Key points of linear algebra

basics:
- basic concepts
- concrete interpretations

core:
- classifications
- structures (Lie theory)
- categorical

etc:
- algorithms
- applications

Basic concepts:
- elements: vectors
- objects:  vector spaces
- maps:     linear maps

Concrete egs:
- K^n
- polynomials and calculus
- analysis (continuous functions etc.)

vector/matrix interpretations
function/integral operators (with integration kernels)
  [and reading matrix as columns, as rows, as total, as entries]

Subspaces, quotient spaces
affine spaces, cones, convex spaces

Classifications:
- vector spaces (dimension)
  span, independent, basis
  [and interpretations in terms of map / matrix]
- maps between spaces (rank; & rank-nullity etc.)
  ...and image, kernel, cokernel (im = ker coker = coker ker)
  (nullity + rank = domain; rank + (conullity?) = target)
- structure of an operator
  (+ Det, trace: --> char poly, Alt^k)
  + structure theorem for fg mods over PID
    (and semisimplicity, simple / indecomposable)
  + structure of an operator (inv factor/primary decomposition)
  + ...and interpretation: eigen values, evectors / espaces / gen espaces
      & Jordan form
- structure of an operator on hermitian space
  + spectral theorems

Lie theory: structures on a space:
(talk about Klein's Ehr... geometrization / symmetry program,
 and origin in study of )
+ R/C/H/O (multiplications!) [and clifford algebras too]

- volume form: sl
- orthogonal form: so
- symplectic form: sp
[exotics]

real structures:
- orient, (spin) (also complex orientation)
- complex structure, quaternionic structure
  ...and (left) regular representations of complexes and quaternions
  (-> representation theory)
- so(p,q): p+q space-time, physics
  [and bases for the Lie algebras: rotations, boosts]
(and 6 degrees of freedom in affine 3-space, and pitch/roll/yaw on planes)
(cross product and physics love of quaternions & ijk basis,
 b/c of Clifford algebras and 3+1 space-time)

[mention Jordan algebras too]

Categorical:
- representability of id (Hom(K,-) = id)
- dual (& covector interpretation!)
- Hom
- Tensor
  (hard, weird; defined by universal property; multi-linear)
  [also do concretely as "list of lists" = lex order basis]
- adjunction! $Hom(V,W) = V^* \otimes W$

- Schur functors
  tensor powers, sym, alt, and others
    --> representation thy of $S_n$, Schur-Weyl
  ...and tensor algebra, symmetric (polynomial) algebra, alt (grassmann) algebra
    --> differential forms
  (vector interpretation in R^3: grad/curl/div;
   of course grad and div are general; curl is the weird one)

Algorithms:
- Gauss-Jordan
[- simplex method]

Applications:
- solving systems of equations
- calculus: derivative (as linear transform of tangent spaces) and chain rule (compose / mult matrices!) and gradient (it's dual: df is what's natural)
- PCA
- ???
