See problem 3.26 in [Etingof's "Introduction to representation theory"][1]. If you have troubles with understanding the hint, feel free to ask me. (The first sentence uses the fact that if a vector space over an infinite field is the union of finitely many subspaces, then one of these subspaces is the whole vector space.)


  [1]: http://math.mit.edu/~etingof/replect.pdf