Drorbn - User contributions [en]

12-240/Navigation

2012-12-08T23:09:02Z

Oguzhancan:

<noinclude>Back to [[12-240]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 10
|[[12-240/About This Class|About This Class]], [[12-240/Classnotes for Tuesday September 11|Tuesday]], [[12-240/Classnotes for Thursday September 13|Thursday]]
|- align=left
|align=center|2
|Sep 17
|[[12-240/Homework Assignment 1|HW1]], [[12-240/Classnotes for Tuesday September 18|Tuesday]], [[12-240/Classnotes for Thursday September 20|Thursday]], [[12-240/HW1 Solutions|HW1 Solutions]]
|- align=left
|align=center|3
|Sep 24
|[[12-240/Homework Assignment 2|HW2]], [[12-240/Classnotes for Tuesday September 25|Tuesday]], [[12-240/Class Photo|Class Photo]], [[12-240/Classnotes for Thursday September 27|Thursday]]
|- align=left
|align=center|4
|Oct 1
|[[12-240/Homework Assignment 3|HW3]], [[12-240/Classnotes for Tuesday October 2|Tuesday]], [[12-240/Classnotes for Thursday October 4|Thursday]]
|- align=left
|align=center|5
|Oct 8
|[[12-240/Homework Assignment 4|HW4]], [[12-240/Classnotes for Tuesday October 09|Tuesday]], [[12-240/Classnotes for Thursday October 11|Thursday]]
|- align=left
|align=center|6
|Oct 15
|[[12-240/Classnotes for Tuesday October 16|Tuesday]], [[12-240/Classnotes for Thursday October 18|Thursday]]
|- align=left
|align=center|7
|Oct 22
|[[12-240/Homework Assignment 5|HW5]], [[12-240/Classnotes for Tuesday October 23|Tuesday]], [[12-240/Term Test|Term Test]] was on Thursday. [[12-240/HW5 Solutions|HW5 Solutions]]
|- align=left
|align=center|8
|Oct 29
|[[12-240/Linear Algebra - Why We Care|Why LinAlg?]], [[12-240/Homework Assignment 6|HW6]], [[12-240/Classnotes for Tuesday October 30|Tuesday]], [[12-240/Classnotes for Thursday November 1|Thursday]], Nov 4 is the last day to drop this class
|- align=left
|align=center|9
|Nov 5
|[[12-240/Classnotes for Tuesday November 6|Tuesday]], [[12-240/Classnotes for Thursday November 8|Thursday]]
|- align=left
|align=center|10
|Nov 12
|Monday-Tuesday is UofT November break, [[12-240/Homework Assignment 7|HW7]], [[12-240/Classnotes for Tuesday November 15|Thursday]]
|- align=left
|align=center|11
|Nov 19
|[[12-240/Homework Assignment 8|HW8]], [[12-240/Classnotes for Tuesday November 20|Tuesday]],[[12-240/Classnotes for Thursday October 22|Thursday]]
|- align=left
|align=center|12
|Nov 26
|[[12-240/Homework Assignment 9|HW9]], [[12-240/Classnotes for Tuesday November 27|Tuesday]] , [[12-240/Classnotes for Thursday November 29|Thursday]]
|- align=left
|align=center|13
|Dec 3
|[[12-240/Classnotes for Tuesday December 4|Tuesday]] UofT Fall Semester ends Wednesday
|- align=left
|align=center|F
|Dec 10
|[[12-240/The Final Exam|The Final Exam]] (time, place, style, office hours times)
|- align=left
|colspan=3 align=center|[[12-240/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:12-240-ClassPhoto.jpg|310px]] [[12-240/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:12-240-Splash.png|310px]]
|}

12-240/HW5 Solutions

2012-12-08T23:07:23Z

Oguzhancan:

[[Image:12-240-Assignment5 solutions.pdf]]

12-240/Navigation

2012-12-08T19:31:19Z

Oguzhancan:

<noinclude>Back to [[12-240]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 10
|[[12-240/About This Class|About This Class]], [[12-240/Classnotes for Tuesday September 11|Tuesday]], [[12-240/Classnotes for Thursday September 13|Thursday]]
|- align=left
|align=center|2
|Sep 17
|[[12-240/Homework Assignment 1|HW1]], [[12-240/Classnotes for Tuesday September 18|Tuesday]], [[12-240/Classnotes for Thursday September 20|Thursday]], [[12-240/HW1 Solutions|HW1 Solutions]]
|- align=left
|align=center|3
|Sep 24
|[[12-240/Homework Assignment 2|HW2]], [[12-240/Classnotes for Tuesday September 25|Tuesday]], [[12-240/Class Photo|Class Photo]], [[12-240/Classnotes for Thursday September 27|Thursday]]
|- align=left
|align=center|4
|Oct 1
|[[12-240/Homework Assignment 3|HW3]], [[12-240/Classnotes for Tuesday October 2|Tuesday]], [[12-240/Classnotes for Thursday October 4|Thursday]]
|- align=left
|align=center|5
|Oct 8
|[[12-240/Homework Assignment 4|HW4]], [[12-240/Classnotes for Tuesday October 09|Tuesday]], [[12-240/Classnotes for Thursday October 11|Thursday]]
|- align=left
|align=center|6
|Oct 15
|[[12-240/Classnotes for Tuesday October 16|Tuesday]], [[12-240/Classnotes for Thursday October 18|Thursday]]
|- align=left
|align=center|7
|Oct 22
|[[12-240/Homework Assignment 5|HW5]], [[12-240/Classnotes for Tuesday October 23|Tuesday]], [[12-240/Term Test|Term Test]] was on Thursday.
|- align=left
|align=center|8
|Oct 29
|[[12-240/Linear Algebra - Why We Care|Why LinAlg?]], [[12-240/Homework Assignment 6|HW6]], [[12-240/Classnotes for Tuesday October 30|Tuesday]], [[12-240/Classnotes for Thursday November 1|Thursday]], Nov 4 is the last day to drop this class
|- align=left
|align=center|9
|Nov 5
|[[12-240/Classnotes for Tuesday November 6|Tuesday]], [[12-240/Classnotes for Thursday November 8|Thursday]]
|- align=left
|align=center|10
|Nov 12
|Monday-Tuesday is UofT November break, [[12-240/Homework Assignment 7|HW7]], [[12-240/Classnotes for Tuesday November 15|Thursday]]
|- align=left
|align=center|11
|Nov 19
|[[12-240/Homework Assignment 8|HW8]], [[12-240/Classnotes for Tuesday November 20|Tuesday]],[[12-240/Classnotes for Thursday October 22|Thursday]]
|- align=left
|align=center|12
|Nov 26
|[[12-240/Homework Assignment 9|HW9]], [[12-240/Classnotes for Tuesday November 27|Tuesday]] , [[12-240/Classnotes for Thursday November 29|Thursday]]
|- align=left
|align=center|13
|Dec 3
|[[12-240/Classnotes for Tuesday December 4|Tuesday]] UofT Fall Semester ends Wednesday
|- align=left
|align=center|F
|Dec 10
|[[12-240/The Final Exam|The Final Exam]] (time, place, style, office hours times)
|- align=left
|colspan=3 align=center|[[12-240/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:12-240-ClassPhoto.jpg|310px]] [[12-240/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:12-240-Splash.png|310px]]
|}

12-240/HW1 Solutions

2012-12-08T19:29:27Z

Oguzhancan:

== Assignment 1 Solutions ==
[[Image:12-240-Assignment1 solutions.pdf]]

12-240

2012-12-08T07:59:07Z

Oguzhancan:

__NOEDITSECTION__
__NOTOC__
{{12-240/Navigation}}
==Algebra I==
===Department of Mathematics, University of Toronto, Fall 2012===

{{12-240/Crucial Information}}

===Text===
Our main text book will be ''Linear Algebra'' (fourth edition) by Friedberg, Insel and Spence, ISBN 0-13-008451-4; it is a required reading. An errata is at http://www.math.ilstu.edu/linalg/errata.html.

===Further Resources===

* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]

* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].

* [http://wiki.math.toronto.edu/TorontoMathWiki/index.php/2011F_MAT240_Algebra_I Marco Gualtieri's 2011 Math 240 web site].

* [http://wiki.math.toronto.edu/TorontoMathWiki/index.php/10-240 Marco Gualtieri's 2010 Math 240 web site].

* [[09-240|My 2009 Math 240 web site]].

* [http://www.math.toronto.edu/murnaghan/courses/mat240/index.html The 2008 MAT240 site].

* [[06-240|My 2006 Math 240 web site]].

* My {{Pensieve Link|Classes/12-240/|12-240 notebook}}.

{{12-240:Dror/Students Divider}}

* Mathematics at Google publication: [http://research.google.com/pubs/pub38331.html (abstract)] [http://research.google.com/pubs/archive/38331.pdf (slides)].
* [http://drorbn.net/index.php?title=12-240/Proofs_in_Vector_Spaces ProofWiki]
===Online Discussion Platform===
* [http://mat240.wordpress.com/ Click to go to the online discussion platform]

12-240/Proofs in Vector Spaces

2012-12-08T07:35:30Z

Oguzhancan: /* Theorems & Proofs */

== Important Note About This Page ==

This page is intended for sharing/clarifying proofs. Here, you might add a proof, correct a proof, or request more detailed explanation of some specific parts of given proofs. To request an explanation for a proof, you may put a sign at that specific part by editing this page. For example:

...generating set as <math>\beta</math>, so <math>k = |L|\leq |\beta| = dimV</math> '''***(explanation needed, why? [or your question])***''' since <math>L</math> is a some linearly independent...

== Theorems & Proofs ==

Theorem: Let <math>W</math> be a subspace of a finite dimensional vector space <math>V</math>. Then <math>W</math> is finite dimensional and <math>dimW \leq dimV</math>

Proof: Let <math>\beta</math> be a basis for <math>V</math>. Then we know that <math>\beta</math> is a finite set since <math>V</math> is a finite dimensional. Then, for given a subspace <math>W</math>, let us construct a linearly independent set <math>L</math> by adding vectors from <math>W</math> such that <math>L=\{w_1,w_2, ... w_k\}</math> is maximally linearly independent. In other words, adding any other vector from <math>W</math> would make <math>L</math> linearly dependent. Here, L has to be a finite set by the Replacement Theorem, if we choose the generating set as <math>\beta</math>, so <math>k = |L|\leq |\beta| = dimV</math> since <math>L</math> is a some linearly independent subset of <math>V</math>. Now we want to show that <math>L</math> is a basis for <math>W</math>. Since <math>L</math> is linearly independent, it suffices to show that <math>span(L)=W</math>. Suppose not:<math>span(L)\neq W</math>. (We know that <math>L \subseteq span(L) \subseteq W</math> since <math>L</math> is made of vectors from <math>W</math>.) Then <math>\exists w_a \in W : w_a \notin span(L)</math> But this means <math>span(L)\cup \{w_a\}</math> is linearly independent, which contradicts with maximally linearly independence of <math>L</math>. Therefore <math>span(L)=W</math> and hence, <math>L</math> is a basis for <math>W</math>

Replacement Theorem: Let <math>V</math> be a vector space generated by <math>G</math> (perhaps linearly dependent) where <math>|G|=n</math> and let <math>L</math> be a linearly independent subset of <math>V</math> such that <math>|L|=m</math>. Then <math>m \leq n</math> and there exists a subset <math>H</math> of <math>G</math> with <math>|H| = n-m</math> and <math>span(H \cup L)=V</math>.

Proof: We will prove by induction hypothesis on <math>m=|L|</math>:

For <math>m = 0</math>: <math>L = \emptyset</math>, <math>0 \leq n</math> and <math>H=G</math> so, <math>span(H \cup L) = span(H) = span(G) = V</math>

Now, suppose true for <math>m</math>:

12-240/Proofs in Vector Spaces

2012-12-08T07:05:18Z

Oguzhancan:

12-240/Navigation

2012-12-01T01:36:05Z

Oguzhancan:

<noinclude>Back to [[12-240]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 10
|[[12-240/About This Class|About This Class]], [[12-240/Classnotes for Tuesday September 11|Tuesday]], [[12-240/Classnotes for Thursday September 13|Thursday]]
|- align=left
|align=center|2
|Sep 17
|[[12-240/Homework Assignment 1|HW1]], [[12-240/Classnotes for Tuesday September 18|Tuesday]], [[12-240/Classnotes for Thursday September 20|Thursday]]
|- align=left
|align=center|3
|Sep 24
|[[12-240/Homework Assignment 2|HW2]], [[12-240/Classnotes for Tuesday September 25|Tuesday]], [[12-240/Class Photo|Class Photo]], [[12-240/Classnotes for Thursday September 27|Thursday]]
|- align=left
|align=center|4
|Oct 1
|[[12-240/Homework Assignment 3|HW3]], [[12-240/Classnotes for Tuesday October 2|Tuesday]], [[12-240/Classnotes for Thursday October 4|Thursday]]
|- align=left
|align=center|5
|Oct 8
|[[12-240/Homework Assignment 4|HW4]], [[12-240/Classnotes for Tuesday October 09|Tuesday]], [[12-240/Classnotes for Thursday October 11|Thursday]]
|- align=left
|align=center|6
|Oct 15
|[[12-240/Classnotes for Tuesday October 16|Tuesday]], [[12-240/Classnotes for Thursday October 18|Thursday]]
|- align=left
|align=center|7
|Oct 22
|[[12-240/Homework Assignment 5|HW5]], [[12-240/Classnotes for Tuesday October 23|Tuesday]], [[12-240/Term Test|Term Test]] was on Thursday.
|- align=left
|align=center|8
|Oct 29
|[[12-240/Linear Algebra - Why We Care|Why LinAlg?]], [[12-240/Homework Assignment 6|HW6]], [[12-240/Classnotes for Tuesday October 30|Tuesday]], [[12-240/Classnotes for Thursday November 1|Thursday]], Nov 4 is the last day to drop this class
|- align=left
|align=center|9
|Nov 5
|[[12-240/Classnotes for Tuesday November 6|Tuesday]], [[12-240/Classnotes for Thursday November 8|Thursday]]
|- align=left
|align=center|10
|Nov 12
|Monday-Tuesday is UofT November break, [[12-240/Homework Assignment 7|HW7]], [[12-240/Classnotes for Tuesday November 15|Thursday]]
|- align=left
|align=center|11
|Nov 19
|[[12-240/Homework Assignment 8|HW8]], [[12-240/Classnotes for Tuesday November 20|Tuesday]],[[12-240/Classnotes for Thursday October 22|Thursday]]
|- align=left
|align=center|12
|Nov 26
|[[12-240/Homework Assignment 9|HW9]], [[12-240/Classnotes for Tuesday November 27|Tuesday]] , [[12-240/Classnotes for Thursday November 29|Thursday]]
|- align=left
|align=center|13
|Dec 3
|UofT Fall Semester ends Wednesday
|- align=left
|align=center|F
|Dec 10
|[[12-240/The Final Exam|The Final Exam]] (time, place, style, office hours times)
|- align=left
|colspan=3 align=center|[[12-240/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:12-240-ClassPhoto.jpg|310px]] [[12-240/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:12-240-Splash.png|310px]]
|}

12-240/Classnotes for Thursday November 29

2012-12-01T01:35:03Z

Oguzhancan:

== Lecture notes upload by [[User:Oguzhancan|Oguzhancan]] ==
{{12-240/Navigation}}

<gallery>
Image:12-240-1129-1.jpg|'''Page 1'''
Image:12-240-1129-2.jpg|'''Page 2'''
Image:12-240-1129-3.jpg|'''Page 3'''
Image:12-240-1129-4.jpg|'''Page 4'''
Image:12-240-1129-5.jpg|'''Page 5'''
</gallery>

File:12-240-1129-5.jpg

2012-12-01T01:32:06Z

Oguzhancan:

File:12-240-1129-4.jpg

2012-12-01T01:31:59Z

Oguzhancan:

File:12-240-1129-3.jpg

2012-12-01T01:31:54Z

Oguzhancan:

File:12-240-1129-2.jpg

2012-12-01T01:31:48Z

Oguzhancan:

File:12-240-1129-1.jpg

2012-12-01T01:31:39Z

Oguzhancan:

12-240/Classnotes for Tuesday October 09

2012-10-21T05:07:11Z

Oguzhancan:

{{12-240/Navigation}}
In this lecture, the professor concentrate on basics and related theorems.
== Definition of basic ==
β <math>\subset \!\,</math> V is a basic if

1/ It generates ( span) V, span β = V

2/ It is linearly independent

== theorems ==
1/ β is a basic of V iff every element of V can be written as a linear combination of elements of β in a unique way.

proof: ( in the case β is finite)

β = {u1, u2, ..., un}

(<=) need to show that β = span(V) and β is linearly independent.

The fact that β span is the fact that every element of V can be written as a linear combination of elements of β, which is given

Assume <math>\sum \!\,</math> ai.ui = 0 ai <math>\in\!\,</math> F, ui <math>\in\!\,</math> β

<math>\sum \!\,</math> ai.ui = 0 = <math>\sum \!\,</math> 0.ui

since 0 can be written as a linear combination of elements of β in a unique way, ai=0 <math>\forall\!\,</math> i

Hence β is linearly independent

(=>) every element of V can be written as a linear combination of elements of β in a unique way.

So, suppose <math>\sum \!\,</math> ai.ui = v = <math>\sum \!\,</math> bi.ui

Thus <math>\sum \!\,</math> ai.ui - <math>\sum \!\,</math> bi.ui = 0

<math>\sum \!\,</math> (ai-bi).ui = 0

β is linear independent hence (ai - bi)= 0 <math>\forall\!\,</math> i

i.e ai = bi, hence the combination is unique.

== Clarification on lecture notes ==

On page 3, we find that <math>G \subseteq span(\beta)</math> then we say <math>span(G) \subseteq span(\beta)</math>. The reason is, the Theorem 1.5 in the textbook.

Theorem 1.5: The span of any subset <math>S</math> of a vector space <math>V</math> is a subspace of <math>V</math>. Moreover, any subspace of <math>V</math> that contains <math>S</math> must also contain <math>span(S)</math>

Since <math>\beta</math> is a subset of <math>V</math>, <math>span(\beta)</math> is a subspace of <math>V</math> from the first part of the Theorem 1.5. We have shown (in the lecture notes page 3) that <math>G \subseteq span(\beta)</math>. From the "Moreover" part of Theorem 1.5, since <math>span(\beta)</math> is a subspace of <math>V</math> containing <math>G</math>, <math>span(\beta)</math> must also contain <math>span(G)</math>.

== Lecture notes scanned by [[User:Oguzhancan|Oguzhancan]] ==
<gallery>
Image:12-240-1009-1.jpg|Page 1
Image:12-240-1009-2.jpg|Page 2
Image:12-240-1009-3.jpg|Page 3
Image:12-240-1009-4.jpg|Page 4
Image:12-240-1009-5.jpg|Page 5
Image:12-240-1009-6.jpg|Page 6
Image:12-240-1009-7.jpg|Page 7
</gallery>

12-240/Class Photo

2012-10-12T02:37:28Z

Oguzhancan: /* 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
!ID wcashore
!e-mail
!Location
!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=Bartnicki|first=Piotr|userid=Peter|email=piotr.bartnicki@ mail.utoronto.ca|location=Left part of the last row sitting directly between two standing guys, *left* of the one in orange (from the camera's perspective) and to the right of one in a black striped shirt |comments=}}
{{Photo Entry|last=Can|first=Oguzhan|userid=Oguzhancan|email=oguzhan.can @ mail. utoronto .ca|location=seventh row from front, fifth from the right, blue tshirt|comments=}}
{{Photo Entry|last=Cashore|first=Walter|userid=wcashore|email=wcashore 12 @ hotmail .com|location=third row back, in the green star wars shirt|comments=great pic guys}}
{{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=Kennedy|first=Christopher|userid=ckennedy|email=christopherpa. kennedy@ mail. utoronto. ca|location=Third row; third from the right in white |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=Le|first=Quan|userid=Quanle|email=quan. le@ mail. utoronto. ca|location=Start bottom right corner, third from right. Go three steps north-west. Directly north-east from there, in blue collar shirt|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=Lue|first=Peter|userid=Peterlue|email=peter. lue@ mail. utoronto. ca|location=On the left edge 3rd from the back in the reddish shirt|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=McGrath|first=Celton|userid=CeltonMcGrath|email=celton. mcgrath@ mail. utoronto. ca|location=4th row front from, centre right, brown sweater|comments=}}
{{Photo Entry|last=Morenz|first=Karen|userid=KJMorenz|email=kjmorenz@ gmail.com|location=3rd-ish row from the back, centre right, purple shirt|comments=}}
{{Photo Entry|last=Pan|first=Li|userid=panli19|email=panli19@gmail.com|location=fourth row, the guy in grey fleece sweater.|comments=}}
{{Photo Entry|last=Ratz|first=Derek|userid=Derek.ratz|email=ratz.derek@gmail.com|location=2nd from the back, 2 in from the far left, yellow shirt|comments=}}
{{Photo Entry|last=Tong|first=Cheng Yu|userid=Chengyu.tong|email=chengyu. tong@ mail. utoronto. ca|location=fourth row from the front on the left side of the picture wearing green sweater and black rimmed glasses |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=Winnitoy|first=Leigh|userid=Leighwinnitoy|email=leigh.winnitoy@ mail. utoronto. ca|location=sixth row, near the middle of the picture|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=Yang|first=Tianlin|userid=Tianlin.yang|email=Tianin.Yang@ mail. utoronto. ca|location=4th row, first from left in blue wind coat.|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=Zhao|first=TianChen|userid=Ericolony|email=zhao_ tianchen@ hotmail. com|location=fourth row, the guy in green shirt.|comments=}}
{{Photo Entry|last=Zibert|first=Vincent|userid=vincezibert|email=vincent. zibert@ mail. utoronto. ca|location=Directly beneath the white notice posted on the door on the right-hand side.|comments=}}
{{Photo Entry|last=Zoghi|first=Sina|userid=sina.zoghi|email=sina.zoghi@ utoronto .ca|location=First row, leftest left.|comments=}}
{{Photo Entry|last=Léger|first=Zacharie|userid=zach.leger8|email=zacharie. leger@ mail. utronto. ca|location= 5th row in a black T-shirt.|comments=}}
{{Photo Entry|last=Wang|first=Minqi|userid=Michael.Wang|email=wangminqi@ yahoo.cn|location=First row, fourth from the left in black oufit) |comments=}}


|}

12-240/Classnotes for Tuesday October 09

2012-10-12T01:54:38Z

Oguzhancan:

{{12-240/Navigation}}
== Lecture notes scanned by [[User:Oguzhancan|Oguzhancan]] ==
<gallery>
Image:12-240-1009-1.jpg|Page 1
Image:12-240-1009-2.jpg|Page 2
Image:12-240-1009-3.jpg|Page 3
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</gallery>

12-240/Classnotes for Tuesday October 09

2012-10-12T01:51:26Z

Oguzhancan:

12-240/Navigation

2012-10-12T01:49:31Z

Oguzhancan:

12-240/Navigation

2012-10-12T01:49:10Z

Oguzhancan:

12-240/Classnotes for Thursday October 11

2012-10-12T01:46:44Z

Oguzhancan: /* Lecture notes scanned by Oguzhancan */

== Lecture notes scanned by [[User:Oguzhancan|Oguzhancan]] ==
<gallery>
Image:12-240-1011-1.jpg|Page 1
Image:12-240-1011-2.jpg|Page 2
Image:12-240-1011-3.jpg|Page 3
</gallery>

File:12-240-1009-7.jpg

2012-10-12T01:43:07Z

Oguzhancan:

File:12-240-1009-6.jpg

2012-10-12T01:42:17Z

Oguzhancan:

File:12-240-1009-5.jpg

2012-10-12T01:42:05Z

Oguzhancan:

File:12-240-1009-4.jpg

2012-10-12T01:41:15Z

Oguzhancan:

File:12-240-1009-3.jpg

2012-10-12T01:38:49Z

Oguzhancan:

File:12-240-1009-2.jpg

2012-10-12T01:38:39Z

Oguzhancan:

File:12-240-1009-1.jpg

2012-10-12T01:38:11Z

Oguzhancan:

12-240/Navigation

2012-10-12T01:26:06Z

Oguzhancan:

12-240/Classnotes for Thursday October 11

2012-10-12T01:24:27Z

Oguzhancan:

== Lecture notes scanned by [[User:Oguzhancan|Oguzhancan]] ==
<gallery>
Image:12-240-1011-1.jpg|Page 1
Image:12-240-1011-2.jpg|Page 2
Image:12-240-1011-3.jpg|Page 2
</gallery>

File:12-240-1011-3.jpg

2012-10-12T01:19:26Z

Oguzhancan:

File:12-240-1011-2.jpg

2012-10-12T01:19:09Z

Oguzhancan:

File:12-240-1011-1.jpg

2012-10-12T01:18:21Z

Oguzhancan:

12-240/Classnotes for Thursday September 27

2012-09-29T20:26:20Z

Oguzhancan: /* Subspaces */

{{12-240/Navigation}}

'''Vector Spaces'''

== Reminders ==

- Tag yourself in the photo!

- Read along textbook 1.1 to 1.4

- Riddle: Professor in ring with lion around the perimeter.
Consider this: http://mathforum.org/library/drmath/view/63421.html

== Vector space axioms ==

''(Quick recap)''

VS1. x + y = y + x

VS2. (x + y) + z = x + (y + z)

VS3. 0 vector

VS4. + inverse → -

VS5. 1x = x

VS6. a(bx) = (ab)x

VS7. a(x + y) = ax + ay

VS8. (a+b)x = ax + bx

== Theorems ==

1.a x + z = y + z ⇒ x = y

1.b ax = ay, a ≠ 0, ⇒ x = y

1.c ax = bx, x ≠ 0, ⇒ a = b

2. 0 is unique.

3. Additive inverse is unique.

4. 0_F ∙ x = 0_V

5. a ∙ 0_V = 0_V

6. (-a) x = -(ax) = a(-x)

7. cx = 0 ⇔ c = 0 or x = 0_V

=== Hints for proofs ===

1.a Same as for fields

1.b. Use similar proof as for fields, but use VS6 NOT F2b. F2b guarantees existence, but VS6 allows algebraic manipulation.

1.c Discussed after proof of 7, harder than you think at first glance.

2. Same as F.

3. Same as F

4. 0_F + 0_F = 0_F => by [VS8]: 0x + 0x = (0+0)x = 0x = 0x + 0 [VS3] = 0 + 0x [VS1]
⇒ 0x + 0x = 0 + 0x ⇒ [Cancellation property] 0x = 0

5. Same as 4 except using 0_V + 0_V = 0_V and using VS7

6. Skip

7. Prove both ways: Easy way is to the left, show left is 0 if either on right is 0.
To the right, Suppose c not= 0, then show x must equal 0.

1.c Add (-bx) to each side, use VS8 then VS6 -> (a-b)x =0, use property 7.

== Subspaces ==

Definition: Let V be a vector space over a field F. A ''subspace'' W of V is a subset of V, has the operations inherited from V and 0_V of V, is itself a vector space.

Theorem: A subset W ⊂ V, W ≠ ∅, is a subspace iff it is closed under the operations of V.

1. ∀ x, y ∈ W, x + y ∈ W

2. ∀ c ∈ F, ∀ x ∈ W, cx ∈ W

== Scanned notes upload by [[User:Starash|Starash]] ==

<gallery>
Image:12-240-0927-1.jpg|Page 1
Image:12-240-0927-2.jpg|Page 2
</gallery>

12-240/Class Photo

2012-09-27T02:37:47Z

Oguzhancan:

12-240

2012-09-27T02:29:03Z

Oguzhancan:

__NOEDITSECTION__
__NOTOC__
{{12-240/Navigation}}
==Algebra I==
===Department of Mathematics, University of Toronto, Fall 2012===

{{12-240/Crucial Information}}

===Text===
Our main text book will be ''Linear Algebra'' (fourth edition) by Friedberg, Insel and Spence, ISBN 0-13-008451-4; it is a required reading. An errata is at http://www.math.ilstu.edu/linalg/errata.html.

===Online Discussion Platform===
* [http://mat240.wordpress.com/ Click to go to the online discussion platform]

===Further Resources===

* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]

* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].

* [http://wiki.math.toronto.edu/TorontoMathWiki/index.php/2011F_MAT240_Algebra_I Marco Gualtieri's 2011 Math 240 web site].

* [http://wiki.math.toronto.edu/TorontoMathWiki/index.php/10-240 Marco Gualtieri's 2010 Math 240 web site].

* [[09-240|My 2009 Math 240 web site]].

* [http://www.math.toronto.edu/murnaghan/courses/mat240/index.html The 2008 MAT240 site].

* [[06-240|My 2006 Math 240 web site]].

* My {{Pensieve Link|Classes/12-240/|12-240 notebook}}.

{{Template:12-240:Dror/Students Divider}}

12-240

2012-09-27T01:28:27Z

Oguzhancan:

12-240

2012-09-27T01:08:30Z

Oguzhancan:

__NOEDITSECTION__
__NOTOC__
{{12-240/Navigation}}
==Algebra I==
===Department of Mathematics, University of Toronto, Fall 2012===

{{12-240/Crucial Information}}

===Text===
Our main text book will be ''Linear Algebra'' (fourth edition) by Friedberg, Insel and Spence, ISBN 0-13-008451-4; it is a required reading. An errata is at http://www.math.ilstu.edu/linalg/errata.html.

===Discussion Platform===
http://mat240.wordpress.com

===Further Resources===

* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]

* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].

* [http://wiki.math.toronto.edu/TorontoMathWiki/index.php/2011F_MAT240_Algebra_I Marco Gualtieri's 2011 Math 240 web site].

* [http://wiki.math.toronto.edu/TorontoMathWiki/index.php/10-240 Marco Gualtieri's 2010 Math 240 web site].

* [[09-240|My 2009 Math 240 web site]].

* [http://www.math.toronto.edu/murnaghan/courses/mat240/index.html The 2008 MAT240 site].

* [[06-240|My 2006 Math 240 web site]].

* My {{Pensieve Link|Classes/12-240/|12-240 notebook}}.

{{Template:12-240:Dror/Students Divider}}

12-240/Classnotes for Tuesday September 11

2012-09-21T19:16:58Z

Oguzhancan: /* 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}\ , a\ \neq 0 , \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?
** Answer by [[User:Drorbn|Drorbn]] 16:51, 13 September 2012 (EDT): A "field" is a set with two operations and 0 and 1 so that some properties hold. In principle, the same set can be made into a field in many different ways - by choosing different operations (so long as they satisfy F1-5). In practice though, there is essentially only one field with 5 elements (but I the word "essentially" here requires an explanation). Many other sets can be considered as fields in many "genuinely different" ways, depending in how you choose the operations <math>+</math> and <math>\times</math>.

*<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}4</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]]

== Lecture notes upload by [[User:Starash|Starash]] ==
Another upload of lecture 1 notes.
<gallery>
Image:Mat240-120911-p01.jpg|page 1
Image:Mat240-120911-p02.jpg|page 2
Image:Mat240-120911-p03.jpg|page 3
</gallery>

12-240/Classnotes for Tuesday September 11

2012-09-21T16:39:14Z

Oguzhancan: /* Existence of Negatives/Inverses */

{{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}\ , a\ \neq 0 , \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?
** Answer by [[User:Drorbn|Drorbn]] 16:51, 13 September 2012 (EDT): A "field" is a set with two operations and 0 and 1 so that some properties hold. In principle, the same set can be made into a field in many different ways - by choosing different operations (so long as they satisfy F1-5). In practice though, there is essentially only one field with 5 elements (but I the word "essentially" here requires an explanation). Many other sets can be considered as fields in many "genuinely different" ways, depending in how you choose the operations <math>+</math> and <math>\times</math>.

*<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]]
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[[Image:12-240-Sept11-Page7.jpeg|250px]]
[[Image:12-240-Sept11-Page8.jpeg|250px]]
[[Image:12-240-Sept11-Page9.jpeg|250px]]

== Lecture notes upload by [[User:Starash|Starash]] ==
Another upload of lecture 1 notes.
<gallery>
Image:Mat240-120911-p01.jpg|page 1
Image:Mat240-120911-p02.jpg|page 2
Image:Mat240-120911-p03.jpg|page 3
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12-240/Classnotes for Tuesday September 11

2012-09-21T16:38:43Z

Oguzhancan: /* Existence of Negatives/Inverses */

{{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}\, a\ \neq 0, \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?
** Answer by [[User:Drorbn|Drorbn]] 16:51, 13 September 2012 (EDT): A "field" is a set with two operations and 0 and 1 so that some properties hold. In principle, the same set can be made into a field in many different ways - by choosing different operations (so long as they satisfy F1-5). In practice though, there is essentially only one field with 5 elements (but I the word "essentially" here requires an explanation). Many other sets can be considered as fields in many "genuinely different" ways, depending in how you choose the operations <math>+</math> and <math>\times</math>.

*<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]]

== Lecture notes upload by [[User:Starash|Starash]] ==
Another upload of lecture 1 notes.
<gallery>
Image:Mat240-120911-p01.jpg|page 1
Image:Mat240-120911-p02.jpg|page 2
Image:Mat240-120911-p03.jpg|page 3
</gallery>