09-240/Classnotes for Thursday September 10 - Revision history

Drorbn at 15:47, 17 September 2009

2009-09-17T15:47:40Z

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			{{09-240/Class Notes Warning}}

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

Drorbn: 09-240/Classnotes for Thursday September 11 moved to 09-240/Classnotes for Thursday September 10

2009-09-13T10:50:51Z

09-240/Classnotes for Thursday September 11 moved to 09-240/Classnotes for Thursday September 10

← Older revision	Revision as of 06:50, 13 September 2009
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Drorbn at 10:50, 13 September 2009

2009-09-13T10:50:38Z

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			{{09-240/Navigation}}
	== Date: Thurs. Sept. 10, 2009 ==		== Date: Thurs. Sept. 10, 2009 ==

Drorbn: 09-240/Lecture Notes moved to 09-240/Classnotes for Thursday September 11

2009-09-13T10:44:58Z

09-240/Lecture Notes moved to 09-240/Classnotes for Thursday September 11

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C8sd: /* Date: Thurs. Sept. 10, 2009 */ typos

2009-09-12T19:48:33Z

Date: Thurs. Sept. 10, 2009: typos

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	* 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 everything that has a mathematical description.

C8sd: line breaks

2009-09-12T19:44:37Z

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	<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>

C8sd: /* Date: Thurs. Sept. 10, 2009 */ format

2009-09-12T19:42:23Z

Date: Thurs. Sept. 10, 2009: format

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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)

Gdjones at 17:05, 12 September 2009

2009-09-12T17:05:36Z

New page

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

• Professor's name: Dror Bar-Natan

• Solve systems of equations

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

• 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.'''

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

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

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

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

<math>AB \neq BA</math>

<math>\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>\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>
#<math>a+b=b+a </math>
#<math>a \cdot b=b \cdot a</math>

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

• 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)