\chapter{Determinant}
it's $\End V \to K$
from
$\End V \to \End \Lambda^n V = K$
determinant is something you take *of a map*
(it's *change* in volume, not *volume* itself)
[like orientation-preserving/reversing
is more basic than an orientation]
vs. n-frame
(action on n-frames is choosing a basis)
vs. volume form
real interpretation: volume and orientation
(coordinate formula comes from basis of $\Lambda^n V$?)
it's a degree n map of K-algebras
[all the endofunctors of Vect are Schur functors;
top exterior power is only non-trivial linear (1-d) one;
there's also $\Lambda^0$ (which yields {0})
and $\Lambda^{-n}$ (which yields empty set)]
\section{det and sign}
Given det, get sign by permutation representation
...but to define det requires sign
(otherwise how do you know size of exterior power?)
[this is a shadow of Schur-Weyl duality?]
[indeed, given map on a set,
get operator on free vector space;
get 0 if not invertible, and sign of permutation if is]
\section{Traces}
BTW, it's the last coefficient in characteristic polynomial
that's these are all the exterior powers
...which correspond to elementary symmetric polynomials
(think of 'em as regular semisimple:
it's symmetric in eigenvalues)
there're other Schur functors
...and really, you're talking about Schur-Weyl duality