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.