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\subsection{The contributions of Grassmann}
The contribution of Grassmann to the user psychology of linear algebra are not well understood, even today. In
this section, we survey the contributions of Grassmann to our terminology.
It's well known that Grassmann introduced the notions of inner and outer products; less known perhaps is the history of the term ``outer product''.
TODO put text about the two kinds of outer product here
Grassmann's
Aus1862 he \textit{Ausdehnungslehre} of 1862\cite[Ch. 2]{Grassmann1862} introduces describes general structures to form products of two or more vectors. In modern language,
Chapter 2 begins by saying that if
you express two vectors
are expressed in some basis
\begin{aligned}
u & = \sum_i c_i e_i \\
v & = \sum_i d_i e_i \\
\end{aligned}
and you have some then any vector product operation $\circ$
you want to define, then it suffices to describe is fully defined if the result of the operation $\circ$ on the basis vectors
$e_i$, $e_i$ is known, since
\[
u \circ v = \sum_{ij} c_i d_i (e_i \circ e_j)
= \sum_{ij} c_i d_i G_{ij}
\]
If $e_i \circ e_j = \delta_{ij}$, then one recovers where we introduce formally $G_{ij}$, the Gramian of the
ordinary vector space with basis ${e_i}$ and inner product
(Grassmann calls it that), i.e. $\circ = \cdot$.
If $e_i \circ e_j = - e_j \circ e_i$, then one recovers Grassmann's exterior product (which Grassmann calls the combinatorial product). Today, we would write $\circ = \wedge$ for this product. $\circ$.
Grassmann
defined the inner product to be the product whose Gramian is the identity, and the outer product by placing new basis vectors in each uniquely defined entry of an antisymmetric $G$. Interestingly, Grassmann does not give the general
product $\circ$ case a
special name, instead referring to it informally in the text defining his notation:
,,Ein Produkt, in welchern die Faktoren $a, b, \cdots$ irgend wie enthalten find, werde ich[...] mit $P_{a,b,...}$ bezeichnen`` \cite[p. 24, \S 43]{Grassmann1862}
``A product in which the factors a, b, ... are included in any way I will[...] denote by $P_{a,b,...}$''~\cite[p. 22, \S 43]{Grassmann2000}
Later in his discussion Gibbs, however, recognized the value of
symmetric tensors, Grassmann refers the general case, bequeathing it the name ``indeterminate product''.~\footnote{Some of the English literature incorrectly attribute the name to
``ein Grassmann; however, it is quite clear from \textit{Multiple Algebra} that the name is his invention. The closest phrase used by Grassmann is ,,ein beliebiges
Produkt'' Produkt $P_{a_1, a_2, ...,
a_n}$ a_n}$`` (``an arbitrary product''~\cite[p. 196, \S 353]{Grassmann2000}), which
Gibbs is not defined in
his Multiple Algebra (collected works, Vol 2, p 109) picked up on a formal, technical sense.} If we interpret this as
being of significance using an unsymmetric Gramian $G$ and
bequeathed the name ``indeterminate product'', being the most general product from which all placing a new unique basis vector in each entry of $G$, then we have the
other products Grassmann discusses can be derived. basic ingredients of a tensor product.
Grassmann's other contribution to this topic is the concept of ,,offne Produkt`` or ``open product''~\footnote{Somewhat confusingly, Grassmann's 1862 book defines open products, or ``product[s] with n \{interchangable\} openings''~\cite[\S\S 353, p. 196]{Grassmann2000}, in a way which we would recognize today as symmetric tensors of rank $n$.}, which he writes in Ausdehnungslehre 1844, Sec 172, p 267 (English pp 271-2) with the notation $[A() . B]$, acting on a vector $P$ by
