Drorbn - User contributions [en]

06-240/Class Photo

2006-12-08T16:45:14Z

Uhoang36: /* Who We Are... */

Our class on September 28, 2006:

[[Image:06-240-ClassPhoto.jpg|thumb|centre|500px|Class Photo: click to enlarge]]
{{06-240/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
|-
!First name
!Last name
!UserID
!Email
!In the photo
!Comments
{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn@ math.toronto.edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank}}
{{Photo Entry|last=Beach|first=Laura|userid=Beacher|email=laura.beach@ utoronto.ca|location=3rd row from front, far right in light shirt, in front of guy in hat.|comments=Lookin' good, everybody!}}
{{Photo Entry|last=Bjorndahl|first=Paolo|userid=Bjorndahl|email=paolo.bjorndahl@ utoronto.ca|location=Back row, 5th form the far right.|comments=}}
{{Photo Entry|last=Bumagin|first=Mark|userid=Bumaginm|email=mark.bumagin@ utoronto.ca|location=Guy in light blue collar shirt, second row, third on the right|comments= }}
{{Photo Entry|last=Cao|first=Yanshuai|userid=yanshuaicao|email=yanshuaicao@gmail.com|location= the Asian guy in jeans and jacket in the left mini photo,second from the left, |comments= Glad to be in this group }}
{{Photo Entry|last=Carberry|first=Mick|userid=MC|email=Mick.Carberry@ utoronto.ca|location=long haired, bearded old guy in back|comments= }}
{{Photo Entry|last=Cerezo|first=Richard|userid=Cerezo|email=richard.cerezo@ utoronto.ca|location=Guy in black jacket and black hat on far right, second from the bottom. |comments= }}
{{Photo Entry|last=Dong|first=Qi (Tom)|userid=houseofglass|email=tom_dongqi@ hotmail.com|location= The only chinese blonde guy with glasses located on the second row in the main picture|comments=testing 1 2 3 4 5 6 7 8 9 }}
{{Photo Entry|last=Dzamba|first=Michael|userid=dzambami|email=michael.dzamba@ utoronto.ca|location=In the middle, sort of, have a bit of extra hair on my head, forms sort of a spherical volume, sometimes visible from a distance|comments="The obvious mathematical breakthrough would be development of an easy way to factor large prime numbers." - Bill Gates , The Road Ahead (his book), P265, This is not a typo on my behalf ;) }}
{{Photo Entry|last=Gokmen|first=Murat|userid=Gokmen|email=uoftmurat@ gmail.com|location=the guy second row @second from the right:with the shining jacket & blue hat|comments=}}
{{Photo Entry|last=Guillon|first=Adrien|userid=adrien_joel|email=guillona_at_remove_me_utoronto.ca|location=Third row of main picture, third from the left with black jacket and brown hair.|comments= }}
{{Photo Entry|last=Halacheva|first=Iva|userid=Haliv|email=iva.halacheva@ utoronto.ca|location=Third row from the front, light brown jacket|comments= }}
{{Photo Entry|last=Kaifosh|first=Patrick|userid=Pat|email=patrick.kaifosh@ utoronto.ca|location=Front row, rightmost.|comments= }}
{{Photo Entry|last=Kim|first=Taehyung|userid=Taehyung Kim|email=taehyung.Kim@utoronto.ca|location= Asian,Second row from the back, Second right most, black shirt,black backpack|comments= }}
{{Photo Entry|last=Koziar|first=John|userid=John.koziar|email=|location=Front row, centre.|comments= }}
{{Photo Entry|last=Matskin|first=Jeffrey|userid=jeff.matskin|email=jeff.matskin@utoronto.ca|location= Bottom right corner of people, farthest to the right, white shirt|comments= }}
{{Photo Entry|last=McIntyre|first=Sean|userid=Smcintyre|email=s.mcintyre@ utoronto.ca|location=mini-picture, fourth from the right|comments= }}
{{Photo Entry|last=Merchant|first=Aliya|userid=amerchant|email=aliya.merchant@gmail.com|location=Main picture, third girl in the third row. The tiny girl who looks like she's hiding.|comments= }}
{{Photo Entry|last=Ng|first=Gilbert|userid=Gilbert|email=gilbert.ng@ utoronto.ca|location=to the right of the pole in the back, last row|comments= }}
{{Photo Entry|last=Nikolaev|first=Nikita|userid=Zimba|email=nnabelka@yahoo.ca|location=Second row, second from the right. A guy in bright-red and dark-blue jacket|comments=Thou shall love squirrels}}
{{Photo Entry|last=Park|first=Philip|userid=Psp|email=philip.park@utoronto.ca|location=2nd row from the back, 3rd from the right, wearing a light-blue shirt with a dark-blue jacket|comments=}}
{{Photo Entry|last=Qiao|first=Li|userid=joy9999|email=li.qiao@ utoronto.ca|location=behind the guy in black, last row|comments= }}
{{Photo Entry|last=Shestakov|first=Ilya|userid=ilya87|email=ilya.shestakov@utoronto.ca|location=In the yellow/green shirt in the lower right corner|comments= }}
{{Photo Entry|last=Soreanu|first=Alla|userid=Alla|email=alla.soreanu@ utoronto.ca|location=mini-picture, first from the left|comments= }}
{{Photo Entry|last=SUN|first=LUYANG|userid=Luyang|email=luyang.sun@ utoronto.ca|location=the guy in the middle, wearing a white coat row3 col4|comments=Hello World}}
{{Photo Entry|last=Tai|first=Johnathan|userid=Zapyre|email=zapyre_1@ hotmail.com|location= At the very back, left hand corner (all blue) |comments= >.<}}
{{Photo Entry|last=Veytsman|first=Maxim|userid=Mveytsman|email=|location=small picture in the bottom left corner. Second one from the right.|comments=}}
{{Photo Entry|last=Wan|first=Mike|userid=wanmike|email=Mike.Wan@utoronto.ca|location=Main picture, front row, second from right.|comments="If I have seen farther, it is by standing on the shoulders of giants." -Sir I N.}}
{{Photo Entry|last=Wong|first=Pak|userid=wongpak|email=pl.wong@ utoronto.ca|location=Third row from the back, left most, black shirt|comments= }}
{{Photo Entry|last=Jaimungal|first=Curt|userid=curtdbz|email=curtdbz@hotmail.com or curt.jaimungal@utoronto.ca|location=The brown guy in the middle of the mini-pic (bottom left) in the middle, without glasses.|comments= Hello.. Newman..!}}
{{Photo Entry|last=Hoang|first=Uyen|userid=uhoang36|email=uhoang36@yahoo.com |location=The Asian girl in the front row.|comments= }}

06-240/Classnotes For Thursday, September 28

2006-10-02T01:06:53Z

Uhoang36: /* Scan of Lecture notes */

===Scan of Lecture notes===

*Image file: week 3 lecture
-note1[[http://i98.photobucket.com/albums/l269/uhoang/1.jpg]]
-note2:[[http://i98.photobucket.com/albums/l269/uhoang/2.jpg]]

===Scan of Lecture notes===

* PDF file by [[User:Alla]]: [[Media:MAT_Lect006.pdf|Week 3 Lecture 2 notes]]

===Scan of Tutorial notes===

* PDF file by [[User:Alla]]: [[Media:MAT_Tut003.pdf|Week 3 Tutorial notes]]

===Linear Combination===

<math>\mbox{Definition: Let }(u_i) = (u_1,u_2,\ldots,u_n)\mbox{ be a sequence of vectors in }V</math>.

<math>\mbox{A sum of the form:}{}_{}^{}</math>

<math> a_i\in F,\sum_{i=1}^n a_i u_i = a_1u_1 + a_2u_2+\ldots+a_nu_n</math>

<math>\mbox{is called a Linear Combination of the }u_i^{ }</math>.

===Span===
<math>\mbox{span}(u_i^{ }):= \lbrace\mbox{ The set of all possible linear combinations of the } u_i^{ }\rbrace</math>

<math>\mbox{If }\mathcal{S} \subset V\ \mbox{ is any subset, }</math>

<math>\mbox{span}(\mathcal{S}):= \lbrace\mbox{The set of all linear combination of vectors in }\mathcal{S}\rbrace=\left\lbrace\sum_{i=0}^n a_i u_i,\quad a_i \in F, u_i \in \mathcal{S}\right\rbrace</math>

<math>\mbox{span}(\mathcal{S})\mbox{ always contains }0\mbox{ even if }\mathcal{S}=\emptyset</math>

'''Theorem'''

<math>\forall\mathcal{S} \subset V\mbox{, span}(\mathcal{S})\mbox{ is a subspace of }V</math>

<math>\mbox{Proof:}{}_{}^{}</math>

1. <math>0 \in\mbox{ span}(\mathcal{S})</math>. 
2. <math>\mbox{Let }x \in \mbox{ span}(\mathcal{S})\Rightarrow x =\sum_{i=1}^n a_iu_i\mbox{, }u_i\in \mathcal{S}\mbox{, }</math>

<math>\mbox{and let }y \in \mbox{ span}(\mathcal{S})\Rightarrow y =\sum_{i=1}^m b_iv_i\mbox{, }v_i\in \mathcal{S}</math>

<math>x+y = \sum_{i=1}^n a_iu_i+ \sum_{i=1}^m b_iv_i = \sum_{i=1}^{\mbox{max}(m,n)} c_iw_i</math>

<math>\qquad\mbox{ where }c_i=(a_1+b_1,a_2+b_2,\ldots,a_{\mbox{max}(m,n)}+b_{\mbox{max}(m,n)})\mbox{ and }w_i\in\mathcal{S}</math>

3.<math>cx= c\sum_{i=1}^n a_iu_i=\sum_{i=1}^n(ca_i)u_i\in\mbox{ span}(\mathcal{S})</math>

''Example''
1.

<math>\mbox{Let } P_3(\mathbb{R})=\lbrace ax^3+bx^2+cx+d\rbrace\subset P(\mathbb{R})\mbox{, where }a, b, c, d \in \mathbb{R}</math>.

<math>\begin{matrix}u_1^{}&=&x^3-2x^2-5x-3\\
u_2^{}&=&3x^3-5x^2-4x-9\\
v_{}^{}&=&2x^3-2x^2+12x-6\end{matrix}</math>

<math>\mbox{Let }W=\mbox{span}(u_1^{},u_2^{})\mbox{,}</math> 

 <math>\mbox{Does/Is } v \in W\mbox{ ?}</math>

<math>v\in W\mbox{ if it is a linear combination of span}(u_1^{},u_2^{})</math>

<math>v=a_1u_1 + a_2u_2 \mbox{ for some }a_1, a_2 \in \mathbb{R}</math> 

 <math>\mbox{If }\exists a_1,a_2\in \mathbb{R}</math>

<math>\begin{matrix}2x^3-2x^2+12x-6&=& a_1^{}(x^3-2x^2-5x-3) + a_2^{}(3x^3-5x^2-4x-9)\\
\ &=&(a_1^{}+3a_2^{})x^3 + (-2a_1^{}-5a_2^{})x^2 + (-5a_1^{}-4a_2^{})x + (-3a_1^{}-9a_2^{})\end{matrix}</math>

<math>\mbox{Need to solve}\begin{cases}
2=a_1^{}+3a_2^{}\\
-2=-2a_1^{}-5a_2^{}\\
12=-5a_1^{}-4a_2^{}\\
-6=-3a_1^{}-9a_2^{}\end{cases}</math>

<math>\mbox{Solve the four equations above and we will get }a_1^{}=-4\mbox{ and }a_2^{}=2</math>

<math>\mbox{Check if }a_1^{}=-4\mbox{ and }a_2^{}=2\mbox{ holds for all 4 equations.}</math>

<math>\mbox{Since it holds, } v\in W</math>

06-240/Classnotes For Tuesday September 26

2006-09-27T03:04:27Z

Uhoang36:

Classnote for Tuesday Sept 26 [http://www.megaupload.com/?d=4L41DERL]