# Field extension similarity

Let $$F \subseteq K$$ Be a field extension.I am Trying to prove that if $$A,B \in M_n(F)$$ are similar as matrices over the field $$K$$ (there exists an invertible matrix $$P \in M_n(K)$$ such that $$PA=BP$$) then they're also similar over the field $$F$$. In order to prove that, I need to prove that if $$P \in M_n(K)$$ is invertible, then there exists matrices $$P_1,..,P_r \in M_n(F)$$ and scalars $$a_1,...,a_r \in K$$ such that $$\{a_1,...a_r\}$$ is F-linearly independent and $$a_1P_1+...+a_rP_r = P$$. Why is that true?

• rational canonical form? – Lord Shark the Unknown Apr 7 at 8:16
• @LordSharktheUnknown I never heard of it, is there more basic solution? – Omer Apr 7 at 13:15
• anyone please... – Omer Apr 7 at 15:43