# Questions tagged [exterior-algebra]

It is a quotient - of the tensor algebra, obtained by taking graded sum over whole numbers $n$ of $n$-fold tensor products - by the ideal generated by elements of the form $a\otimes a$. We write the residue class of $a\otimes b$ in this algebra, as $a\wedge b$ and call it the wedge product.

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### Canonical isomorphism between $\Lambda^2 V$ and $\mathbb{R}$

Let us take a two-dimensional real vector space $V$, and define $$\det \colon V \times V \rightarrow \mathbb{R} \colon (u,v) \mapsto \det(u\vert v).$$ Since this map is bilinear, the universal ...
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### How do I see if $f \in V \wedge W$?

Let $V,W \leq X'$, where $X$ is a vector space and $X'$ its dual. Let $f \in X'$. How do I check if $f \in V \wedge W$? To make it concrete, Let $X$ be a real vector space with complex structure $J$, ...
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### Defining the exterior algebra

To define a quotient algebra $\mathcal{A} / I$, with $\mathcal{A}$ an algebra over a field $K$, I thought that $I \subset A$ is a condition which had to be required. However, I read that the exterior ...