Fundamentals of exterior algebra?

What are some fundamental concepts of exterior algebra that everyone should
know? I'm not necessarily only talking about those things which are on the
wiki or which have immediately known applications in music theory, I mean
those things which are considered "fundamental" to understand the subject
as it is commonly understood from an algebraic standpoint. I'd like to make
sure I have all of the fundamentals covered, so that I have a general
toolbox of exterior algebra in my arsenal to use when I tackle new music
theory problems.

Things I've found useful:
1) The exterior product
2) The interior product
3) Duality
4) The exterior powers of a vector space
5) The actual definition of the exterior algebra, i.e. the direct sum of
all the exterior powers, with the 0th exterior power being the field of
scalars
6) Multivectors, and the concept of multivector "rank" and "grade"
7) The exterior powers of a linear map, compound matrices (not on the wiki
but still very useful)
8) The Grassmannian and its embedding into the various exterior powers
9) Abstract algebra stuff: its relationship to the tensor algebra (it's a
"temperament" of it!), the fact that it's a ring, I dunno what else
10) Linear algebra stuff: the relationship between the exterior algebra and
the deterministic and characteristic polynomial of a matrix, I dunno what
else

Is there anything else that's crucial to understand this subject?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia wrote:
>
> What are some fundamental concepts of exterior algebra that everyone should
> know?

It's a good list. You might add the Gramian and its relationship to the exterior product, and Plucker relations. Also the definition of the wedge product in terms of tensors if that's not already implicitly included below