# Matrices and isomorphisms

I am having difficulty interpreting this problem. $M_2(\mathbb R)$ and $\mathbb R^4$ are isomorphic as real vector spaces. Consider the explicit isomorphism $T$ induced by $$e_1 \mapsto e_{11}, e_2 \mapsto e_{21}, e_3 \mapsto e_{12}, e_4 \mapsto e_{22}$$ Define a linear map $S_A$ from $M_2(\mathbb R)$ to itself by taking $A,B \in M_2(\mathbb R)$ and setting $S_A(B)=AB$. Under the isomorphism $T$, $S_A$ becomes a linear map from $\mathbb R^4$ to $\mathbb R^4$, hence $S_A$ is represented by an element of $M_4(\mathbb R)$.

Find the matrix of $S_A$ under this isomorphism with respect to the standard basis, and check that it's correct.

Thanks.

I try to help with notation. Let $T:\mathbb R^4\rightarrow M_2(\mathbb R)$ be the isomorphism given in the OP. We denote by $$\operatorname{End}_{\mathbb R}(M_2(\mathbb R)):=\operatorname{Hom}_\mathbb R(M_2(\mathbb R),M_2(\mathbb R)),$$ resp. $$\operatorname{End}_{\mathbb R}(\mathbb R^4):=\operatorname{Hom}_\mathbb R(\mathbb R^4,\mathbb R^4 ),$$ the (linear) space of $\mathbb R$-linear endomorphisms of $M_2(\mathbb R)$ resp. $\mathbb R^4$, i.e. linear maps from $M_2(\mathbb R)$ to $M_2(\mathbb R)$ and similarly for the second case.
$T$ induces an isomorphism
$$\mathcal T:\operatorname{End}_{\mathbb R}(M_2(\mathbb R))\rightarrow \operatorname{End}_{\mathbb R}(\mathbb R^4)$$
given by $\mathcal T(\rho):= T^{-1} \circ \rho \circ T,$ for all $\rho \in \operatorname{End}_{\mathbb R}(M_2(\mathbb R))$. Now let $S_A\in \operatorname{End}_{\mathbb R}(M_2(\mathbb R))$ be given as in the OP; we need to apply the linear operator $\mathcal T(S_A)$ on the basis $\{e_i\}_{i=1,2,3,4}$ of $\mathbb R^4$ to arrive at the answer. For example,
$$\mathcal T(S_A)(e_1):=T^{-1}( S_A( T(e_1)))=T^{-1}(S_A(e_{11}))=T^{-1}(Ae_{11})=...$$