Let $V$ be a vector space with dimension greater than 1 over the field $F$ and $Sim = \{(f\in \operatorname{Lin}(V))\mapsto gf(g^{-1}) : g\in \operatorname{GL}(V)\}$, ie $Sim$ is the set of all similarity transforms on $\operatorname{Lin}(V)$. Let $Aut$ be the set of all automorphisms of the $F$-algebra $\operatorname{Lin}(V)$.


What is $Aut\setminus Sim$? In particular, is it empty?



Are all automorphisms of Lin(V) given by similarity transforms?
Let $M$ denote the $F$-algebra $\hom(V, V)$. An $F$-algebra homomorphism $M \to \hom(V, V)$ is tantamount to an $M$-module structure on $V$ in the category of $F$-vector spaces. But the category of $M$-modules is equivalent to $Vect\_F$ (there is a Morita equivalence between the categories given by the functor $- \otimes\_F V: Vect\_F \to Mod\_M$). That is to say, every $M$-module is isomorphic to a direct sum of copies of $V$, and in particular, any module structure on $V$ is isomorphic to the standard $M$-module structure on $V$. 


So, if $\mu': M \otimes\_F V \to V$ is any module structure, and $\mu: M \otimes\_F V \to V$ denotes the standard structure, there is an isomorphism $g: V \to V$ such that $g \circ \mu' = \mu \circ (M \otimes\_F g)$. Putting all this together, any $F$-algebra *homomorphism* $M \to \hom(V, V)$ is obtained by conjugating the standard homomorphism (the identity) by an isomorphism $g: V \to V$. 


Edit: If this proof seems too categorical, there is a proof in slightly alternative language given on page 401 of Bilinear algebra: An Introduction to the algebraic theory of quadratic forms by Szymiczek (Google books [here](http://books.google.com/books?id=CcM8_iiGPxAC&pg=PA401&lpg=PA401&dq=automorphisms+of+a+matrix+algebra&source=bl&ots=Ba5Y2F3AbT&sig=qX8ef4eQVkLoFN-_5CtHGX4s8h8&hl=en&ei=7hm5TJyVOs6s8AauyfH-Dg&sa=X&oi=book_result&ct=result&resnum=7&ved=0CDQQ6AEwBjgK#v=onepage&q=automorphisms%20of%20a%20matrix%20algebra&f=false). 


