is a linear transformation and
Is it true that ?
Here is what I did, but unfortunately it was graded as incorrect.
On the k-tensor powers the induced map is which on pure tensors is
If is a permutation, then and thus we can immediately verify the identity
because both sides will equal
Because pure tensors span the k-tensor power space, we conclude that
Now let's get back to the problem.
By definition, which is
Now by two consecutive applications of (1), and that is, by definition, - which again by definition equals