Consider a vector space V over $\displaystyle \mathbb{C}$ mapped to itself by the operator L. The characteristic polynomial is $\displaystyle c(\lambda)=(\lambda-1)^4(\lambda-2)^4(\lambda-3)^4$ and the minimal polynomial is $\displaystyle m(\lambda)=(\lambda-1)^3(\lambda-2)^2(\lambda-3)$.

Determine whether or not each of the following properties can be determined uniquely from these 2 polynomials. If so, state it and argue why it is unique. If not, give a counterexample of distinct operators with the given minimal and characteristic polynomials:

a. dimension of V

c. geometric multiplicity of $\displaystyle \lambda$=1, 2, and 3

d. the number of linearly independent eigenvectors for L.