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.

If you enjoy playing around with quaternions, you may also have fun with geometric algebra.

Geometric Algebra for Physicists is very clear and gently paced. http://www.amazon.com/Geometric-Algebra-Physicists-Chris-Dor...

This one also looks good: http://www.amazon.com/Linear-Geometric-Algebra-Alan-Macdonal...