Coordinate free Geometric Algebra vs. Linear Algebra

I think I know what coordinate free means. But I never found in ANY text a good explanation of it or something like: This is the problem solved with coordinates and this is the problem solved without coordinates, etc. Since the philosophy of GA is that everything should be coordinate free, I would like to see an example of something that can be done in GA without coordinates but you have to use coordinates with usual linear algebra. To be specific, in GA you can make something like this in, let's say $\mathbb{G_3}$:

$c=(bab)I$

The vector $a$ is rotated about the vector $b$, and then you take the dual plane of it. Is there an efficient way to do this without coordinates in linear algebra? This is just an example that I made up spontaneously; maybe there are better examples.

• taking a step back, do you know of some coordinate-free Linear Algebra ideas? Apr 11, 2016 at 15:13
• I mean, since the clifford algebra can be described as a quotient of the tensor algebra, any products can be written in terms of the tensor product, so I'm not sure what you're asking for exists. Apr 12, 2016 at 3:12

In linear algebra and geometric algebra both, you can talk about "transformations," "bilinear forms" and "vector spaces" all without referring to coordinates. They are all abstract ideas.

It is only when you begin to identify $V=F^n$ for some $n$ and field $F$, transformations with matrices, bilinear forms with Gram matrices, etc., that you start to get coordinates.

So, one example of a theorem of linear algebra that you could solve without coordinates is this:

If $A^2=A$, then $V=Im A\oplus Ker A$.

You need never refer to bases and coordinates, you only need to use the abstract properties of $V$ and $A$.

Another one is

Given a unit vector $v$, the transformation $x\mapsto x-2v(v^\ast x)$ is a reflection in the plane normal to $v$.

In fact, in the example you gave, you said everything entirely in terms of abstract vectors and geometric notions, never referring once to coordinates. Whether or not you prove what you are doing works using coordinates is another matter.

It is nice to be able to prove things in a coordinate-free way sometimes, but it is not always desirable. For one thing, the coordinate definition of "determinant" is much less complicated than the coordinate free version.