06-240/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.

Some Technical Remarks

The Term Test

Our one and only Term Test is coming up. It will take place in class on Tuesday October 24 2006, starting promptly at 1:10PM and ending at 3:00PM sharp. It will consist of 4-5 questions (each may have several parts) on everything that we will have covered in class by October 18: the axiomatic definition of fields and vector spaces, {\mathbb Z}/p and other examples, spans, linear combinations and linear equations, linear dependence and independence, bases, the replacement lemma and its consequences, a bit about linear transformations and a few smaller topics that we touched but don't deserve their own headers.

Note my comments from 06-240/Homework Assignment 5:

Will there be "proof questions"?
Sure. What else have we done so far?
Do we need to know the proofs from class?
Sure. There's a reason why these proofs are in class to start with; if they weren't valuable, we wouldn't have covered them.

Note also that there may be some computations, but nothing that will require a calculator. Note also that I may include some questions from the homework assignments verbatim or nearly verbatim.

No electronic devices capable of displaying text or sounding speech will be allowed.

Extra Class Time

On Thursday, October 26th, we will have a 2-hour class (1-3PM) to make up for the class time lost on the Term Test, and just one hour of tutorials (3-4PM) as no HW assignment is due on that week anyway.