Question

let V be a finite dimensional vector space and \(A_0\in L(V)\) be given. What is the dimension of the following subspaces in terms of dim V, \(\nu (A)\) and rank (A)?
a) U ={B\(\in L(V) : A_0 B=0\)}
b) S = {B\(\in L(V) : BA_0=0\)}
c) Prove that if dim V= 10 and rank \((A_0)\)=4, then there are non -zero operators B, such that \(BA_0=A_0 B=0\)
Please, help