Why is dimension well-defined?
The key lemma is the "exchange" lemma (name?)
This is abstracted in the notion of a matroid:
a set with a collection of subsets, called "independent sets", s.t.
- empty set is independent
- closed under subset heredity
- exchange: given A, B, w/ #A > #B, can augment B by an element of A
and get another independent set
Thus:
matroids have well-defined rank
All the "can exchange between two bases" etc.
properties in linear algebra are *combinatorial*
(not that they're that interesting, afaict)
alternatively, a matroid is a set with a subadditive rank function:
* $r(A) \leq \abs{A}$
* $A \subseteq B \implies r(A) \leq r(B)$
* $r(A \cup B) + r(A \cap B) \leq r(A) + r(B)$
*matroids are what you get if you study dimension and independence deeply*