One important tool for getting a handle on invariants is the Reynolds operator, which essentially is a projection map onto the space of invariants. In this video, we show how the averaging trick can be used to construct this Reynolds operator. We define abstractly the notion of linearly reductive groups to be those where a well-behaved Reynolds operator exists. We give Weyl's unitary trick which give examples of linearly reductive groups. Finally, we give Hilbert's proof of the finite generation of invariants using the Reynolds operator.
