- classifying operators up to iso is -exactly- classify K[x]
  modules
  ...representing them as -matrices- is secondary.

  Hence the -nice proof- of Jordan normal form is:
  V is a f.d. K[x] module
  Take primary decomposition.
  A Jordan block gives a representative (of each primary component).
  
  [BTW, what about "preserves a flag"?
   Again, try to do everything abstractly,
   at level of K[x] modules.
   It has a filtration?]

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Cool:
classifying operators -> classifying modules
  ...is a categorical trick,
and lets us get rather more abstract,
and much, much more powerful.
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What's the connection between the operator,
its char poly, and its min poly?

It's -very- instructive to think of an operator as a finite abelian
group. (an operator yields a finitely generated torsion K[T]-module)
In that dictionary,
- char poly = order
- min poly = exponent

That is, a f.g. mod over a PID has a few canonical decompositions:
Primary
  R/q1 + ... + R/qk
...where the components are primes to some power
(Divisible?)
  R/p1 + ... + R/pl
  where pi \divides p_{i+1}
    [Note: this -ends- in the free part, since everything divides 0,
     but zero doesn't divide anything (so it doesn't -start- with 0)]

EG:
Z/2 + Z/4
  (is in both such forms)
Z/2 + Z/6 = Z/2 + Z/2 + Z/3
  (is divisible?/primary)

A module is determined by its -factors-
  (its primary factors or its -invariant- factors)

"elementary divisors vs. invariant factors"

The characteristic polynomial is the -product- of these factors;
the minimal polynomial is the -lcm- of these factors.
  (in particular, it's the top factor in the divisible? decomposition,
   and indeed is -zero- if there's any free part)
  More intrinsically, it's the -annihilator- of the space/operator.
  (hey, Cayley-Hamilton!)
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Algebraically closed?

The invariant? & primary decomposition exist over any PID.
For K[x], the primes are simply (x-a), so the primary decomposition is
very nice.
For R[x], there's also irreducible quadratics, which make life a
little messier.
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Regular, semisimple -in these terms-

- regular = cyclic
  V is K[T]-cyclic; V = K[x]/c(x)
    (in primary decomposition, only 1 copy of each prime,
     by Chinese Remainder Theorem)
    (geometrically, no repeated points)
    (better: geometrically, -connected?-) (what's the right word?)
- semisimple = splits up into 1-dimensional pieces
    (in primary decomposition, each prime to -1st power-)
    (Equiv., no repeated roots in minimal poly)
    (geometrically, -reduced-)
so: regular says 1 -copy-; semisimple says -to the power 1-
  semisimple means V = K[x]/(x-\lambda_1) + K[x]...
  semisimple = diagonalizable ("basis of eigenvectors")
  regular semisimple = diagonalizable with distinct char
    poly
    (reduced, connected!)
  [in primary decomposition, each prime one time to power 1]

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Canonical forms
  various canonical forms are simply nice ways to write the
  matrices
  (I've realized this in some form before)
  + Frobenius normal form / rational canonical form
    write a companion matrix
    for each component of the
    invariant decomposition
    This is called "rational" b/c it exists over the starting field.
  + Jordan normal form
    write a Jordan block
    for each component of the
    primary decomposition
    This may require going to the algebraic closure,
    though there's an analog over R (it's not pretty though)
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proofs:
  + existence of eigenvector
    ...and for commuting family
    ...Schur form
    ...Jordan form
  + spectral theorem
- commuting operators have common eigenvector:
  this is -exactly- the Nullstellensatz!
  A vector space with $k$ commuting operators is -exactly-
  a K[x_1,\dots,x_k]-module.
  The Nullstellensatz says that this actually has a point,
  say $(\lambda_1,\dots,\lambda_k)$, and this is exactly
  an eigenvalue (and it has an eigenvector).

  [This is a much nicer algebraic proof than the "topological
   generator" argument. Also, it shows that it's -exactly- the natural
   higher dimensional analog of "has a root/algebraically closed"]
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Pedagogy

The core content of linear algebra
is -very- dry and algebraic.
It's cool, but -very- austere.

The -point- of linear algebra is that it's hugely applicable,
and it really should be introduced -in that context-.

Cool applications include:
- eigenfunctions in physics
  (esp. waves & music! -> PDEs, Laplacian)
- stats
  (ok, regressions are kinda complicated)
- algebraic geometry!
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Linear Algebra → Algebraic Geometry

Linear algebra leads naturally to algebraic geometry, in several ways.
- classify a single operator
  This leads to studying modules over K[x],
  and the spectrum of this operator.

[Nomenclature: Spectrum of an operator -is- Spec K[T]/c_T;
 for spectral sequences, it's b/c Leray though of the filtration as
 like a "spectrum" (like a prism, it splits up a complex object into
 simple pieces). C.f. the "chromatic" spectral sequence, where
 the angled wedges are like the spectrum of a rainbow.]
[Actually, it is kinda the same thing:
 the spectrum of an operator -is- the action on the associated graded,
 i.e., coming from a filtration
 ...and note that the spectrum of an operator is a -lot- like the
 physical spectrum; indeed, for operators on Hilbert spaces, you get a
 continuous spectrum.]

- geometric objects,
  like the rational normal curve

- GL, and other algebraic groups:
  these are -algebraic- but not -linear- subsets

- 2 commuting operators: this is a K[x,y]-module

Indeed, classification of 2 or more -non-commuting- operators exactly
leads to non-commutative algebraic geometry.
