TITLE: Why is the exterior algebra called the "exterior algebra?" What makes it "exterior?"
QUESTION [42 upvotes]: Why is the exterior algebra called the "exterior algebra?" What makes it "exterior?" Is it just because a module can be universally embedded into its exterior algebra, so one could view the exterior algebra as surrounding the module? Why is it not just called the "alternating algebra?"

REPLY [52 votes]: It was Grassmann that called it exterior because to have a non-null product the multiplicands must be geometrically one to the exterior of the other. For instance $$\mathbf{x}\wedge\mathbf{y}\wedge\mathbf{z}=0$$
if $\mathbf{x}$ lies in (is not exterior of) the subspace spanned by the $\mathbf{y}$ and $\mathbf{z}$. So the product is called exterior product, and consequently the algebra with this product is called exterior algebra.