. key point of "classification of maps":
can extend a basis.
More abstractly, VectLin is a semisimple category
(indeed, it's the archetype),
[and the simple object is K]
so get transversals, and it breaks up into blocks:
dom ~= ker + im
tar ~= im + coker
(need to choose splittings)
This is not an idle point;
K[G] is also semisimple
(for measure of G rel. prime to char of K)
...and a key example, where the pieces are more interesting.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
beyond rank
rank is only invariant of a map
(between given spaces)
[NB: injective map can be regularized by row ops (target);
epi map can be regularized by col ops (source)]
...but given k vectors, question is "configurations of vectors",
aka, orbits of GL(V) acting on V^k (or V choose k)
[so given k vectors,
or a matrix, what else can you say by way of classification?
"vectors" corresponds to columns of a matrix, natch,
and the action on the target.
if allow action on target and source, get classification of maps;
if don't allow either, just get all matrices. duh.]
EG, given 2 points in plane,
can either be same or not;
given 3 points, can all be same, 2 same and 1 different, all different
and colinear, or not