In a comment to the top answer of [this](https://mathoverflow.net/questions/7584/what-are-the-most-misleading-alternate-definitions-in-taught-mathematics/7952) question Darij Grinberg says that 



> 
> the problem with the dynamical perspective is that it is way harder to
>  grasp for algebraic/combinatorial-minded people than any formula,
>  however complicated it is. I still don't get the difference between a
>  transformation of points and a transformation of coordinates; for me,
>  they're all endomorphisms of a vector space.
> 
> 
> 


Since apparently I'm also an algebraic minded person - I neither can see a difference between those transformations and also view only as endomorphisms - I would very much like to know what their difference consists of (even if the difference manifests itself only on the level of intuition and not of formal mathematics).



Explaining a comment: Difference between a transformation of points and a transformation of coordinates
Let $V$ be a finite-dimensional real vector space. Choosing a basis of $V$ amounts to giving an isomorphism $\phi : \mathbb{R}^n \to V$. Changing basis amounts to hitting $\mathbb{R}^n$ with an automorphism, but transforming points amounts to hitting $V$ with an automorphism. 


