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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!

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Sure, just add matrix representatives and consider the equivalence class of the sum. Then check that it's well-defined. – AsinglePANCAKE Oct 9 '12 at 4:48
What is $L(V)$? – Qiaochu Yuan Oct 9 '12 at 6:53
@AsinglePANCAKE: you'll find that pretty difficult, as the sum is in fact not well-defined. – Qiaochu Yuan Oct 9 '12 at 6:53
@QiaochuYuan Oops. And methinks $L(V)=Hom_F(V,V)$ here... – AsinglePANCAKE Oct 9 '12 at 6:59
@AsinglePANCAKE: and what is $V$? (I'm being intentionally dense here.) – Qiaochu Yuan Oct 9 '12 at 7:00

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.

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I see what you are saying under the comments. I should have been more specific. Does this change if I consider only infinite fields like $\mathbb{R}$? – Bachmaninoff Oct 10 '12 at 3:04
@Bachmaninoff it does, but not in a useful way. – Alexei Averchenko Oct 10 '12 at 3:05

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