Let $K$ be an extension of the field $F$ of degree $n$, $K = \{c_0 b0 + c_1 b_1 + \dots + c_{n-1} b_{n-1} : c_i \in F \}$. Then for any $a \in K$, $\phi_a : K \to K : x \mapsto ax$ is an $F$-linear transformation of $K$, so the matrix for $\phi_a$ would be $A_a = \begin{pmatrix} \phi_a(b_0) & \phi_a(b_1) & \dots & \phi_a(b_{n-1})\end{pmatrix}$ where $\phi_a(b_k)$ is understood to be the column vector at the basis vector $b_k$. I've shown that $\Phi : a \to A_a$ is additive, but how do you show that it is multiplicative?
The purpose is to show that $K$ is isomorphic to a subfield of the ring $F^{n\times n}$.
My attempt: $\Phi(ad) = \begin{pmatrix} ad b_0 & ad b_1 &\dots & ad b_{n-1}\end{pmatrix}$. I've also tried working with a $2\times 2$ example and don't see where to take it.
Thanks.
EDIT:
The answers in the comments don't answer my general question, they only give an example. Therfore, this question is NOT a duplicate! In fact, I was already aware of the content of those answers! (See above!)