\chapter{Dimension of a vector space}
[The definitions are really "compare to a model space";
you can give more abstract algebraic definitions,
but b/c there actually -are- model spaces,
they amount to the same thing.]
Recall the philosophy (also included in measure theory):
to measure something unknown, -compare- it with known things.
From the categorical POV, this means:
see if some standard objects map -to- it (or conversely),
and see if these maps are -mono- or -epi-
(suggesting "at least as big as/no bigger than")
For linear algebra, the model objects are $K^n$.
The -correct- definition of dimension of a vector space is:
- whichever model space it corresponds to (whichever $n$ s.t. $V \iso K^n$)
...and then it's a -theorem- to show that:
- this is unique if it exists
(this is the crux)
[BTW, if it doesn't exist, you're infinite dimensional;
this also requires some showing]
- it agrees with the other definitions (below)
The standard definition of dimension is:
"maximum cardinality of lin. indep. set"
(max num of lin. indep. vects)
...or
"min card of spanning set"
These are precisely:
"maximum $K^n \inj V$"
and
"min $K^n \surj V$":
approximating it via -packing- or -covering- (inner/outer measure)
Hey, you can dualize these!
The -dual- definitions are:
"min $V \inj K^n$"
and
"max $V \surj K^n$"
...which can be interpreted as:
"min number of -coordinates- to distinguish all vectors"
and
"max number of -coordinates- that can be -realized-"
...of course, for finite dimensional vector spaces,
these are just the dimension of the dual,
and agree with the dimension of the original space.
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Geometry
These dual definitions are not at all insane!
The "realize coordinates" definition
is the traditional definition of dimension of a variety!
(max space s.t. projection maps onto an open set)
...and actually also works for manifolds,
so long as you use Diff or PL (or PDiff) maps!
(mapping onto an open set, namely a ball;
proof is to just map a nbhd to ball,
and map the outside to some junk; basically the cone on bdry
yields a sphere, which you flatten)
BTW, why do these dual definitions work so well for spaces?
B/c you can dualize 'em to maps of -rings of functions-:
the dimension of a space is the dim of the local ring of reg. funcs.
(The "distinguish" coordinates definition
is embedding dimension,
and is rather subtler and topological;
think about why.)
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enough projectives/enough injectives
The categorical notions of "projectives" and "injectives"
are basically "good space for measuring with".
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modules
for vector spaces, all of these agree;
for modules, it's much more complicated/interesting:
for finitely -generated- abelian groups,
$\bZ$ is enough for mapping -in-
(covering = # of gens; filling = rank = size of free)
you can also cover $\bZ$ (recovering rank)
...but not every group fits -into- $\bZ^k$,
due to torsion -- so you need $\bQ/\bZ$
(and $\bQ$ isn't bad for fitting $\bZ$ into)
...which leads to divisibility and injectives.
NB: you can always fit $\bZ^k$ -in-,
or -cover- $\bZ^k$, b/c $\bZ^0$ is -small-
(it's a zero object, in fact; need initial/terminal
object to be one of your models!)
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cone/affine/convex
You can do -exactly- the same thing with cones, affine spaces, and
convex spaces.
(I discuss more in their areas)