Why do linear algebra and calc together?

- functions as first order objects
  For me, this is the key reason:
  a deep goal of calc is to get people to think of functions as
  objects themselves -- that you can plug into maps etc. (EG, function
  spaces, operators, etc.)
  If you just have a function -set-, it's kinda boring.
  OTOH, if you do -rich- stuff with 'em, like look at operators on
  function spaces, it's -easier- to learn.
  (recall: easier to learn more than less)
  By -working- with them, they become familiar.

- function spaces are familiar and rich vector spaces
  (and often don't have natural bases: they're thus a good
   EG in addition to finite dimensional ones)

- diff/int are linear,
  and discussion of linear maps, kernels, affine spaces,
  etc., clarify much of calculus

- the jacobian is a linear transform
  to discuss diff. in several variables, you need linear transforms:
  what do you -mean- by differentiable?

- they're both totally fundamental
  calc, lin.alg., and some stats are fundamental to applications of math,
  and (dropping stats) fundamental to pure math
  (which suggests teaching both, but not nec. together)

- the point is that they -support- each other:
  you can do better lin.alg. with calc,
  and better calc with lin.alg.
