I am postponing problems 4(c) and 4(d) until HW2, since we haven’t defined the necessary concepts yet.
In problem 3, all you need to know about is that we defined by setting for , and then extending linearly.
In problem 5, just as is decomposed as a direct sum of (tensors of order k) for each k, there is a similar decomposition for and . The Hilbert series for these other two algebras is defined analogously. (You can start with a basis , find a basis of each of these three spaces. In each case, count how many basis vectors are order k tensors for each k, and then find the resulting generating function.)