I'd say alternative is an unlucky choice of words. I'd rather say geometric algebra (GA) is an extension of linear algebra (LA). In order to really understand GA you need first to firmly understand LA. Then it becomes clear that all that GA does is to turn a Hilbert space into an algebra called a Clifford algebra, and to examine the geometric semantics of the various operations that pop up in the process.

Here are three great sources that helped me to understand GA:

1. https://www.amazon.co.uk/Geometric-Algebra-Computer-Science-...

2. https://www.amazon.co.uk/Linear-Geometric-Algebra-Alan-Macdo...

3. https://www.amazon.co.uk/Algebra-Graduate-Texts-Mathematics-... , pages 749-752

The first source gives great motivation and intuition for GA and its various products. Its mostly coordinate free approach is very refreshing and makes the subject feel exciting and magical. This is also the problem of the book, it's easy to end up confused and disoriented after working through it for a while. The second source is great because it grounds GA firmly on LA, and makes everything very clear and precise. The third source gives a short and concise definition of what a Clifford algebra is.

I found Geometric Algebra for Computer Science (Revised Edition): An Object-Oriented Approach to Geometry to be eye-opening.

https://www.amazon.com/Geometric-Algebra-Computer-Science-Re...

I've had good luck with this one: http://www.amazon.com/Geometric-Algebra-Computer-Science-Rev... They definitely explain why the cross product is an ugly hack :)

The authors have a website - geometricalgebra.net - where you can download a program which will display all the diagrams from the book and allow you to manipulate them. You can also render arbitrary low-dimensional geometric algebra constructions which helps tremendously with improving your intuition about how things work. I'd say it's at a middle ground of mathematical sophistication - it's a good mix of proofs and practical usage. It's almost entirely dedicated to applying geometric algebra to computer graphics, so you won't get as much out of it if you're interested in applications to physics, or just pure mathematics.

If nothing else, I finally know what a quaternion really is after going through this book.

Geometric algebra: http://www.amazon.com/Geometric-Algebra-Computer-Science-Rev...

It is to linear algebra what Lisp is to assembly.