Special Lecture - The Octionions

Forming real vector spaces, we have the linear maps, which preserve the vector space structure L(n; R) = { M : R → R ∣∣M(a~x+ b~y) = aM(~x) + bM(~y) } (1) This is a monoid, not a group, as inverses may not exist. The first group is the general linear group, which preserve dimensionality GL(n; R) = { M ∈ L(n) ∣∣M is invertible } . (2) Further, we have the area preserving maps SL(n; R) = { M ∈ L(n) ∣∣ det(M) = 1 } . (3)