09-240/Classnotes for Thursday September 10

Date: Thurs. Sept. 10, 2009

• Professor's name: Dror Bar-Natan

• Solve systems of equations

[math]\displaystyle{ 5x_{1} - 2x_{2} + x_{3} = 9 }[/math]

[math]\displaystyle{ -x_{1} + x_{2} - x_{3} = 2 }[/math]

[math]\displaystyle{ 2x_{1} + 9x_{2} - 3x_{3} = -4 }[/math]

• how? when? one/many?

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

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

[math]\displaystyle{ \begin{pmatrix} 5 & -2 & 1\\ -1 & 1 & -1\\ 2 & 9 & -3 \end{pmatrix} }[/math]

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

[math]\displaystyle{ \mathbf{A}=\begin{pmatrix} 5 & -2 & 1\\ -1 & 1 & -1\\ 2 & 9 & -3 \end{pmatrix}, \mathbf{B}=\cdots }[/math]

[math]\displaystyle{ \begin{pmatrix} 5 & -2 & 1\\ -1 & 1 & -1\\ 2 & 9 & -3 \end{pmatrix}+\mathbf{B} }[/math]

[math]\displaystyle{ \mathbf{AB} \neq \mathbf{BA} }[/math]

[math]\displaystyle{ \mathbf{A}^{2009} }[/math]

• 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 [math]\displaystyle{ \mathbb{R} }[/math] with two binary operations [math]\displaystyle{ \,\!+ }[/math], [math]\displaystyle{ \times }[/math](2 inputs, one output) and also with two distinguished elements [math]\displaystyle{ 0,1\epsilon\mathbb{R} }[/math] with the following properties:

R1 [math]\displaystyle{ \forall a,b }[/math]

1. [math]\displaystyle{ \,\!a + b = b + a }[/math]
2. [math]\displaystyle{ a \cdot b=b \cdot a }[/math]

Aside: The [math]\displaystyle{ \perp }[/math] character used for additon:

• Prof. Dror asked why [math]\displaystyle{ + }[/math] is sometimes written as [math]\displaystyle{ \perp }[/math]?

• 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 [math]\displaystyle{ + }[/math] symbol, which resembles a Christian cross. (reference: http://en.wikipedia.org/wiki/Plus_and_minus_signs#Alternative_plus_sign)