A lil' nomenclature point:
I prefer the terms "linear map" or "linear function"
The term "transform" should really be reserved for -isomorphisms-
(-invertible- maps; not necessarily with the same domain and target).
Transform suggests that you can be transformed -back-.
Think "Fourier Transform".
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Why is it called the `general linear group'?
B/c if you think of endomorphisms as a set of $n$ vectors,
then they're in the general linear group iff these vectors are in
-general linear- position, meaning they have no linear relations.
I.e., they're linearly independent.
Special linear is presumably b/c they have 1 condition
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symplectic is a pun on complex;
sym- is the Latin analog of com-,
and symplectic/complex geometry are rather closely related
(especially in dimension 1)
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"inner product"
Given a vector $v$ & a covector $w^*$,
you can combine them in two ways:
either $w^*(v) = \left \in \bR$, the -inner- product,
or $v\otimes w^* \in \Hom$, the -outer- product.
This nomenclature is deceivingly symmetric:
you can take the outer product of -any- vector & covector,
(b/c they factor through scalars)
but can only take the inner product of a vector & covector of the same
space.
AFAIK, this is not related to ``exterior product'' (aka, wedge) as in
differential forms.