## Some Natural Identifications (Pt. III)

**Point of Post:** This is a continuation of this post.

We begin by defining a map by having map to the map defined by the following: . In words the map associated to takes a linear functional and maps it to the element of gotten by evaluating at and then multiplying this by . This is map is clearly bilinear and so by the universal characterization of tensor products we can lift this to a map with the property that .

*Remark: *We obviously index with to show which spaces the map is defined on and mapped to (since we have such a map for every pair of spaces), but when the context is clear we shall omit this and just write .

Now, to prove this is an isomorphism it suffices to prove that is surjective since both spaces are of the same finite dimension. To do this it suffices to show that a basis of is hit. To do this we let be a basis for and a basis for . We know then that a basis for is the set of maps defined by . That said, note that

Thus we see that and agree on a basis and thus must be equal. Thus, we see that hits a basis of and thus (as previously mentioned) must be an isomorphism.

It remains to show that this isomorphism is natural in both variables. Let’s show that it is natural in . This amounts to showing that if is a linear map the following diagram commutes

where is the usual map (which acts on simple tensors as ) and is the God-awful map defined by taking and mapping it to where . Ok, so we take an arbitrary (it suffices to check commutativity on simple tensors since they are a generating set)and and check that

and

Yay! So naturality in the more difficult variable is verified. Now, to verify that there is naturality in the -variable we need to suppose that we have map and check that the following diagram commutes

where is the usual map which acts on simple tensors as and takes a map and maps it to . To check this we let (for the same reasons) and be arbitrary, we see then that

and

where the last step follows from the fact that is linear!

Thus, summing this all up we may conclude the following:

**Theorem: ***Let be a field and and finite dimensional -spaces. Then, is naturally isomorphic to (in both variables) via the map which acts on simple tensors as . *

**References:**

[1] Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print.

[2] Rotman, Joseph J. *Advanced Modern Algebra*. Providence, RI: American Mathematical Society, 2010. Print.

[3] Blyth, T. S. *Module Theory.* Clarendon, 1990. Print.

[4] Lang, Serge. *Algebra*. Reading, MA: Addison-Wesley Pub., 1965. Print.

[5] Grillet, Pierre A. *Abstract Algebra*. New York: Springer, 2007. Print.

[6] Conrad, Keith. “Exterior Powers.” *Www.math.uconn.edu/~kconrad*. Web. <http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/extmod.pdf>.

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