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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!!

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  • (i) Your understanding is correct.
  • (ii) The significance of the ordering is mainly bookkeeping; in particular they provide an easy way to start from a basis $e_1,\dots,e_n$ for the vector space $V$ and extend it to a basis for the spaces of $p$-vectors and the exterior algebra.
  • (iii) An algebra is a vector space with a product (that satisfies some rules). This product is supposed to be the 'wedge' $\wedge$. But ${\wedge^p}V$ is not closed under the wedge operation, so it is merely a vector space. To make a vector space that admits the wedge as its product, you must consider all $p$-vectors, for all $p$ at the same time. And this becomes the exterior algebra.
  • (iv) Of course you can (and should!) think about the exterior algebra as an object in its own right! Of course, it is mainly an algebraic object, that is why it is called an algebra. And you can think of the vectorspaces of $p$-vectors a providing a decomposition for this algebra ${\wedge}V = \bigoplus_{i=0}^{\mathrm{dim}V} {\wedge^i}V$. (In fact this decomposition 'respects' the product, in the sense that it provides a grading for the algebra. (This last thing is just saying that the product of a $p$-vector and a $q$-vector is a $p+q$-vector.)
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  • $\begingroup$ The $\bigwedge^p V$ can be consider as a subspace of $\bigwedge V$, also in terms of basic linear algebra. $\endgroup$ – janmarqz Jan 5 '15 at 17:23
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    $\begingroup$ Just what I needed, thank you! $\endgroup$ – Janet the Physicist Jan 5 '15 at 17:35

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