Camped outside the walls of a maze that surrounded a golden city are the gatekeepers. Their self appointed job is to give seekers an experience of the maze. Like tourist information centres they attempt to guide visitors to the city to the important places, and to cream off as much cudos as they can. Most visitors never never make it to the city, finding the gatekeepers a trip in itself.
The golden city stands aloof, full of untouched treasures while tourists go away happy with trinkets
The work of Grassman Hamiilton and Clifford(GHC) is obscured by detail and innovation, and in many cases not accessible without hard intensive cogitation. Peano was the most famous to rework the material in italian. Sartre was a beligerent side issue. Maxwell found the algebra he was looking for and wrote his Equations in these terms. meanwhile less great minds found interesting tidbits and made innovative variations. The introduction of Nabla gradually helped. The products in Grassmann found uses and counterparts. The dot product was defined, the cross product the determinant. These were all defined and utilised in and around general mathematical exploration.http://darkwing.uoregon.edu/~vitulli/441.sp04/LinAlgHistory.html
Gradually this notation became accepted and mathematicians were surprised to find that these definitions were implied in the work of GHC.
Many scientists were torn between Hamilton's active campaigning for quaternions and Gibbs and Heavside promotion of Gibb's version of vectors, and had no knowledge of Grassmann's contribution and foundational work. Others like Clifford Pickover rejoiced in the inspiring work of Hamilton and derived many vector operations from it.
The American scientists eventually turned their back on Hamilton and effectively killed its academic credibility in higher education.
Slowly the research of mathematicians developed group theory and algebraic groups became topics of interest. Linear algebra afforded the researchers with untrammeled ground, and matrices were undeveloped as a tool. Then a particular phd student proved that certain matrices had a group structure and the topic took off. Matrices were shown to be an algebra.
For matrix algebra to fruitfully develop one needed both proper notation and the proper definition of matrix multiplication. Both needs were met at about the same time and in the same place. In 1848 in England, J.J. Sylvester first introduced the term ''matrix,'' which was the Latin word for womb, as a name for an array of numbers. Matrix algebra was nurtured by the work of Arthur Cayley in 1855. Cayley studied compositions of linear transformations and was led to define matrix multiplication so that the matrix of coefficients for the composite transformation ST is the product of the matrix for S times the matrix for T. He went on to study the algebra of these compositions including matrix inverses. The famous Cayley-Hamilton theorem which asserts that a square matrix is a root of its characteristic polynomial was given by Cayley in his 1858 Memoir on the Theory of Matrices. The use of a single letter A to represent a matrix was crucial to the development of matrix algebra. Early in the development the formula det(AB) = det(A)det(B) provided a connection between matrix algebra and determinants. Cayley wrote ''There would be many things to say about this theory of matrices which should, it seems to me, precede the theory of determinants.''
Cantor and Boyle defined the rules of combination of sets and predicate statements. This lead to the development of modern set theory, honed into an analytical tool by A N Whitehead and his student Bertrand Russel. Thus subspaces were well understood by the time mathematicians revisited Grassmann. Grassmann in particular, but also Hamilton had intimated set theoretic notation underlay there thinking. Both intimated large sets would be possible for referencing space. Grassmann particularly mentions subspaces.
Matrices have been shown to be an algebra, Grassmann vector theory has been shown to be a vector algebra. Thus GHC vector algebras are shown to be constructed on sub algebras, and subspaces, and this is an indication of the fractal nature of space. We would expect the same algebraic structures at all scales and involved in all ways with a theory that apprehends space.. This goes deepre. Grassmann started with units that are real "numbers", but in fact "numbers" are based on trigonometric ratios which themselves are fractally distributed.
One of the simple things about Grassmann unit vectors is that they are the basis vectors of a reference frame. Vectors are not over any set of numbers they are over a set of basis vectors. These vectors are extensions measurements actions of sighting, lining up,linking, relatings. They are independent therefore non parallel and they are arvitrary but related to an observers subjective gnomon.
In a reference frame system a system of reference frames the unit vectors are disposed according to my subjective gnomon. Thus one of the unit vectors relates to me. The other two unit vectors in 3d space relate to the local gnomon of the object or person. Another unit vector can be placed in the object to give a full three unit description but the unit vector relating to me remains, so the system can have three arbitrary fixed unit vectors and one that rotates relative to me. : this is a quaternion description