Thanks for the response.

I now understand the terms “vector” and “co-vectors” as relative and not absolute. Thanks.

I think what bothered me most, and still bothers me a bit, is how implicit it is in the document that an element of the algebra can represent a geometric figure. Because to me, no matter how I look at it, a geometric figure is always a set of elements (called points), not a single one.

Take for instance a line. In \mathbf{P}(\mathbb{R}^*_{2,0,1}), it is represented by a vector \mathbf{a} = a^i\mathbf{e_i}. But as I understand it, this means that such vector, which is actually a linear functional since we’re in dual space, is used to define first the hyperplane \mathcal{P}(\mathbf{a}) = \{\mathbf{x}\in\mathbb{R}_{2,0,1}|\langle\mathbf{a},\mathbf{x}\rangle = 0\} , and then, because we’re in projective space, the line represented by \mathbf{a} is the intersection of \mathcal{P}(\mathbf{a}) with the plane \{x\in\mathbb{R}_{2,0,1}|\langle\mathbf{e_0},\mathbf{x}\rangle =1\}.

I’m only assuming though, but I need something like that if I ever want to understand for instance how the outer product of two lines, \mathbf{a}\wedge\mathbf{b}, can represent their intersection.

Regarding the two different meanings of \langle\mathbf{a},\mathbf{b}\rangle, I think it’s unfortunate. IMHO it would be less confusing if this notation was reserved to the applied mapping of the linear functional \mathbf{a} on the vector \mathbf{b}, while the inner product would always be noted \mathbf{a}\cdot\mathbf{b} where \mathbf{a} and \mathbf{b} are two vectors.

Finally, I’d like to point out that raised indices are used at some point, but only once in the document (in the caption of Figure 6), and with no explanation.