The wedge product of $p$ vectors in vector space $V$ is called a $p$-vector and the vector space generated by all $p$-vectors is denoted $\bigwedge^p V$ with the basis $e_I:=e_{i1}\wedge\dots\wedge e_{ip}$ where the subscript $I$ has $p$ indices and indicates an ordered collection, and $\dim \bigwedge ^pV=\binom{n}{p}$ where $n$ is the dimension of a vector space $V$.
A general vector in $\bigwedge ^p V$ is written as $\omega$ where
\begin{equation} \omega =\sum _{I} \beta ^{I} e_I \end{equation} where $\beta^I\in \mathbb F$ are the components of $\omega$ in the basis $e_I$.
The collection of all the p-vector spaces $\bigwedge ^pV$ is called an exterior algebra denoted $\bigwedge V$.
Firstly, is my understanding of all this correct?
Secondly, what is the significance of ordering the indices by $I$, is it just to keep a note of all the vectors that should be kept together, a sort of housekeeping?
Thirdly is there an analogy of the relationship between $\bigwedge ^pV$ and $\bigwedge V$ in terms of linear algebra? I am trying to understand the concept of a collection of vector spaces being called an algebra due to the properties of the wedge product!
And most importantly, would it be correct to think that an algebra is a geometrical object in its own right or is instead a class of geometrical object which the collection of vector spaces now qualifies for due to the wedge product? Or neither!
Many thanks!!
