linear algebra in calculus
calculus in linear algebra

- dot product/cross product
  dot product works for all $\bR^n$, is square of $L^2$ (pythagorean) norm
  (Sylvester's law of inertia/quadratic forms)
  (for number systems, comes from conjugation)
  cross product is unusual: it's actually a Lie Bracket,
    coming from lie algebra of ($SU_2$ = unit norm quaternions)
    I -think- it arose from people trying to find a multiplication
    on $\bR^3$ -- it certainly arises if you look at quaternions
    (that is, it does have a generalization, but it's rather more
     complicated and subtler than dot product)
    (it's certainly more interesting!)
- differential equations
  (linear differential equations can be solved easily
   by linear algebra)
- integral as bilinear:
  functions,measures -> scalars
- number systems:
  most (all?) early vector spaces were -concrete spaces-:
  only after you have several examples do you feel the need to abstract
- duality of spaces (this is getting to be hard core real analysis)
