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According to Hermann Grassmann and the creation of linear algebra by Desmond Sander, Grassmann was able to identify all the important notions in linear algebra in his book "Ausdehnungslehre". For him, vector spaces are freely generated by "units" $e_1, e_2, \cdots$ over real or complex numbers.

Then he embarked on studying something he called "linear products" by setting $$\left(\sum\alpha_ie_i\right)\left(\sum\beta_ie_j\right)=\sum(\alpha_i\beta_j)e_ie_j$$ subject to the condition $\color{red}{\sum\xi_{ij}e_ie_j=0.}$ According to the paper, to be invariant under change of bases, only possibilities are $$\color{blue}{e_ie_j=0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, e_ie_j=e_je_i,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, e_ie_j=-e_je_i.}$$

  1. What are his motivations for the relation "$\sum\xi_{ij}e_ie_j=0$"?
  2. What was his derivation for those commutative relations?

Since I donot read German, I wasn't able to his original book nor I would be able to understand what he has written there.

Bumblebee
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  • Please confirm that the publication you are interested in is reference [11] in Sander's papers (it appears in the section 4. Products). If so, the publication is in French, not German: Hermann Grassmann, "Sur les différents genres de multiplication." Journal für die reine und angewandte Mathematik, Vol. 49, No. 2, 1855, pp. 123-141 (scan) – njuffa Aug 27 '23 at 09:42
  • Grassman does not use the same representation used by Sander. The closest I can spot is at the bottom of p. 128: $S(a_{r,s} u_{r}u_{s}) = 0$. – njuffa Aug 27 '23 at 09:55
  • Understanding Grassmann's train of thought will require working through about five pages of French text, way too much for my very limited French language skills. From perusing bits and pieces, $S(a_{r,s} u_r us_s) = 0$ seems to be just a convenient canonical representation, as he notes notes earlier that a linear combination (my terminology, not his) $A= \beta B + \gamma C + \ldots$ can be "canonicalized" to $\alpha A + \beta B + \gamma C + \ldots = 0$. Hopefully someone fluent in French can help us out; I have not been able to locate an English translation of this publication. – njuffa Aug 27 '23 at 10:46

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