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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5x_{1} - 2x_{2} + x_{3} = 9 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -x_{1} + x_{2} - x_{3} = 2 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix} 5 & -2 & 1\\ -1 & 1 & -1\\ 2 & 9 & -3 \end{pmatrix}}

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{A}=\begin{pmatrix} 5 & -2 & 1\\ -1 & 1 & -1\\ 2 & 9 & -3 \end{pmatrix}, \mathbf{B}=\cdots}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix} 5 & -2 & 1\\ -1 & 1 & -1\\ 2 & 9 & -3 \end{pmatrix}+\mathbf{B}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{AB} \neq \mathbf{BA}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \,\!+} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \times} (2 inputs, one output) and also with two distinguished elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0,1\epsilon\mathbb{R}} with the following properties:

R1 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall a,b}

1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \,\!a + b = b + a }
2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \cdot b=b \cdot a}

Aside: The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \perp} character used for additon:

• Prof. Dror asked why Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +} is sometimes written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +} symbol, which resembles a Christian cross. (reference: http://en.wikipedia.org/wiki/Plus_and_minus_signs#Alternative_plus_sign)