I am OK with keeping the default extend=True. But i would like it to
be the opposite in the case of endomorphisms (i know it is not a good
idea to have different default behaviours for matrices and
endomorphisms, so i won't argue if its decided to keep it alwais true
as default).

My reason to prefear extend=False in the case of endomorphisms is
preciselly the teaching linear algebra. I define to my students a
linear map emphasizing the ideas of domain and range, even doing some
exercises about deciding where an element is in the domain or not. If
after all that effort to make the groundings very precise, sage does
preciselly what i told them not to do (that is, forgetting about these
rigurous definitions and just care about the computations), well it is
not the ideal scenario.

Anyways, just having the option gives a solution to my problem.

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