Maths?

[codimension4]

Using each of the following n x n matrices in turn, regard C^n as a C[x]-module: find all C[x]-submodules of C^n, and find a generating set which is as small as possible (C is the set of complex numbers; I can't type it with the bar it should have

Comments

[quarks]

A module is like a vector space except the scalars come from a separate ring rather than the base field. C[x] refers in the general to the set of polynomials over the complex numbers. In this context, I believe you have to apply the complex polynomials to the given matrices to get the ring; that is the ring (and hence the module) is defined for a particular linear operator. I'm not sure how to analyze the submodules and spanning sets of the modules; I suspect it's going to involve looking at the eigenvalues and eigenvectors of the matrices.