Geometric Algebra

Geometric Algebra (GA) is a mathematical framework for manipulating geometrical entities. It is similar to linear algebra, but more unified and less quirky.

In Linear Algebra, we manipulate vectors and matrices in all sorts of ways to solve equations, perform transformations (like translations, rotations), and even distinguish dogs from cats. It’s a very powerful framework, but it does have some quirks, like how the cross product is only well-defined in 3 and 7 dimensions, or how vectors are routinely used to represent high-dimensional things even though vectors are really only just one dimensional.

In Geometric Algebra, there exist multidimensional vectors, and vector products work for arbitrary dimensions. Additionally, algebraic operations in GA directly correspond with geometric operations, like rotations, reflections, and intersections. I don’t want to spoil too much at this point, so that’s all I will say for now.

Reading

After reading a tutorial section, I encourage you to also read specific sections from the “Geometric Algebra Primer”. The accompanied reading for this notebook is “Chapter 1: Introduction”.

What’s Next?

Next, we discuss subspaces.