12-240/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.
06-240-QuiltBeforeMap.png \longrightarrow 06-240-QuiltAfterMap.png
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 QuiltPlot.

Lecture notes scanned by marie