09-240/Classnotes for Thursday September 10

Date: Thurs. Sept. 10, 2009

• Professor's name: Dror Bar-Natan

• Solve systems of equations

$5x_{1} - 2x_{2} + x_{3} = 9$

$-x_{1} + x_{2} - x_{3} = 2$

$2x_{1} + 9x_{2} - 3x_{3} = -4$

• how? when? one/many?

• This describes the small scale behaviour of almost everthing that has a mathematical description.

1. A matrix is a square or rectangular array of numbers.

$\begin{pmatrix} 5 & -2 & 1\\ -1 & 1 & -1\\ 2 & 9 & -3 \end{pmatrix}$

• we will learn addition, multiplication, and powers of matrices

$\mathbf{A}=\begin{pmatrix} 5 & -2 & 1\\ -1 & 1 & -1\\ 2 & 9 & -3 \end{pmatrix}, \mathbf{B}=\cdots$

$\begin{pmatrix} 5 & -2 & 1\\ -1 & 1 & -1\\ 2 & 9 & -3 \end{pmatrix}+\mathbf{B}$

$\mathbf{AB} \neq \mathbf{BA}$

$\mathbf{A}^{2009}$

• describes the approximate long-term behaviour of almost anything...

• Do all this without choosing coordinates.

2. Do everything over other “systems of numbers”

1. real numbers
2. rational numbers
3. complex numbers (things like alternating current, circuit)
4. {0,1} (binary, computer science)

3. Hidden Agenda

• Learn the basic pure-math processes of: abstraction, generalizations, definitions, theorems, proofs, notation logic

4. Administration

• can add things to wiki (so long as relevant to course material)

• any page added to wiki must start with 09-240- or 09-240/

• HW assigned on Tuesday, due in tutorial 9 days later.

• HW graded and returned by following tutorial

5. Classwork done today

• The Real Numbers: a set $\mathbb{R}$ with two binary operations $\,\!+$, $\times$(2 inputs, one output) and also with two distinguished elements $0,1\epsilon\mathbb{R}$ with the following properties:

R1 $\forall a,b$

1. $\,\!a + b = b + a$
2. $a \cdot b=b \cdot a$

Aside: The $\perp$ character used for additon:

• Prof. Dror asked why $+$ is sometimes written as $\perp$?

• This is a Jewish tradition that dates back to at least the 19th century, and is still used today in Israeli elementary schools. It avoids the writing of the $+$ symbol, which resembles a Christian cross. (reference: http://en.wikipedia.org/wiki/Plus_and_minus_signs#Alternative_plus_sign)