09-240/Classnotes for Thursday September 10: Difference between revisions

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== Date: Thurs. Sept. 10, 2009 ==
== Date: Thurs. Sept. 10, 2009 ==


• Professor's name: Dror Bar-Natan
* Professor's name: Dror Bar-Natan


• Solve systems of equations
* Solve systems of equations


<math>5x_{1} - 2x_{2} + x_{3} = 9 </math>
<math>5x_{1} - 2x_{2} + x_{3} = 9 </math>
<math>-x_{1} + x_{2} - x_{3} = 2 </math>
<math>-x_{1} + x_{2} - x_{3} = 2 </math>
<math>2x_{1} + 9x_{2} - 3x_{3} = -4 </math>
<math>2x_{1} + 9x_{2} - 3x_{3} = -4 </math>


• how? when? one/many?
* how? when? one/many?


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




Line 19: Line 19:
5 & -2 & 1\\
5 & -2 & 1\\
-1 & 1 & -1\\
-1 & 1 & -1\\
2 & 9 & -3\end{pmatrix}</math>
2 & 9 & -3 \end{pmatrix}</math>


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


:<math>\mathbf{A}=\begin{pmatrix}
:<math>\mathbf{A}=\begin{pmatrix}
5 & -2 & 1\\
5 & -2 & 1\\
-1 & 1 & -1\\
-1 & 1 & -1\\
2 & 9 & -3\end{pmatrix}</math>, <math>B=\cdots</math>
2 & 9 & -3 \end{pmatrix}, \mathbf{B}=\cdots</math>


:<math>\begin{pmatrix}
:<math>\begin{pmatrix}
5 & -2 & 1\\
5 & -2 & 1\\
-1 & 1 & -1\\
-1 & 1 & -1\\
2 & 9 & -3\end{pmatrix}+\mathbf{B}</math>
2 & 9 & -3 \end{pmatrix}+\mathbf{B}</math>


<math>AB \neq BA</math>
<math>\mathbf{AB} \neq \mathbf{BA}</math>


<math>\mathbf{A}^{2009}</math>
<math>\mathbf{A}^{2009}</math>


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


• Do all this without choosing coordinates.
* Do all this without choosing coordinates.




'''2. Do everything over other “systems of numbers”'''
'''2. Do everything over other “systems of numbers”'''


1. real numbers
# real numbers
⚫
# rational numbers

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

⚫
3. complex numbers (things like alternating current, circuit)

⚫
4. {0,1} (binary, computer science)



'''3. Hidden Agenda'''
'''3. Hidden Agenda'''


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




'''4. Administration'''
'''4. Administration'''


• can add things to wiki (so long as relevant to course material)
* 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/
* any page added to wiki must start with 09-240- or 09-240/


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


• HW graded and returned by following tutorial
* HW graded and returned by following tutorial




'''5. Classwork done today'''
'''5. Classwork done today'''


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


R1 <math>\forall a,b</math>
R1 <math>\forall a,b</math>
#<math>a+b=b+a </math>
#<math>\,\!a + b = b + a </math>
#<math>a \cdot b=b \cdot a</math>
#<math>a \cdot b=b \cdot a</math>


'''Aside: The <math>\perp</math> character used for additon:'''
'''Aside: The <math>\perp</math> character used for additon:'''


• Prof. Dror asked why <math>+</math> is sometimes written as <math>\perp</math>?
* Prof. Dror asked why <math>+</math> is sometimes written as <math>\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>+</math> symbol, which resembles a Christian cross. (reference: http://en.wikipedia.org/wiki/Plus_and_minus_signs#Alternative_plus_sign)
* 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>+</math> symbol, which resembles a Christian cross. (reference: http://en.wikipedia.org/wiki/Plus_and_minus_signs#Alternative_plus_sign)

Revision as of 14:42, 12 September 2009

Date: Thurs. Sept. 10, 2009

  • Professor's name: Dror Bar-Natan
  • Solve systems of equations

  • 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.

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

  • 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 with two binary operations , (2 inputs, one output) and also with two distinguished elements with the following properties:

R1

Aside: The character used for additon:

  • Prof. Dror asked why is sometimes written as ?
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