How to convert a permutation group into linear transformation matrix? is there any example about apply isomorphism to permutation group
and how to convert linear transformation matrix to permutation group and convert back to linear transformation matrix

$$\varphi:\begin{array}{ll}\mathcal S_n \to \mathcal L\left(\Bbb R^n\right)\\ \sigma \mapsto \left(x = x_1b_1 + \dots + x_n b_n \mapsto x_1b_{\sigma(1)} + \dots + x_n b_{\sigma(n)} \right)\end{array} $$
You can prove it is an injective morphism so if you restrict its codomain to its image, you get and isomorphism.
And then you just need to have an isomorphism from $\mathcal L\left(\Bbb R^n\right)$ to $\mathcal M_n\left(\Bbb R\right)$ to get the isomorphism you want.
