09-240/Linear Algebra - Why We Care

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Linear Algebra is the Small Scale Theory of Everything

• To study the large, start with the small.

• In small scales, every space is a vector space.
  • Indeed if you walk a mile east, a mile north, a mile west and a mile south, you're back were you started, but if you fly a 1,000 miles east, a 1,000 miles north, a 1,000 miles west and a 1,000 miles south, you're not back were you started (where will you be?).

• In small scales, every function is a linear function.

	$\longrightarrow$
$z$	$\mapsto$	$z^{2}$	(code)

• The world doesn't come with coordinates.
  • Hence whenever we can we work without a basis, and when we do study bases, we study all of them.

See also The Mathematica notebook 240AlgebraI-QuiltPlot.