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My professor demonstrated that in vector calculus that you can construct basis vectors for one, two, and three forms using the vectors $dx$, $dx$ and $dy$, as well as $dx \wedge dy$, $dy \wedge dz$, and $dx \wedge dz$...but he never explained the process thoroughly. I need help constructing a real vector space, but I don't know how.

From his assignment:

  1. Define the real vector space $\bigwedge^p {\bf R}^n$ for all integers $p\geq 0$. Check that your definition agrees for the cases $p=1, 2, 3$. - 1 form, 2 form, and 3 forms in vector space.
  2. Compute the dimension of the vector space $\bigwedge^p {\bf R}^n$.
  3. For a set $E\subseteq {\bf R}^n$, define the set $\Omega^p (E)$ of $p$-forms defined on $E$.

The problem is that I do not know what the omega sign and the bigwedge sign is. Could anyone please give me some hints so that I can do this by myself? I'm not hounding orders to anyone, I just need help from a different perspective.

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    The all-knowing Wikipedia helps [here](h$t$tp://en.wikipedia.org/wiki/Exterior_algebra), too :)2011-04-25

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