Are all isomorphisms between matrices of the form $ T(A)=UA$ for some matrix $U$? Let $U\in M_{nxn}(\mathbb{R})$ be an invertible matrix and T:$ M_{nxn}(\mathbb{R})\mapsto M_{nxn}(\mathbb{R})$ is a linear map defined by T(A)=UA. 
My question is, are all isomorphisms between $M_{nxn}(\mathbb{R})$ and $M_{nxn}(\mathbb{R})$ of this form? Say, if S:$M_{nxn}(\mathbb{R})\mapsto M_{nxn}(\mathbb{R})$ is an isomorphism, does an invertible matrix M$\in M_{nxn}(\mathbb{R})$ exist such that S(A)=MA.   $\forall A\in M_{nxn}(\mathbb{R})$

No, $A \mapsto A^T$ is not of this form.
