When we considered the representation theory of the algebra of polynomials, we saw that all it takes to specify such a representation is choosing a single endomorphism . That is, once we pick a transformation we get a whole algebra of transformations , corresponding to polynomials in one variable over the base field . Today, I want to outline one useful fact about these: that their kernels are invariant subspaces under the action of .
First, let’s remember what it means for a subspace to be invariant. This means that if we take a vector then its image is again in . This generalizes the nice situation about eigenspaces: that we have some control (if not as complete) over the image of a vector.
So, we need to show that if then , too. But since this is a representation, we can use the fact that , because the polynomial algebra is commutative. Then we calculate
Thus if is a linear transformation which is built by evaluating a polynomial at , then its kernel is an invariant subspace for .