Geometric algebra has a definition that is more specific than any use of geometry to teach Elementary Algebra on real numbers.

https://en.m.wikipedia.org/wiki/Geometric_algebra

https://en.m.wikipedia.org/wiki/Clifford_algebra

For teaching geometric algebra, there are curricula and textbooks available...

http://geometry.mrao.cam.ac.uk/home/introduction-to-ga/

https://arxiv.org/abs/1205.5935v1

http://www.faculty.luther.edu/~macdonal/GA&GC.pdf

http://assets.cambridge.org/052148/0221/sample/0521480221WS....

http://geometricalgebra.org/downloads/ga4cs_toc.pdf

https://www.amazon.com/dp/0792355148/

My personal recommendation for a book on geometric algebra is the one by Hestenes, New Foundations for Classical Mechanics (https://www.amazon.com/dp/0792355148/). I was disappointed by Geometric Algebra for Computer Science and I recently got rid of my copy when I moved to a new apartment, but I have a mathematics background and tend to prefer denser books.

I would say that "alternative" is a viable word here. Yes, you'll need a foundation in linear algebra to understand geometric algebra, but our classes and books on linear algebra go beyond what is necessary for understanding geometric algebra and introduce concepts (like the cross product) which have more natural equivalents in geometric algebra. I'm not even convinced that it's necessary to have a good understanding of matrixes in order to work with geometric algebra.