Drorbn - User contributions [en]

File:09-240 HW3-12.png

2009-10-19T06:00:07Z

Gdjones:

File:09-240 HW3-11.png

2009-10-19T05:59:57Z

Gdjones:

File:09-240 HW3-10.png

2009-10-19T05:59:46Z

Gdjones:

File:09-240 HW3-9.png

2009-10-19T05:59:34Z

Gdjones:

File:09-240 HW3-8.png

2009-10-19T05:59:23Z

Gdjones:

File:09-240 HW3-7.png

2009-10-19T05:59:12Z

Gdjones:

File:09-240 HW3-6.png

2009-10-19T05:59:01Z

Gdjones:

File:09-240 HW3-5.png

2009-10-19T05:58:48Z

Gdjones:

File:09-240 HW3-4.png

2009-10-19T05:58:36Z

Gdjones:

File:09-240 HW3-3.png

2009-10-19T05:58:23Z

Gdjones:

File:09-240 HW3-2.png

2009-10-19T05:58:11Z

Gdjones:

File:09-240 HW3-1.png

2009-10-19T05:57:58Z

Gdjones:

File:09-240 HW3-0.png

2009-10-19T05:56:41Z

Gdjones:

Template:09-240/Navigation

2009-10-19T05:55:46Z

Gdjones:

{| cellpadding="0" cellspacing="0" style="clear: right; float: right"
|- align=right
|<div class="NavFrame"><div class="NavHead">[[09-240]]/[[Template:09-240/Navigation|Navigation Panel]]  </div>
<div class="NavContent">
{| border="1px" cellpadding="1" cellspacing="0" width="220" style="margin: 0 0 1em 0.5em; font-size: small"
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 7
|<s>Tue</s>, [[09-240/About This Class|About]], [[09-240/Classnotes for Thursday September 10|Thu]]
|- align=left
|align=center|2
|Sep 14
|[[09-240/Classnotes for Tuesday September 15|Tue]], [[09-240:HW1|HW1]], [[09-240:HW1 Solution|HW1 Solution]], [[09-240/Classnotes for Thursday September 17|Thu]]
|- align=left
|align=center|3
|Sep 21
|[[09-240/Classnotes for Tuesday September 22|Tue]], [[09-240:HW2|HW2]], [[09-240:HW2 Solution|HW2 Solution]], [[09-240/Classnotes for Thursday September 24|Thu]], [[09-240/Class Photo|Photo]]
|- align=left
|align=center|4
|Sep 28
|[[09-240/Classnotes for Tuesday September 29|Tue]], [[09-240:HW3|HW3]], [[09-240:HW3 Solution|HW3 Solution]], [[09-240/Classnotes for Thursday October 1|Thu]]
|- align=left
|align=center|5
|Oct 5
|[[09-240:HW4|HW4]]
|- align=left
|align=center|6
|Oct 12
|
|- align=left
|align=center|7
|Oct 19
|[[09-240:HW5|HW5]], [[09-240/Term Test|Term Test on Thu]]
|- align=left
|align=center|8
|Oct 26
|[[09-240:HW6|HW6]]
|- align=left
|align=center|9
|Nov 2
|
|- align=left
|align=center|10
|Nov 9
|[[09-240:HW7|HW7]], <s>Thu</s>
|- align=left
|align=center|11
|Nov 16
|[[09-240:HW8|HW8]]
|- align=left
|align=center|12
|Nov 23
|[[09-240:HW9|HW9]]
|- align=left
|align=center|13
|Nov 30
|[[09-240/On The Final Exam|On the final]]
|- align=left
|align=center|S
|Dec 7
|
|- align=left
|align=center|F
|Dec 14
|[[09-240/The Final Exam|Final on Dec 16]]
|- align=left
|colspan=3 align=center|[[09-240/To do|To Do List]]
|- align=left
|colspan=3 align=center|[[09-240/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:09-240-ClassPhoto.jpg|180px]]<br>[[09-240/Class Photo|Add your name / see who's in!]]
|}
</div></div>
|}

File:09-240 HW2-21.png

2009-10-19T05:54:10Z

Gdjones:

File:09-240 HW2-20.png

2009-10-19T05:53:58Z

Gdjones:

File:09-240 HW2-19.png

2009-10-19T05:53:44Z

Gdjones:

File:09-240 HW2-18.png

2009-10-19T05:53:30Z

Gdjones:

File:09-240 HW2-17.png

2009-10-19T05:53:19Z

Gdjones:

File:09-240 HW2-16.png

2009-10-19T05:53:07Z

Gdjones:

File:09-240 HW2-15.png

2009-10-19T05:52:56Z

Gdjones:

File:09-240 HW2-14.png

2009-10-19T05:52:43Z

Gdjones:

File:09-240 HW2-13.png

2009-10-19T05:52:32Z

Gdjones:

File:09-240 HW2-12.png

2009-10-19T05:52:21Z

Gdjones:

File:09-240 HW2-11.png

2009-10-19T05:52:08Z

Gdjones:

File:09-240 HW2-10.png

2009-10-19T05:51:56Z

Gdjones:

File:09-240 HW2-9.png

2009-10-19T05:51:43Z

Gdjones:

File:09-240 HW2-8.png

2009-10-19T05:51:31Z

Gdjones:

File:09-240 HW2-7.png

2009-10-19T05:51:18Z

Gdjones:

File:09-240 HW2-6.png

2009-10-19T05:51:07Z

Gdjones:

File:09-240 HW2-5.png

2009-10-19T05:50:56Z

Gdjones:

File:09-240 HW2-4.png

2009-10-19T05:50:45Z

Gdjones:

File:09-240 HW2-3.png

2009-10-19T05:50:34Z

Gdjones:

File:09-240 HW2-2.png

2009-10-19T05:50:22Z

Gdjones:

File:09-240 HW2-1.png

2009-10-19T05:50:10Z

Gdjones:

File:09-240 HW2-0.png

2009-10-19T05:49:58Z

Gdjones:

Talk:09-240:HW1 Solution

2009-10-13T04:48:25Z

Gdjones:

Added solutions for questions 1 and 2, will add rest later tonight sometime.

UPDATE: Apologies for the delay, will upload answers to 3 and 4 by the end of the week.

Template:09-240/Navigation

2009-10-10T20:15:20Z

Gdjones:

{| cellpadding="0" cellspacing="0" style="clear: right; float: right"
|- align=right
|<div class="NavFrame"><div class="NavHead">[[09-240]]/[[Template:09-240/Navigation|Navigation Panel]]  </div>
<div class="NavContent">
{| border="1px" cellpadding="1" cellspacing="0" width="220" style="margin: 0 0 1em 0.5em; font-size: small"
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 7
|<s>Tue</s>, [[09-240/About This Class|About]], [[09-240/Classnotes for Thursday September 10|Thu]]
|- align=left
|align=center|2
|Sep 14
|[[09-240/Classnotes for Tuesday September 15|Tue]], [[09-240:HW1|HW1]], [[09-240:HW1 Solution|HW1 Solution]], [[09-240/Classnotes for Thursday September 17|Thu]]
|- align=left
|align=center|3
|Sep 21
|[[09-240/Classnotes for Tuesday September 22|Tue]], [[09-240:HW2|HW2]],[[09-240:HW2 Solution|HW2 Solution]], [[09-240/Classnotes for Thursday September 24|Thu]], [[09-240/Class Photo|Photo]]
|- align=left
|align=center|4
|Sep 28
|[[09-240/Classnotes for Tuesday September 29|Tue]], [[09-240:HW3|HW3]], [[09-240/Classnotes for Thursday October 1|Thu]]
|- align=left
|align=center|5
|Oct 5
|[[09-240:HW4|HW4]]
|- align=left
|align=center|6
|Oct 12
|
|- align=left
|align=center|7
|Oct 19
|[[09-240:HW5|HW5]], [[09-240/Term Test|Term Test on Thu]]
|- align=left
|align=center|8
|Oct 26
|[[09-240:HW6|HW6]]
|- align=left
|align=center|9
|Nov 2
|
|- align=left
|align=center|10
|Nov 9
|[[09-240:HW7|HW7]], <s>Thu</s>
|- align=left
|align=center|11
|Nov 16
|[[09-240:HW8|HW8]]
|- align=left
|align=center|12
|Nov 23
|[[09-240:HW9|HW9]]
|- align=left
|align=center|13
|Nov 30
|[[09-240/On The Final Exam|On the final]]
|- align=left
|align=center|F
|
|[[09-240/The Final Exam|Final]]
|- align=left
|colspan=3 align=center|[[09-240/To do|To Do List]]
|- align=left
|colspan=3 align=center|[[09-240/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:09-240-ClassPhoto.jpg|180px]]<br>[[09-240/Class Photo|Add your name / see who's in!]]
|}
</div></div>
|}

Talk:09-240:HW1 Solution

2009-10-10T20:12:23Z

Gdjones:

Added solutions for questions 1 and 2, will add rest later tonight sometime.

09-240/Classnotes for Thursday September 17

2009-09-17T23:33:41Z

Gdjones:

{{09-240/Navigation}}

==NSERC - CMS Math in Moscow Scholarships==

The Natural Sciences and Engineering Research Council (NSERC) and the Canadian Mathematical Society (CMS) support scholarships at $9,000 each. Canadian students registered in a mathematics or computer science program are eligible.

The scholarships are to attend a semester at the small elite Moscow Independent University.

Math in Moscow Program http://www.mccme.ru/mathinmoscow/

Application details http://www.cms.math.ca/Scholarships/Moscow

For additional information please see your department or call the CMS at 613-733-2662.

Deadline September 30, 2009 to attend the Winter 2010 semester.

==Some links==
* Dori Eldar's work on "mechanical computations": {{Home Link|People/Eldar/thesis/linkfunc.htm|Machines as Calculating Devices}} and {{Home Link|People/Eldar/thesis/squaring.htm|Computing the function <math>W=Z^2</math> the hard way}}.
* The "Dimensions" video on "Nombres complexes", is at http://dimensions-math.org/Dim_reg_AM.htm (and then go to "Dimensions_5".

==Class notes for today==

• Convention for today: <math>x,y,a,b,c,d,...</math> will be real numbers; <math>z,w,u,v,...</math> will be complex numbers

Dream: Find a field <math>\mathbb C</math> that contains <math>\mathbb R</math> and also contains an element <math>i</math> such that <math>i^2=-1</math>

'''Implications:'''

• <math>b \in \mathbb R \Rightarrow bi \in \mathbb C</math>

• <math>a \in \mathbb R \Rightarrow a+bi \in \mathbb C</math>

• <math>c,d \in \mathbb R \Rightarrow c+di \in \mathbb C</math>

• <math>\Rightarrow (a+bi)+(c+di)</math> must be in <math>\mathbb C</math>

:<math>=(a+c)+(bi+di)</math>

:<math>=(a+c)+(b+d)i</math>

:<math>=e+fi</math>

<math>(a+bi)(c+di)=(a+c)+(b+d)i</math>

:<math>=a(c+di)+bi(c+di)</math>

:<math>=ac+adi+bic+bidi</math>

:<math>=ac+bdi^2 + adi+bci</math>

:<math>=(ac-bd)+(ad+bc)i</math>

:<math>=e+fi</math>

:<math>0_C=0+0i</math>

:<math>1_C=1+0i</math>

:<math>(a+bi)+(c+di)=0+0i</math>

:<math>-(a+bi)=(-a)+(-b)i</math>

:<math>a+bi \neq 0 \Rightarrow (a,b) \neq 0</math>

• Find another element of <math>\mathbb C</math>, <math>x+yi</math> such that <math>(a+bi)(x+yi)=(1+0i)</math>

:<math>(a+bi)(x+yi)=(ax-by)+(ay+bx)i=1+0i</math>

:<math>ax-by=1</math> (1)

:<math>bx+ay=0</math> (2)

:<math>a,b</math> are given

:<math>x,y</math> unknowns

• <math>b \times (1)</math> <math>abx-b^2y=b</math>

• <math>a \times (2)</math> <math>abx+a^2y=0</math>

:<math>\Rightarrow a^{2}y+b^{2}y=-b</math>

:<math>y=\frac{-b}{a^{2}+b^{2}}</math>

:<math>x=\frac{a}{a^{2}+b^{2}}</math>

• (Note: We can divide since we assumed that <math>(a,b) \neq 0</math>

:<math>(a+bi)^{-1}=\frac{a}{a^{2}+b^{2}}+\frac{-b}{a^{2}+b^{2}}i=\frac{a-bi}{a^{2}+b^{2}}=\frac{\overline{a+bi}}{|a+bi|^{2}}=\frac{\mbox{conjugate}}{\mbox{norm squared }}</math>

Def: Let <math>\mathbb C</math> be the set of all pairs of real numbers <math>{(a,b)}={a+bi}</math>

with <math>+: (a,b)+(c,d)=(a+c,b+d)</math>

:<math>(a+bi)+(c+di)=(a+c)+(b+d)i</math>

<math>\times :(a+bi)(c+di)=</math>...you know what

• 0 = you know what

• 1 = you know what

'''Theorem:'''

#:<math>\mathbb C</math> is a field

#:<math>(0+1i)^2=(0,1)^2=i^2=-1_{C}=(-1,0)</math>

#:<math>\mathbb R \rightarrow \mathbb C</math> by <math>a \rightarrow a+0i</math>

Proof: <math>F_{1},F_{2},F_{3},...</math>

'''Example:''' <math>F_{5}</math> (distributivity)

• Show that <math>z(u+v)=zu+zv</math>

Let <math>z=(a+bi)</math>

:<math>u=(c+di)</math>

:<math>v=(e+fi)</math>

When <math>a,b,c,d,e,f \in \mathbb R</math>

:<math>(a+bi)[(c+di)+(e+fi)]=(a+bi)(c+di)+(a+bi)(e+fi)=(ac-bd)+\ldots</math>

• NEXT WEEK: Complex numbers have geometric meaning, geometric interpretation (waves)

09-240/Classnotes for Thursday September 17

2009-09-17T23:18:42Z

Gdjones:

{{09-240/Navigation}}

==NSERC - CMS Math in Moscow Scholarships==

The Natural Sciences and Engineering Research Council (NSERC) and the Canadian Mathematical Society (CMS) support scholarships at $9,000 each. Canadian students registered in a mathematics or computer science program are eligible.

The scholarships are to attend a semester at the small elite Moscow Independent University.

Math in Moscow Program http://www.mccme.ru/mathinmoscow/

Application details http://www.cms.math.ca/Scholarships/Moscow

For additional information please see your department or call the CMS at 613-733-2662.

Deadline September 30, 2009 to attend the Winter 2010 semester.

==Some links==
* Dori Eldar's work on "mechanical computations": {{Home Link|People/Eldar/thesis/linkfunc.htm|Machines as Calculating Devices}} and {{Home Link|People/Eldar/thesis/squaring.htm|Computing the function <math>W=Z^2</math> the hard way}}.
* The "Dimensions" video on "Nombres complexes", is at http://dimensions-math.org/Dim_reg_AM.htm (and then go to "Dimensions_5".

==Class notes for today==

• Convention for today: <math>x,y,a,b,c,d,...</math> will be real numbers; <math>z,w,u,v,...</math> will be complex numbers

• Dream: Find a field <math>\mathbb C</math> that contains <math>\mathbb R</math> and also contains an element <math>i</math> such that <math>i^2=-1</math>

'''Implications:'''

• <math>b \in \mathbb R \Rightarrow bi \in \mathbb C</math>

• <math>a \in \mathbb R \Rightarrow a+bi \in \mathbb C</math>

• <math>c,d \in \mathbb R \Rightarrow c+di \in \mathbb C</math>

• <math>\Rightarrow (a+bi)+(c+di)</math> must be in <math>\mathbb C</math>

• <math>=(a+c)+(bi+di)</math>

• <math>=(a+c)+(b+d)i</math>

• <math>=e+fi</math>

• <math>(a+bi)(c+di)=(a+c)+(b+d)i</math>

• <math>=a(c+di)+bi(c+di)</math>

• <math>=ac+adi+bic+bidi</math>

• <math>=ac+bdi^2 + adi+bci</math>

• <math>=(ac-bd)+(ad+bc)i</math>

• <math>=e+fi</math>

• <math>0_C=0+0i</math>

• <math>1_C=1+0i</math>

• <math>(a+bi)+(c+di)=0+0i</math>

• <math>-(a+bi)=(-a)+(-b)i</math>

• <math>a+bi \neq 0 \Rightarrow (a,b) \neq 0</math>

• Find another element of <math>\mathbb C</math>, <math>x+yi</math> such that <math>(a+bi)(x+yi)=(1+0i)</math>

• <math>(a+bi)(x+yi)=(ax-by)+(ay+bx)i=1+0i</math>

• <math>ax-by=1</math> (1)

• <math>bx+ay=0</math> (2)

• <math>a,b</math> are given

• <math>x,y</math> unknowns

• <math>b \times (1)</math> <math>abx-b^2y=b</math>

• <math>a \times (2)</math> <math>abx+a^2y=0</math>

• <math>\Rightarrow a^{2}y+b^{2}y=-b</math>

• <math>y=\frac{-b}{a^{2}+b^{2}}</math>

• <math>x=\frac{a}{a^{2}+b^{2}}</math>

• (Note: We can divide since we assumed that <math>(a,b) \neq 0</math>

• <math>(a+bi)^{-1}=\frac{a}{a^{2}+b^{2}}+\frac{-b}{a^{2}+b^{2}}i=\frac{a-bi}{a^{2}+b^{2}}=\frac{\overline{a+bi}}{|a+bi|^{2}}=\frac{\mbox{conjugate}}{\mbox{norm squared }}</math>

• Def: Let <math>\mathbb C</math> be the set of all pairs of real numbers <math>{(a,b)}={a+bi}</math>

• with <math>+: (a,b)+(c,d)=(a+c,b+d)</math>

• <math>(a+bi)+(c+di)=(a+c)+(b+d)i</math>

• <math>x:(a+bi)(c+di)=</math>...you know what

• 0 = you know what

• 1 = you know what

• Thm:

• 1. <math>\mathbb C</math> is a field

• 2. <math>(0+1i)^2=(0,1)^2=i^2=-1_{C}=(-1,0)</math>

• 3. <math>\mathbb R \rightarrow \mathbb C</math> by <math>a \rightarrow a+0i</math>

• Proof: <math>F_{1},F_{2},F_{3},...</math>

• Example: <math>F_{5}</math> (distributivity)

• Show that <math>z(u+v)=zu+zv</math>

• Let <math>z=(a+bi)</math>

• <math>u=(c+di)</math>

• <math>v=(e+fi)</math>

• When <math>a,b,c,d,e,f \in \mathbb R</math>

• <math>(a+bi)[(c+di)+(e+fi)]=(a+bi)(c+di)+(a+bi)(e+fi)=(ac-bd)+\ldots</math>

• NEXT WEEK: Complex numbers have geometric meaning, geometric interpretation (waves)

09-240/Classnotes for Thursday September 17

2009-09-17T21:59:00Z

Gdjones:

Talk:09-240/Classnotes for Thursday September 10

2009-09-12T17:36:26Z

Gdjones:

I can't get the "alignat" tags working correctly. They're what I would prefer to use for systems of equations, instead of having each individual equation enclosed in "math" tags. They seem to work fine on Wikipedia (see http://en.wikipedia.org/w/index.php?title=System_of_linear_equations&action=edit).

Template:09-240/Navigation

2009-09-12T17:11:40Z

Gdjones:

09-240/Classnotes for Thursday September 10

2009-09-12T17:05:36Z

Gdjones:

== 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)