Matrix and field extension

Question

It is given that $F\subset K$ are fields. $A$ is a matrix of size $n\times n$ over $K$. I need to prove that there exist $c_1,\ldots,c_k\in K$, linearly independent over $F$, and matrices $A_1,\ldots,A_k$ of size $n\times n$ over $F$ such that $A=\sum\limits_{i=1}^kc_iA_i$.

Answer 1

Set $A=(a_{ij})$, $a_{ij}\in K$. Then $A=\sum_{i,j}a_{ij}E_{ij}$ where $E_{ij}$ is the $n\times n$ matrix which has $1$ on the $(i,j)$th entry and $0$ otherwise. Now consider the $F$-subspace of $K$ generated by all $a_{ij}$ and choose $c_1,\dots,c_m$ an $F$-basis for this subspace. Then $a_{ij}=\sum_{k}f_{ij,k}c_k$ with $f_{ij,k}\in F$. Thus $$A=\sum_{i,j}(\sum_{k}f_{ij,k}c_k)E_{ij}=\sum_kc_k(\sum_{i,j}f_{ij,k}E_{ij})=\sum_kc_kA_k,$$ where $A_k=\sum_{i,j}f_{ij,k}E_{ij}\in M_n(F)$.