I am wondering given a $K$-vector space $E$ of dimension $n$ if $GL_n(K)=Aut(E,+)$.
For a finite field it seems to be true. For example $Aut(\mathbb Z/2\mathbb Z^3,+)=GL_3(\mathbb Z/2\mathbb Z)$.
If $K=\mathbb Q$ it also seems to be true. Is this correct in general?
I have a another question. The axioms of the external law of a $K$-vector space E are equivalent to the given of a ring morphism from $(K,+,\times)$ to $(Aut(E),+,\circ)$. For instance on a $\mathbb R^3$, does it exist another morphism than the standard one $\lambda \mapsto (x \mapsto\lambda x)$? Do you know some intersting non trivial examples of external laws?
Thanks for your help.