geometry, differential forms, Pauli and Dirac matrices, Lie theory, representation theory, lattice theory, etc.)

– The geometric product is the basic operation. Inner (“dot”) and outer (“wedge”) products of pairs of vectors are derived products. Deriving other subsidiary products from the basic geometric product for various niche purposes and learning to reason about their various features or discussing how you might use one or another to solve particular problems in various ways is a good exercise.

If a and b are vectors, then by definition a∧b = ½(ab – ba) = –b∧a. (We’ve intentionally cancelled out the commutative parts of the product, leaving us something purely anti-commutative.)

For two vectors, by definition a·b = ½(ab + ba) = b·a. (We’ve intentionally cancelled out the anti-commutative parts of the product, leaving something purely commutative.)

That the geometric product of two vectors can be thus split apart by grade (i.e. ab = a∧b + a·b) is a consequence derived from the basic axioms. It so happens that the purely anti-commuting part of the geometric product of two vectors has the dimension/orientation of the plane they lie in, whereas the purely commuting part has the dimension of a scalar. This just falls right out of the basic definition of the geometric product.

– The initial presentation should not focus on coordinates. Coordinates are an arbitrary choice, not inherent to the geometry.

Ideally lessons should start with a vector space (or affine space) where we assume we know how to add or multiply two vectors (probably the most useful way to start building intuition is to just show vectors as arrows in a picture rather than written down as some explicit expression), and then develop a full set of multivectors on top. Figuring out how to stick some arbitrary coordinate system into place for making concrete computations is then straightforward to understand. But if students get used to working with coordinates, they will generally have great difficulty ditching them to think more abstractly. David Hestenes calls this a “math virus”: http://geocalc.clas.asu.edu/pdf/MathViruses.pdf

If you start with an affine space (e.g. the Euclidean affine plane), you can pick an arbitrary origin point “zero”, and then treat the other points in your space as zero plus some vector displacement, and get to work with the displacement vectors in the derived vector space. You can then pick an arbitrary direction and magnitude to be your e₁ direction, and an arbitrary orientation of the plane (I) or an arbitrary orthogonal e₂ vector of the same magnitude. This lets you keep always in mind that your chosen origin and basis were arbitrary, and can be changed at will to suit the problem.

– This presentation doesn’t discuss much of any of the geometrical relationships between vectors, bivectors, lines, planes, etc., or how you might go about using geometric algebra to solve practical geometry (or Newtonian mechanics, or computer graphics, or whatever) problems.

The difficult part about geometric algebra is not the basic definitions, but rather learning to think with it fluently, and practicing combining its abstractions. I personally don’t feel like I have reached a level of real fluency, but only basic familiarity. Mainly because I habitually reach for more standard tools which I know better from using them for years of schoolwork.

The “sunset geometry” blog post which didn’t get any discussion here a week or two ago was IMO a better introduction. https://news.ycombinator.com/item?id=13162368 http://www.shapeoperator.com/2016/12/12/sunset-geometry/