1
$\begingroup$

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!

  • 0
    @Bach: I am trying to get at the fact that you talked about $M_n(F)$ and then talked about $L(V)$ without specifying the relationship between these two things.2012-10-10

1 Answers 1

2

No. If you compute the sizes of the two sets when $F$ is a finite field, you will find that they disagree. If you try to define addition on equivalence classes, you will find that it is not well-defined.

  • 1
    @Bachmaninoff it does, but not in a useful way.2012-10-10