Grassmann's story is semi-tragic. He was a school teacher who created a powerful algebra of space where you could compute with not just points and vectors but with lines, planes, surfaces and other vector subspaces without being hampered by a particular coordinate system. Geometric Algebra can subsume complex numbers, quaternions and plane geometry - projections, rotations etc and easily extend them to higher surfaces. Matrices play less a role here. Hestens also puts forward that Geometric Calculus can contain matrix calculus, linear algebra, lie algebra and groups and Differential forms, see [1]. But because Grassmann was so ahead of his time and did himself no favours by writing a dense, super long text with lots of philosophy interwoven not many bothered to read or could understand it. Later, Clifford came and made extensions, cleaned it up and the subject gets named after him as Clifford Algebras, though not by his choice. But Clifford died early so the more geometrically unwieldy Vector Analysis won out. A setback for physics?...Angular momentum makes so much more sense as a bivector. From [2]:
Fearnley-Sander writes in [27]:-
All mathematicians stand, as Newton said he did, on the shoulders of giants, but few have come closer than Hermann Grassmann to creating, single-handedly, a new subject.
To read more about Grassmann I recommend: A History of Vector Anaylsis by Michael Crowe, well written, I couldn't put it down.
[1] http://geocalc.clas.asu.edu/pdf/DIF_FORM.pdf
[2] http://www.gap-system.org/~history/Biographies/Grassmann.htm...