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What does it mean if we say isomorphism $f:V\to W$ of vector spaces $\mathbb{V}$ and $\mathbb{W}$ depends on basis?

As far as I know basis is introduced after linear mappings. It makes no sense to me how can such notion be formally defined, can someone provide example and counterexample?

Punga
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    It depends on what you mean by "depends." – Randall Apr 30 '18 at 18:16
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    Who says that it depends on the basis? – José Carlos Santos Apr 30 '18 at 18:17
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    I saw in it this question: https://math.stackexchange.com/questions/234127/natural-isomorphism-in-linear-algebra

    I'm not concerned about category theory part, just what does it mean in this context in language of linear algebra.

     – Punga Apr 30 '18 at 18:21
  • I followed that link; what is says (correctly) is that an isomorphism does not depend on bases. – Lee Mosher Apr 30 '18 at 19:05
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    It is the process of constructing the isomorphism the one that can depend on bases or not. Consider this problem: Given a vector space $V$, construct an isomorphism between $V\to V$. The construction in this case can be done without any extra input. Define $f(x)=x$. It can also be done in a way that does depend on the basis. Given two ordered bases of $V$, send the corresponding elements of one to those of the other according to their ordering. Now, if you are given arbitrary vector spaces of the same dimension, it is not possible to construct the isomorphism independent on the choice of bases –  Apr 30 '18 at 19:19

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