^^Foundations of Linear Algebra.

Foundations of Linear Algebra  .pdf  (in The History of Numerical Linear Algebra)

  1. first systems in which geometrical entities (points) were operated on directly.
  2. Möbius showed how to add collinear line-segments, but gave no addition rules for non-collinear segments and no multiplication.
  3. 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”).
  4. In the late 1920s axiomatization finally unified the field, which nowadays holds a central position in Mathematics.
  5. Many of Graßmann’s Linear Algebra results were reestablished independently by others and now are associated with them (e.g. Steinitz’ Exchange Thm.).
  6. Graßmann’s first major work: The Ausdehnungslehre (Theory of Extension) of 1844.
    1. “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.”
    2. “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.”