# 06-240/Linear Algebra - Why We Care

## Linear Algebra is the Small Scale Theory of Everything

• 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.
 ${\displaystyle \longrightarrow }$ ${\displaystyle z}$ ${\displaystyle \mapsto }$ ${\displaystyle z^{2}}$ (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, ${\displaystyle {\mathbb {Z} }/p}$ 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.