Drorbn - User contributions [en]

12-240/Class Photo

2012-09-27T18:33:51Z

Sina.zoghi: /* Who We Are... */

Our class on September 25, 2012:

[[Image:12-240-ClassPhoto.jpg|thumb|centre|800px|Class Photo: click to enlarge]]
{{12-240/Navigation}}

Please identify yourself in this photo! There are two ways to do that:

* [[Special:Userlogin|Log in]] to this Wiki and edit this page. Put your name, userid, email address and location in the picture in the alphabetical list below.
* Send [[User:Drorbn|Dror]] an email message with this information.

The first option is more fun but less private.

===Who We Are...===

{| align=center border=1 cellspacing=0
|-
!First name
!Last name
!User ID
!Email
!Place in photo
!Comments

{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn@ math. toronto. edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank. For better line-breaking, leave a space next to the "@" in email addresses.}}
{{Photo Entry|last=Frailich|first=Rebecca|userid=Rebecca.frailich|email=rebecca.frailich@ mail.utoronto.ca|location=Last row, in between two guys standing at the back (one in red, one in black) |comments=}}
{{Photo Entry|last=Hoover|first=Ken|userid=Khoover|email=ken.hoover@ mail.utoronto.ca|location=First row, fourth from the right.|comments=}}
{{Photo Entry|last=Klingspor|first=Josefine|userid=Josefine|email=josefine. klingspor@ mail. utoronto. ca|location=First row, second from left.|comments=}}
{{Photo Entry|last=Liu|first=Zhaowei|userid=tod|email=tod.liu@mail.utoronto .ca|location=First row, third from the right|comments=}}
{{Photo Entry|last=Millson|first=Richard|userid=Richardm|email=r.millson@ mail. utoronto. ca|location=Seventh row from the front, fourth from the right, blue sweater|comments=}}
{{Photo Entry|last=Vicencio-Heap|first=Felipe|userid=Heapfeli|email=felipe. vicencio. heap@ mail. utoronto. ca|location=Second row from the front, furthest to the right.|comments=}}
{{Photo Entry|last=Wamer|first=Kyle|userid=kylewamer|email=kyle. wamer @ mail. utoronto. ca|location=Second row, fifth from the left in the red shirt.|comments=}}
{{Photo Entry|last=Yang|first=Chen|userid=chen|email=neochen. yang@ mail. utoronto. ca|location=sixth row, first from the right in the black pull-over.|comments=}}
{{Photo Entry|last=Zhang|first=BingZhen|userid=Zetalda|email=bingzhen.zhang@ mail. utoronto. ca|location=Second last row, third from left.|comments=}}
{{Photo Entry|last=Zoghi|first=Sina|userid=sina.zoghi|email=sina.zoghi@ utoronto .ca|location=First row, leftest left.|comments=}}

|}

12-240/Class Photo

2012-09-26T04:44:45Z

Sina.zoghi: /* Who We Are... */

12-240/Classnotes for Tuesday September 11

2012-09-13T17:34:28Z

Sina.zoghi: /* Examples of Fields */

{{12-240/Navigation}}

In this course, we will be focusing on both a practical side and a theoretical side.

== Practical Side ==

1.
Solving complicated systems of equations, such as:

:<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>

2.
We can turn the above into a matrix!
:<math>
\begin{pmatrix}
5 & -2 & 1 \\
-1 & 1 & -1 \\
2 & 9 & -3
\end{pmatrix} = A
</math>

== Theory Side ==

3.
"The world doesn't come with coordinates."
We will learn to do all of this in a coordinate-free way.

4.
We'll learn to do all of this over other sets of numbers and fields.

== Hidden Agenda ==

5.
We'll learn the process of pure mathematics by doing it.
We'll learn about:
*Abstraction
*Generalization
*Definitions
*Theorems
*Proofs
*Notation
*Logic

----

A number system has specific properties of the real numbers.

== Real Numbers ==

A set, <math>\mathbb{R}\!</math>, with:
*Two binary operations, addition and multiplication.
*Two special elements, 0 and 1.

The real numbers have some special properties:

=== Commutative Laws ===
<math>\mathbb{R}1</math>
:<math>\forall\ a, b\ \epsilon\ \mathbb{R} \quad a+b = b+a\!</math>
:<math>\forall\ a, b\ \epsilon\ \mathbb{R} \quad ab = ba\!</math>

=== Associative Laws ===
<math>\mathbb{R}2</math>
:<math>\forall\ a, b, c\ \epsilon\ \mathbb{R} \quad (a + b) + c = a + (b + c)\!</math>
:<math>\forall\ a, b, c\ \epsilon\ \mathbb{R} \quad (ab) \cdot c = a \cdot (bc)\!</math>

=== Existence of "Units" ===
<math>\mathbb{R}3</math>
:<math>\forall\ a\ \epsilon\ \mathbb{R} \quad a + 0 = a\!</math>
:<math>\forall\ a\ \epsilon\ \mathbb{R} \quad a \cdot 1 = a\!</math>

=== Existence of Negatives/Inverses ===
<math>\mathbb{R}4</math>
:<math>\forall\ a\ \epsilon\ \mathbb{R}\ \exists\ b\ \epsilon\ \mathbb{R} \quad a + b = 0\!</math>
:<math>\forall\ a\ \epsilon\ \mathbb{R}\ \exists\ b\ \epsilon\ \mathbb{R} \quad a \cdot b = 1\!</math>

=== Distributive Law ===
<math>\mathbb{R}5</math>
:<math>\forall\ a, b, c\ \epsilon\ \mathbb{R} \quad (a+b) \cdot c = ac + bc\!</math>

==== An example of a property that follows from the earlier ones: ====
:<math>a^2 - b^2 = (a + b)(a - b)\!</math>
We can define subtraction and squaring from the properties covered above.

==== An example of a property that does not follow from the earlier ones: ====
The existence of square roots:

:<math>\forall\ a\ \exists\ b\ \quad b^2 = a\ or\ b^2 = -a\!</math>
We can construct a set that has all of the 5 properties described above, but for which this property does not follow.

This set is the rational numbers.

There is a rational number <math>a\!</math> where there is no <math>b</math> in the set.

This is because<math>\sqrt{2}</math> is irrational.

== Fields ==

The properties we have been discussing aren't restricted to only the real numbers.

They are also properties of:
*Rational numbers
*Complex numbers
*Others

Let's construct an abstract universe where these properties hold.

Definition: Field
*A field is a set, <math>\mathbb{F}</math>, with:
**Two binary operations, addition and multiplication.
**Two special elements, 0 and 1, where 0 does not equal 1.
**All of the above mentioned properties hold.

Now, instead of speaking of <math>\mathbb{R}1,\ \mathbb{R}2,\ \mathbb{R}3,\ \mathbb{R}4,\ \mathbb{R}5</math>, we can speak of <math>\mathbb{F}1,\ \mathbb{F}2,\ \mathbb{F}3,\ \mathbb{F}4,\ \mathbb{F}5</math>.

We have abstracted!

== Examples of Fields ==
*Take <math>\mathbb{F} = \mathbb{R}</math>

*Take <math>\mathbb{F} = \mathbb{Q}</math> (Rational numbers)

*The complex numbers. <math>\mathbb{C} = \lbrace a + bi \quad a, b\ \epsilon\ \mathbb{R} \rbrace</math>

The above fields have an infinite number of elements. We can also have finite fields:

*<math>\mathbb{F} = \mathbb{F}_2 = \mathbb{Z}/2 = \lbrace 0, 1 \rbrace</math>
**There are only 2 elements.
**You can think of 0 as even and 1 as odd, which will help you construct the tables below.
**You can also think of the results below as the remainder of the operations when divided by 2. (mod 2)

::{| border="1"
! scope="col" | +
! scope="col" | 0
! scope="col" | 1
|-
! scope="row" | 0
| 0 || 1
|-
! scope="row" | 1
| 1
| 0
|}

::{| border="1"
! scope="col" | x
! scope="col" | 0
! scope="col" | 1
|-
! scope="row" | 0
| 0 || 0
|-
! scope="row" | 1
| 0
| 1
|}

*Question: Are addition and multiplication defined here only arbitrary? Can we define many other ways to add or multiply, for a set, as long as the result satisfies F1-5 to show that F is indeed a field?

*<math>\mathbb{F} = \mathbb{F}_3 = \mathbb{Z}/3 = \lbrace 0, 1, 2 \rbrace</math>
::{| border="1"
! scope="col" | +
! scope="col" | 0
! scope="col" | 1
! scope="col" | 2
|-
! scope="row" | 0
| 0 || 1 || 2
|-
! scope="row" | 1
| 1
| 2
| 0
|-
! scope="row" | 2
| 2
| 0
| 1
|}

::{| border="1"
! scope="col" | x
! scope="col" | 0
! scope="col" | 1
! scope="col" | 2
|-
! scope="row" | 0
| 0 || 0 || 0
|-
! scope="row" | 1
| 0
| 1
| 2
|-
! scope="row" | 2
| 0
| 2
| 1
|}

*<math>\mathbb{F} = \mathbb{F}_5 = \mathbb{Z}/5 = \lbrace 0, 1, 2, 3, 4 \rbrace</math>
**Not going to bother making the tables here.

*<math>\mathbb{F}_4</math> is '''not a field.'''
**It does not have the property <math>\mathbb{R}5</math>.
:::<math>2 \cdot 0 = 0</math>
:::<math>2 \cdot 1 = 2</math>
:::<math>2 \cdot 2 = 0</math>
:::<math>2 \cdot 3 = 2</math>
:::We never got a 1.

*If the subscript is a prime number, it is a field.

----

Theorem:

1.

:Let F be a field.
:<math>\forall a, b, c\ \epsilon\ \mathbb{F} \quad a+b = c+b</math>
::"Cancellation Lemma"
:<math>\Rightarrow a = c</math>

2.

:<math>ab = cb, b \ne 0</math>
:<math>\Rightarrow a = c</math>

We'll cover 3-11 next class!

Proof of 1:

:Let <math>a, b, c\ \epsilon\ \mathbb{F}</math>
:by <math>\mathbb{F} 4\ \exists\ d\ \epsilon\ \mathbb{F} \quad b+d = 0</math>
:so with this d, <math>a+b = c+b\!</math>
:and so <math>(a+b)+d = (c+b)+d\!</math>
:so by <math>\mathbb{F} 2</math>, <math>a+(b+d) = c+(b+d)\!</math>
:so <math>a+0 = c+0\!</math>
:so by <math>\mathbb{F} 3 \quad a = c\!</math>
:<math>\Box</math>

==Scanned Notes by [[User:Sina.zoghi|Sina.zoghi]]==
[[User:Sina.zoghi|Sina.zoghi]] - Thanks for improving on the previously-uploaded scans - though there is still too much "white space" around each page. It is probably not worth your while to fix it for these scans, but it is something to keep in mind for later ones. [[User:Drorbn|Drorbn]] 10:50, 13 September 2012 (EDT)

[[Image:12-240-Sept11-Page1.jpeg|250px]]
[[Image:12-240-Sept11-Page2.jpeg|250px]]
[[Image:12-240-Sept11-Page3.jpeg|250px]]
[[Image:12-240-Sept11-Page4.jpeg|250px]]
[[Image:12-240-Sept11-Page5.jpeg|250px]]
[[Image:12-240-Sept11-Page6.jpeg|250px]]
[[Image:12-240-Sept11-Page7.jpeg|250px]]
[[Image:12-240-Sept11-Page8.jpeg|250px]]
[[Image:12-240-Sept11-Page9.jpeg|250px]]

File:12-240-Sept11-Page9.jpeg

2012-09-12T02:34:15Z

Sina.zoghi:

File:12-240-Sept11-Page8.jpeg

2012-09-12T02:34:02Z

Sina.zoghi:

File:12-240-Sept11-Page7.jpeg

2012-09-12T02:33:54Z

Sina.zoghi:

File:12-240-Sept11-Page6.jpeg

2012-09-12T02:33:47Z

Sina.zoghi:

File:12-240-Sept11-Page5.jpeg

2012-09-12T02:33:37Z

Sina.zoghi:

File:12-240-Sept11-Page4.jpeg

2012-09-12T02:33:24Z

Sina.zoghi:

File:12-240-Sept11-Page3.jpeg

2012-09-12T02:33:16Z

Sina.zoghi:

File:12-240-Sept11-Page2.jpeg

2012-09-12T02:33:07Z

Sina.zoghi:

File:12-240-Sept11-Page1.jpeg

2012-09-12T02:32:58Z

Sina.zoghi: