Drorbn - User contributions [en]

12-240/Homework Assignment 6

2012-11-04T14:50:35Z

Pratyush: /* 4x4 Matrices In OpenGL */

{{12-240/Navigation}}
This assignment is due at the tutorials on Thursday November 15. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

'''Task 1.''' '''Read''' sections 2.1 through 2.3 in our textbook. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, '''preread''' sections 2.4 and 2.5, just to get a feel for the future.

'''Task 2.''' '''Solve''' problems 20, 21, 24, 25, 26, and 27 on pages 76-77 and problems 1, 5a, 5b, 5c-g, 9, 15 and 16 on pages 84-86, but submit only your solutions of the underlined problems.

'''Just for fun.''' Do a web search for the words "OpenGL" and "matrix" (or "matrices") and find out why graphics accelerators in PCs and game consoles do <math>4\times 4</math> (!) matrix multiplications ''in hardware''. (The full explanation is quite involved; perhaps you should just aim to get a taste of it).

[[Image:06-240-Mario.jpg|thumb|center|176px|A famed creature that owes its mobility to 4x4 matrices.]]

== 4x4 Matrices In OpenGL ==
If anyone is interested, here is a good site about 4x4 matrices in OpenGL: [http://www.opengl-tutorial.org/beginners-tutorials/tutorial-3-matrices/ Go To Link]

12-240/Homework Assignment 6

2012-11-04T14:49:56Z

Pratyush: adding a link

{{12-240/Navigation}}
This assignment is due at the tutorials on Thursday November 15. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

'''Task 1.''' '''Read''' sections 2.1 through 2.3 in our textbook. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, '''preread''' sections 2.4 and 2.5, just to get a feel for the future.

'''Task 2.''' '''Solve''' problems 20, 21, 24, 25, 26, and 27 on pages 76-77 and problems 1, 5a, 5b, 5c-g, 9, 15 and 16 on pages 84-86, but submit only your solutions of the underlined problems.

'''Just for fun.''' Do a web search for the words "OpenGL" and "matrix" (or "matrices") and find out why graphics accelerators in PCs and game consoles do <math>4\times 4</math> (!) matrix multiplications ''in hardware''. (The full explanation is quite involved; perhaps you should just aim to get a taste of it).

[[Image:06-240-Mario.jpg|thumb|center|176px|A famed creature that owes its mobility to 4x4 matrices.]]

== 4x4 Matrices In OpenGL ==
If anyone is interested, here is a good site about 4x4 matrices in OpenGL: {{Go To Link|http://www.opengl-tutorial.org/beginners-tutorials/tutorial-3-matrices/}}

12-240/Classnotes for Thursday October 18

2012-10-19T18:24:41Z

Pratyush: /* Riddle Along */ editting typo

{{12-240/Navigation}}
===Riddle Along===
The game of 15 is played as follows. Two players alternate choosing cards numbered between 1 and 9, with repetitions forbidden, so that the game ends at most after 9 moves (or four and a half rounds). The first player to have within her/his cards a set of precisely 3 cards that add up to 15 wins.

Does this game have a winning strategy? What is it? Who wins, the first to move or the second? Why am I asking this question at this particular time?

[[Image:12-240-DeckOfCards.png|center]]

See also [https://media.library.utoronto.ca/play.php?DJ6CPFxByy2J&id=8503 a video] and the [https://cmc.math.ca/home/videos/game-of-15-and-isomorphisms/ transcript] of that video.

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

== Theorems ==
1. If G generates, |G| <math>\ge \!\,</math> n and G contains a basis, |G|=n then G is a basis

2. If L is linearly independent, |L| <math>\le \!\,</math> n and L can be extended to be a basis. |L|=n => L is a basis.

3.W <math>\subset \!\,</math> V a subspace then W is finite dimensioned and dim W <math>\le \!\,</math> dim V

If dim W = dim V, then V = W
If dim W < dim V, then any basis of W can be extended to be a basis of V

Proof of W is finite dimensioned:

Let L be a linearly independent subset of W which is of maximal size.

Fact about '''N'''
: Every subset A of '''N''', which is:

1. Non empty

2. Bounded : <math>\exist \!\,</math> N <math>\in \!\,</math> '''N''', <math>\forall \!\,</math> a <math>\in \!\,</math> A, a <math>\le \!\,</math> N

has a maximal element: an element m <math>\in \!\,</math> A, <math>\forall\!\,</math> a <math>\in \!\,</math> A, a <math>\le \!\,</math> m ( m + 1 <math>\notin \!\,</math> A )

== class note ==

<gallery>
Image:12-240-Oct-15-Page-1.jpg |page1
Image:12-240-Oct-15-Page-2.jpg |page2
Image:12-240-Oct-15-Page-3.jpg |page3
</gallery>

== lecture note on oct 18, uploaded by [[User:starash|starash]]==

<gallery>
Image:12-240-1018-1.jpg |page1
Image:12-240-1018-2.jpg |page2
</gallery>

12-240/Class Photo

2012-10-18T01:21:34Z

Pratyush: /* 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=Sarkar|first=Pratyush|userid=Pratyush|email=pratyush.sarkar@ mail.utoronto.ca|location=Somewhere at the back. I think I was in the <math>8^{th}</math> row, <math>3^{rd}</math> from the right. There were some empty seats so I am closer to the middle along the row.|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=}}


|}



</math>

12-240/Class Photo

2012-10-18T01:18:59Z

Pratyush: /* 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=Sarkar|first=Pratyush|userid=Pratyush|email=pratyush.sarkar@ mail.utoronto.ca|location=Somewhere at the back. <math>8^{th}</math>|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=}}


|}



</math>

12-240/Class Photo

2012-10-18T01:18:36Z

Pratyush: /* 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=Sarkar|first=Pratyush|userid=Pratyush|email=pratyush.sarkar@ mail.utoronto.ca|location=Somewhere at the back. 8 <math>^{th}</math>|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=}}


|}



</math>

12-240/Class Photo

2012-10-18T01:17:29Z

Pratyush: /* 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=Sarkar|first=Pratyush|userid=Pratyush|email=pratyush.sarkar@ mail.utoronto.ca|location=Somewhere at the back. 8 <math>^{th}<\math>|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=}}


|}



</math>

12-240/Class Photo

2012-10-18T01:13:52Z

Pratyush: /* Who We Are... */ editing my entry

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=Sarkar|first=Pratyush|userid=Pratyush|email=pratyush.sarkar@ mail.utoronto.ca|location=Somewhere at the back|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/Class Photo

2012-10-18T01:12:17Z

Pratyush: /* Who We Are... */ Adding myself

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=Sarkar|first=Pratyush|userid=Pratyush|email=pratyush. sarkar@ mail. utoronto. ca|location=Somewhere at the back|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 Thursday October 4

2012-10-11T12:42:19Z

Pratyush: /* Recap */ fixing errors

{{12-240/Navigation}}
== Reminders ==
Web Fact: No link, doesn't exist!

Life Fact: Dror doesn't do email math!

Riddle: Professor and lion in a ring with <math>V_p = V_l</math>, help the professor live as long as possible.

== Recap ==

Base - what were doing today

Linear combination (lc) - We say <math>v</math> is a linear combination of a set <math>S = \{u_1, \dots, u_n\}</math> if <math>v = a_1u_1, \dots, a_nu_n</math> for scalars from a field <math>F</math>.

Span - <math>\operatorname{span}(S)</math> is the set of all linear combination of set <math>S</math>.

Generate - We say <math>S</math> generates a vector space <math>V</math> is <math>\operatorname{span}(S) = V</math>.

== Pre - Basis ==

'''Linear dependance'''

'''Definition''' A set S ⊂ V is called linearly dependant if you can express the zero vector as a linear combination of distinct vectors from S, excluding the non-trivial linear combination where all of the scalars are 0.

Otherwise, we call S '''linearly independant.'''

=== Examples ===

1. In '''R'''^3, take S = {u1 = (1,4,7), u2 = (2,5,8), u3 = (3,6,9)}

u1 - 2u2 + u3 = 0

S is linearly dependant.

2. In R^n, take {e_i} = {0, 0, ... 1, 0, 0, 0} where 1 is in the ith position, and (ei) is a vector with n entries.

Claim: This set is linearly independent.

Proof: Suppose (∑ ai*ei) = 0 ({ei} is linearly dependant.)

(∑ ai*ei) = 0 ⇔ a1(1, 0, ..., 0) + ... + an(0, ..., 0, 1) = 0 ⇔ (a1, 0, ... , 0) + ... + (0, ... , an) = 0

⇒ a1 = a2 = ... = an = 0!

===Comments ===

1. {u} is linearly independant.
Proof:
⇐ If u≠0, suppose au =0
By property (a*u = 0, a = 0 or u = 0), a = 0. Thus, {u} is linearly independent.

⇒ By definition, au = 0 for {u} only when a = 0.

2. ∅ is linearly independant.

Exercise: Prove: '''Theorem''' Suppose S1 ⊂ S2 ⊂ V.

-> If S1 is linearly dependant, then S2 is dependant.

-> If S2 is linearly independent, then S1 is linearly independent. (Hint: contrapositive)

== Basis ==

'''Definition''': A subset β is called a basis if 1. β generates V → span(β) = V and 2. β is linearly independent.

===Examples===

1. V = {0}, β = {}

2. {ei} for F^n, this is what we call the '''standard basis'''

3. B = {(1,1),(1, -1)} is a basis for R^2

4. P_n(F) β = {x^n, x^n-1, ... , x^1, x^0}

5. P(F), β = (x^0, x^1 ... and on} ('''Infinite basis'''!)

== Interesting inequality ==

This holds is true if the field does not have characteristic 2. Can you see why?

(a,b) = (a+b)/2 * (1, 1) + (a-b)/2 * (1, -1)

== Lecture notes scanned by [[User:starash|starash]] ==
<gallery>
Image:12-240-1004-1.jpg|Page 1
Image:12-240-1004-2.jpg|Page 2
</gallery></math>

12-240/Classnotes for Thursday October 4

2012-10-11T12:41:23Z

Pratyush: /* Recap */ changing math to math mode

{{12-240/Navigation}}
== Reminders ==
Web Fact: No link, doesn't exist!

Life Fact: Dror doesn't do email math!

Riddle: Professor and lion in a ring with <math>V_p = V_l</math>, help the professor live as long as possible.

== Recap ==

Base - what were doing today

Linear combination (lc) - We say <math>v</math> is a linear combination of a set <math>S = \{u_1, \dotsc, u_n\}</math> if <math>v = a_1u_1, \dotsc, a_nu_n</math> for scalars from a field <math>F</math>.

Span - <math>\operatorname{span}(S)</math> is the set of all linear combination of set <math>S</math>.

Generate - We say <math>S</math> generates a vector space <math>V</math> is <math>\operatorname{span}(S) = V</math>.

== Pre - Basis ==

'''Linear dependance'''

'''Definition''' A set S ⊂ V is called linearly dependant if you can express the zero vector as a linear combination of distinct vectors from S, excluding the non-trivial linear combination where all of the scalars are 0.

Otherwise, we call S '''linearly independant.'''

=== Examples ===

1. In '''R'''^3, take S = {u1 = (1,4,7), u2 = (2,5,8), u3 = (3,6,9)}

u1 - 2u2 + u3 = 0

S is linearly dependant.

2. In R^n, take {e_i} = {0, 0, ... 1, 0, 0, 0} where 1 is in the ith position, and (ei) is a vector with n entries.

Claim: This set is linearly independent.

Proof: Suppose (∑ ai*ei) = 0 ({ei} is linearly dependant.)

(∑ ai*ei) = 0 ⇔ a1(1, 0, ..., 0) + ... + an(0, ..., 0, 1) = 0 ⇔ (a1, 0, ... , 0) + ... + (0, ... , an) = 0

⇒ a1 = a2 = ... = an = 0!

===Comments ===

1. {u} is linearly independant.
Proof:
⇐ If u≠0, suppose au =0
By property (a*u = 0, a = 0 or u = 0), a = 0. Thus, {u} is linearly independent.

⇒ By definition, au = 0 for {u} only when a = 0.

2. ∅ is linearly independant.

Exercise: Prove: '''Theorem''' Suppose S1 ⊂ S2 ⊂ V.

-> If S1 is linearly dependant, then S2 is dependant.

-> If S2 is linearly independent, then S1 is linearly independent. (Hint: contrapositive)

== Basis ==

'''Definition''': A subset β is called a basis if 1. β generates V → span(β) = V and 2. β is linearly independent.

===Examples===

1. V = {0}, β = {}

2. {ei} for F^n, this is what we call the '''standard basis'''

3. B = {(1,1),(1, -1)} is a basis for R^2

4. P_n(F) β = {x^n, x^n-1, ... , x^1, x^0}

5. P(F), β = (x^0, x^1 ... and on} ('''Infinite basis'''!)

== Interesting inequality ==

This holds is true if the field does not have characteristic 2. Can you see why?

(a,b) = (a+b)/2 * (1, 1) + (a-b)/2 * (1, -1)

== Lecture notes scanned by [[User:starash|starash]] ==
<gallery>
Image:12-240-1004-1.jpg|Page 1
Image:12-240-1004-2.jpg|Page 2
</gallery></math>

12-240/Classnotes for Thursday October 4

2012-10-11T12:36:00Z

Pratyush: /* Reminders */ Putting math stuff in math mode

{{12-240/Navigation}}
== Reminders ==
Web Fact: No link, doesn't exist!

Life Fact: Dror doesn't do email math!

Riddle: Professor and lion in a ring with <math>V_p = V_l</math>, help the professor live as long as possible.

== Recap ==

Base - what were doing today

Linear combination (lc) - We say v is a linear combination of a set S = {u1 ... un} if v = a1u1 ... anun for scalars from a field F.

Span - span(S) is the set of all linear combination of set S

Generate - We say S generates a vector space V is span(S) = V

== Pre - Basis ==

'''Linear dependance'''

'''Definition''' A set S ⊂ V is called linearly dependant if you can express the zero vector as a linear combination of distinct vectors from S, excluding the non-trivial linear combination where all of the scalars are 0.

Otherwise, we call S '''linearly independant.'''

=== Examples ===

1. In '''R'''^3, take S = {u1 = (1,4,7), u2 = (2,5,8), u3 = (3,6,9)}

u1 - 2u2 + u3 = 0

S is linearly dependant.

2. In R^n, take {e_i} = {0, 0, ... 1, 0, 0, 0} where 1 is in the ith position, and (ei) is a vector with n entries.

Claim: This set is linearly independent.

Proof: Suppose (∑ ai*ei) = 0 ({ei} is linearly dependant.)

(∑ ai*ei) = 0 ⇔ a1(1, 0, ..., 0) + ... + an(0, ..., 0, 1) = 0 ⇔ (a1, 0, ... , 0) + ... + (0, ... , an) = 0

⇒ a1 = a2 = ... = an = 0!

===Comments ===

1. {u} is linearly independant.
Proof:
⇐ If u≠0, suppose au =0
By property (a*u = 0, a = 0 or u = 0), a = 0. Thus, {u} is linearly independent.

⇒ By definition, au = 0 for {u} only when a = 0.

2. ∅ is linearly independant.

Exercise: Prove: '''Theorem''' Suppose S1 ⊂ S2 ⊂ V.

-> If S1 is linearly dependant, then S2 is dependant.

-> If S2 is linearly independent, then S1 is linearly independent. (Hint: contrapositive)

== Basis ==

'''Definition''': A subset β is called a basis if 1. β generates V → span(β) = V and 2. β is linearly independent.

===Examples===

1. V = {0}, β = {}

2. {ei} for F^n, this is what we call the '''standard basis'''

3. B = {(1,1),(1, -1)} is a basis for R^2

4. P_n(F) β = {x^n, x^n-1, ... , x^1, x^0}

5. P(F), β = (x^0, x^1 ... and on} ('''Infinite basis'''!)

== Interesting inequality ==

This holds is true if the field does not have characteristic 2. Can you see why?

(a,b) = (a+b)/2 * (1, 1) + (a-b)/2 * (1, -1)

== Lecture notes scanned by [[User:starash|starash]] ==
<gallery>
Image:12-240-1004-1.jpg|Page 1
Image:12-240-1004-2.jpg|Page 2
</gallery></math>

12-240/Classnotes for Thursday October 4

2012-10-11T12:34:49Z

Pratyush: /* Reminders */

{{12-240/Navigation}}
== Reminders ==
Web Fact: No link, doesn't exist!

Life Fact: Dror doesn't do email math!

Riddle: Professor and lion in a ring with <math>V_p = V_l<\math>, help the professor live as long as possible.

== Recap ==

Base - what were doing today

Linear combination (lc) - We say v is a linear combination of a set S = {u1 ... un} if v = a1u1 ... anun for scalars from a field F.

Span - span(S) is the set of all linear combination of set S

Generate - We say S generates a vector space V is span(S) = V

== Pre - Basis ==

'''Linear dependance'''

'''Definition''' A set S ⊂ V is called linearly dependant if you can express the zero vector as a linear combination of distinct vectors from S, excluding the non-trivial linear combination where all of the scalars are 0.

Otherwise, we call S '''linearly independant.'''

=== Examples ===

1. In '''R'''^3, take S = {u1 = (1,4,7), u2 = (2,5,8), u3 = (3,6,9)}

u1 - 2u2 + u3 = 0

S is linearly dependant.

2. In R^n, take {e_i} = {0, 0, ... 1, 0, 0, 0} where 1 is in the ith position, and (ei) is a vector with n entries.

Claim: This set is linearly independent.

Proof: Suppose (∑ ai*ei) = 0 ({ei} is linearly dependant.)

(∑ ai*ei) = 0 ⇔ a1(1, 0, ..., 0) + ... + an(0, ..., 0, 1) = 0 ⇔ (a1, 0, ... , 0) + ... + (0, ... , an) = 0

⇒ a1 = a2 = ... = an = 0!

===Comments ===

1. {u} is linearly independant.
Proof:
⇐ If u≠0, suppose au =0
By property (a*u = 0, a = 0 or u = 0), a = 0. Thus, {u} is linearly independent.

⇒ By definition, au = 0 for {u} only when a = 0.

2. ∅ is linearly independant.

Exercise: Prove: '''Theorem''' Suppose S1 ⊂ S2 ⊂ V.

-> If S1 is linearly dependant, then S2 is dependant.

-> If S2 is linearly independent, then S1 is linearly independent. (Hint: contrapositive)

== Basis ==

'''Definition''': A subset β is called a basis if 1. β generates V → span(β) = V and 2. β is linearly independent.

===Examples===

1. V = {0}, β = {}

2. {ei} for F^n, this is what we call the '''standard basis'''

3. B = {(1,1),(1, -1)} is a basis for R^2

4. P_n(F) β = {x^n, x^n-1, ... , x^1, x^0}

5. P(F), β = (x^0, x^1 ... and on} ('''Infinite basis'''!)

== Interesting inequality ==

This holds is true if the field does not have characteristic 2. Can you see why?

(a,b) = (a+b)/2 * (1, 1) + (a-b)/2 * (1, -1)

== Lecture notes scanned by [[User:starash|starash]] ==
<gallery>
Image:12-240-1004-1.jpg|Page 1
Image:12-240-1004-2.jpg|Page 2
</gallery></math>

12-240/Classnotes for Tuesday October 2

2012-10-04T15:28:27Z

Pratyush: /* Subspace */

{{12-240/Navigation}}
The "vitamins" slide we viewed today is [http://drorbn.net/AcademicPensieve/Classes/12-240/index.html?im=FoodsHandout.jpg here].

Today, the professor introduces more about subspace, linear combination, and related subjects.

== Subspace ==

Remind about the theorem of subspace: a non-empty subset W ⊂ V is a subspace iff is is closed under the operations of V

Proof: First direction:

== Class Notes ==

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