Linear Algebra: many things

Q: how much of this is -detailed- stuff about operators
(and esp., not very mneumonic?).
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K[T] is a PID;
minimal poly is generator of annihilating ideal (name?)
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Given v \in V (or better, W < V), have an ideal in K[T] that kills it.
Non-empty b/c 1...A^{n^2} have a linear relation.

More concretely,
v,Av,... after at most A^n, you get a relation
further, this works on the -whole space spanned by v,Av, etc.-

Given W, W', the annihilator (name?) for W + W' is LCM ann(W),ann(W')
(by linearity)

so: start with some vector, and find annihilating ideal for this;
this has degree k, so span(v,Av,...A^{k-1}v) has dim k
okay -- pick another one, and repeat.
Eventually you get a poly of degree n that annihilates the whole
space.

Cayley-Hamilton
pf 1: look at jordan form; then it's "like duh"
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This actually gives a -defn- of dimension;
1: an associative algebra A acts on itself faithfully;
  this gives a linear representation (into End A).
  However, if A=End(K^n), then A has dim n^2 (as a vector space),
  so this representation has A acting on an n^2-dim'l space.
2: By Cayley-Hamilton, we know that every element of End(V^n) has
  "algebraic degree" (some better name?) at most n:
  it satisfies a polynomial of degree at most n.
  This suggests a sort of "dimension" for an algebra:
  the minimal "algebraic degree" of the algebra.
  Dunno if this is a useful POV.
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state/prove statements about bases, spanning, linear independence;
rewrite them in modern language, saying "vector spaces are all
-projective- and -injective-".
(lifting property, direct sum)
the point being: the statements about bases are all about injective
w/r/t K^n.