Field extension similarity

Question

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?