Similarity of matrices gives an equivalence relation on $M_n(F)$, so I can define $S$ to be the set of equivalences classes. Can I define a bijective function $\Phi$ from $S$ to $\mathcal{L}(V)$? (My gut says 'yes' - perhaps I can map an equivalence class of similar matrices to the linear operator that they represent?) And if so, is there a way to define a binary operation on $S$ that turns $\Phi$ into an isomorphism, with addition on $\mathcal{L}(V)$ defined as usual?
Thanks in advance!