^^Foundations of Linear
Algebra.
- first systems in which geometrical entities (points) were operated on
directly.
- Möbius’ Barycentric Calculus (1827)
- Giusto Bellavitis’ (1803-1880) “Calculus of Equipollences”.
- Möbius showed how to add collinear line-segments, but gave no addition
rules for non-collinear segments and no multiplication.
- Peano’s axiomatic definition of a vector space (1888) remained
largely unknown until Hermann Weyl stated it in the context of Relativity
Theory in 1918 (referring only to Graßmann’s “epoch making work”).
- In the late 1920s axiomatization finally unified the field,
which nowadays holds a central position in Mathematics.
- Many of Graßmann’s Linear Algebra results were reestablished
independently by others and now are associated with them (e.g. Steinitz’
Exchange Thm.).
- Graßmann’s first major work: The Ausdehnungslehre (Theory of Extension)
of 1844.
- “I had also realized that there must be a branch of mathematics which
yields in a purely abstract way laws similar to those that in geometry seem
bound to space.
I soon realized that I had come upon the domain of a new science of which
geometry itself is only a special application.”
- “Every displacement of a system of m-th order can be represented as a
sum of m displacements belonging to m given independent methods of evolution
of the system, the sum being unique for each such set.”