Hello everyone, 


I wanted to lecture on Grassmann and his works , and I have been reading the collected works of Grassmann " Die Lineale Ausdehnungslehre ". There Grassmann introduced something called " Interior product " ( Left and Right interior products ) . So I was completely stuck up there, the Bourbaki papers don't speak on the reason or urge in creating such manipulations, but its understood by someone who really mastered them. 


Can anyone suggest me a good definition of the left and right interior products and explain the purpose of introducing them along with the intuition ? 


I would be really honored to hear that. 


Thank you.



Urge/reason for inventing  interior product ( Grassmann algebra )
Here is an elementary motivation for interior products . 


Suppose $V$ is an $n$-dimensional vector space .  

To every non-zero vector $ v\in V$ you can associate the complex $$ 0\to V\to...\to \Lambda ^kV\to \Lambda ^{k+1}V\to...\to \Lambda ^nV\to 0 \\quad (\star)$$ 
where the linear map $ext\_k(v):\Lambda ^kV\to \Lambda ^{k+1}V$ is the map $\omega \mapsto v\wedge \omega$.  

But how do you prove that it is exact? Answer: with interior products! 


Choose a linear form $f\in V^\*$ such that $f(v)=1$ and introduce the interior product map $int\_k(f):\Lambda ^kV\to \Lambda ^{k-1}V$ which on a decomposable vector reduces to $$int\_k(f)(v\_1\wedge...\wedge v\_k)=\sum\_{j=1}^k (-1)^{j+1}f(v\_j)v\_1\wedge...\wedge \widehat {v\_j}\wedge v\_k$$ 
It is then easy to show the relation $$int\_{k+1}(f)\circ ext\_k(v)+ext\_{k-1}(v)\circ int\_k(f)=Id\_k: \Lambda ^k(V)\xrightarrow {=} \Lambda ^k(V) $$ 
which immediately implies that the complex $(\star)$ is exact at the $k$-th slot since $$ext\_k(v)(\omega)=0\implies \omega = ext\_{k-1}(v)[int\_{k-1}(f)(\omega)] $$
 Differential geometers make essentially the same calculation in the context of the De Rham complex of differential forms: see Exercise 4 of Chapter 7 (Volume I) in Spivak's wonderful [*A Comprehensive Introduction to Differential Geometry*](http://www.mathpop.com/bookhtms/dg1.htm)


**NB** The above is the toy version of the theory of the Koszul complex.  

It can always be souped up to any desired degree of unintelligibility (in my case not much souping up is necessary) .


