## Facts about alternating bilinear maps on vector spaces

Let be a field, let be an -dimensional -vector space, and let be a bilinear map, where is an -vector space.

- Prove that if is an alternating -bilinear map on then for all .
- Suppose . Prove that is an alternating bilinear map on if and only if for all .
- Suppose . Prove that every alternating bilinear form on is symmetric. (I.e. for all .) Prove that there exist symmetric bilinear maps which are not alternating.

Let , and suppose is an alternating bilinear map. Now , so that .

Suppose ; in particular, 2 is a unit in . If is bilinear such that for all , then in particular we have , so that . Thus for all , and so is alternating. Conversely, if is alternating then by part (1) above we have for all .

Now suppose . Note that . Mod , we have . In particular, the submodule generated by all tensors of the form is contained in .

We have already seen that and . Thus the containment is proper.

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## Comments

In 2, the bilinearity of f seems to be an hypothesis to prove the equivalence.

Thanks! I think I understand and fixed the problem.