Consider a vector space V over mapped to itself by the operator L. The characteristic polynomial is and the minimal polynomial is .

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 =1, 2, and 3

d. the number of linearly independent eigenvectors for L.