exterior power

The exterior power of a vector space is closely related to the tensor product.

We can think of them as antisymmetric tensors.

Definition

The exterior power of a vector space over a field is a -vector space and an antisymmetric multi-linear map

that satisfies the following universal property:

For every -vector space , and an antisymmetric multi-linear map

there exists a unique linear (NOT MULTI-LINEAR) map such that this diagram commutes:

Notation: we denote

Explicit construction

Using the tensor product (with its explicit construction) we can build the wedge product. Note is a multi-linear map, so by the universal property of the tensor product, there exists a linear map such that

commutes.

is surjective, and

Therefore, by the first isomorphism theorem

Other identifications

multi-linear maps from dual

We can identify the exterior product with the vector-space of all multi-linear maps

This looks like

Alternative constructions/ definitions

We can define this using the free vector space just like the tensor product just using a different quotienting set.

todo add more here about how to construct it.

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References