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
    Sure, just add matrix representatives and consider the equivalence class of the sum. Then check that it's well-defined.2012-10-09
  • 0
    What is $L(V)$?2012-10-09
  • 0
    @AsinglePANCAKE: you'll find that pretty difficult, as the sum is in fact _not_ well-defined.2012-10-09
  • 0
    @QiaochuYuan Oops. And methinks $L(V)=Hom_F(V,V)$ here...2012-10-09
  • 0
    @AsinglePANCAKE: and what is $V$? (I'm being intentionally dense here.)2012-10-09
  • 0
    @QiaochuYuan. Ah, okay. Then I shall intentionally stop...2012-10-09
  • 0
    @QiaochuYuan A vector space over some field $F$...I know you're trying to get at something, but I don't know what!2012-10-10
  • 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