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In a comment to the top answer of this 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).

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  • $\begingroup$ I was hesitant to ask this here, instead of math.SE, since the question seems to elementary, but have finally decided to ask it here, since (1) it orginated from a question from this site and (2) if the author of the statement doesn't know the difference (although he may have been jocular when he said it) an answer to this question may also be worthwhile to other algebraists on this site. $\endgroup$
    – tanktop
    Jul 14, 2013 at 14:09

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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.

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  • $\begingroup$ This makes the difference perfectly clear, thank you. Though now I wonder why this difference runs under the description " dynamical perspective", since it seems to me to be a purely algebraic issue...(do you have any ideas concerning this ?) $\endgroup$
    – tanktop
    Jul 14, 2013 at 17:28
  • $\begingroup$ @tanktop: as JDH mentions in the answer you link to, one can think of an automorphism of $V$ as something which "stretches, skews, reflects or rotates" $V$, while a change of of basis leaves $V$ as it was, it only moves the basis (of course that might also be considered as something dynamic). $\endgroup$
    – Michael Bächtold
    Jul 14, 2013 at 19:40

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