09-240/Classnotes for Thursday September 10

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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 everything 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?