Drorbn - User contributions [en]

06-240/Class Photo

2006-10-01T23:50:22Z

Haliv: /* Who We Are */

{{06-240/Navigation}}

Our class on September 28, 2006:

[[Image:06-240-ClassPhoto.jpg|thumb|center|500px|Class Photo: click to enlarge]]

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 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=Carberry|first=Mick|userid=MC|email=Mick.Carberry@utoronto.ca|location=long haired, bearded old guy in back|comments= }}
{{Photo Entry|last=Kaifosh|first=Patrick|userid=Pat|email=patrick.kaifosh@utoronto.ca|location=large picture, farthest to the right on sidewalk|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=Soreanu|first=Alla|userid=Alla|email=alla.soreanu@utoronto.ca|location=mini-picture, first from the left|comments= }}
{{Photo Entry|last=Wong|first=Pak|userid=wongpak|email=plwong@utoronto.ca|location=Third row from the back, left most, black shirt|comments= }}
{{Photo Entry|last=Halacheva|first=Iva|userid=Haliv|email=iva.halacheva@utoronto.ca|location=Third row from the front, light brown jacket|comments= }}

|}

06-240/Classnotes For Tuesday September 26

2006-10-01T23:45:00Z

Haliv:

{{06-240/Navigation}}

===Links to Classnotes===
* Classnote for Tuesday Sept 26 [http://www.megaupload.com/?d=4L41DERL]
* PDF file by [[User:Alla]]: [[Media:MAT_Lect005.pdf|Week 3 Lecture 1 notes]]
----
===Vector Spaces===

'''Example 5.''' 

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

<math>7x^3+9x^2-2x+\pi\ </math> 

<math>\mbox{Let } \mathcal{F }\ \mbox{be a field.}</math> 

<math>P(\mathcal{F})=\bigg\{ \sum_{i=1}^n a_i x^i :n \in \mathbb{Z}\,\ n\ge 0\ \forall i\ \ a_i \in \mathcal{F} \bigg\} {}_{}^{} </math>

<math> \mbox{Addition of polynomials is defined in the expected way:}{}_{}^{} </math> 

<math> \sum_{i=0}^n a_i x^i + \sum_{i=1}^m b_i x^i =\sum_{i=0}^{max(m,n)}{(a_i+b_i)} x^i </math> 

'''Theorem 1.'''(Cancellation law for vector spaces) 

<math> \mbox{If in a vector space x+z=y+z then x=y.}{}_{}^{} </math> 

'''Proof:''' 

<math> \mbox{Add w to both sides of a given equation where w is an element}{}_{}^{} </math>
<math> \mbox{for which z+w=0 (exists by VS4)}{}_{}^{} </math> 

<math>(x+y)+w=(y+z)+w \ </math> 

<math> x+(z+w)=y+(z+w)\ \mbox{(by VS2)} {}_{}^{} </math> 

<math> x+0=y+0\ \mbox{(by the choice of w)} {}_{}^{} </math> 

<math> x=y\ \mbox{(by VS3)} {}_{}^{} </math> 

'''Theorem 2.''' "0 is unique" 

<math> \mbox{If some z}\in\mbox{V satisfies x+z=0 for some x}\in \mbox{V then z=0.} {}_{}^{} </math> 

'''Proof:" 

<math>x+z=x+0\ </math> 

<math>z+x=0+x\ </math> 
<math>z=0\ </math> 

'''Theorem 3.''' "negatives are unique" 

<math> \mbox{If x+y=0 and x+z=0 then y=z.} {}_{}^{} </math> 

'''Theorem 4.''' 

a)<math>0_F.x=0_V\ </math> 

b)<math>a.0_V=0_V\ </math> 

c)<math>(-a)x=a(-x)=-(ax)\ </math> 

'''Theorem 5.''' 

<math> \mbox{If } x_i\ \mbox{ i=1,...,n are in V then } \sum {x_i}=x_1+x_2+...+x_n\ \mbox{ makes sense whichever way you parse it.} {}_{}^{} </math> 

<math> \mbox{(From VS1 and VS2)} {}_{}^{} </math> 
----
===Subspaces===

'''Definition''' 

<math> \mbox{Let V be a vector space. A subspace of V is a subset W of V which is a vector space in itself under the operations is inherits from V.}{}_{}^{} </math> 

'''Theorem''' 

<math>W\subset V\ \mbox{is a subspace of V iff}{}_{}^{} </math> 

#<math>\forall x,y\in W\ \ x+y\in W \ </math>
#<math> \forall a\in F,\ \forall x\in W\ \ ax\in W\ </math>
#<math>0 \in W\ </math> 

'''Proof''' 
<math>\Rightarrow </math> 

<math>\mbox{Assume W is a subspace. If x,y} \in \mbox{W then x+y} \in \mbox{W because W is a vector space in itself. Likewise for a.w.}{}_{}^{} </math> 

<math>\Leftarrow </math> 

<math>\mbox{Assume W}\subset \mbox{V for which } x,y\in W\Rightarrow x+y\in W\ ; x\in W, a\in F \Rightarrow ax\in W.\ {}_{}^{} </math> 

<math>\mbox{We need to show that W is a vector space. Addition and multiplication are clearly defined on W so we just need to check VS1-VS8.}{}_{}^{} </math> 

<math> \mbox{Indeed, VS1, VS2, VS5, VS6, VS7, and VS8 hold in V hence in W.}{}_{}^{} </math> 

<math> \mbox{VS3-pick any x}\in W\ \ 0=0.x\in W\ by\ 2.\ {}_{}^{} </math> 

<math> \mbox{VS4-given x in W, take y=(-1).x}\in W\ and\ x+y=0.\ {}_{}^{} </math> 

Examples 

'''Example 1.''' 

'''Definition''' 

<math> \mbox{If A}\in M_{m\times n}(F) \mbox{ the transpose of A, } A^t \mbox{ is the matrix } (A^t)_{ij}:=A_{ji}. {}_{}^{} </math> 

<math> \begin{pmatrix} 2 & 3 & \pi\ \\ 7 & 8 & -2 \end{pmatrix}^t = \begin{pmatrix} 2 & 7 \\ 3 & 8 \\ \pi\ & -2 \end{pmatrix} </math> 

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

#<math>A^t \in M_{n\times m}(F)\ </math> 
#<math>(A^t)^t=A\ </math> 
#<math>(A+B)^t=A^t+B^t\ </math> 
#<math>(cA)^t=c(A^t)\ \forall c\in F\ </math> 

'''Definition''' 

<math>A\in M_{n\times n}(F) \mbox{ is called symmetric if } A^t=A. \ {}_{}^{} </math> 

Claim 

<math>V=M_{n\times n}(F) \ \mbox{ is a vector space. Let } \ W=\big\{ \mbox{symmetric A-s in V}\big\} = \big\{ A\in V: A^t=A \big\}\ \mbox{ then W is a subspace of V.} {}_{}^{} </math> 

Proof 

1.<math> \mbox{Need to show that if } A\in W and\ B\in W\ then\ A+B\in W. \ {}_{}^{} </math> 

<math>A^t=A,\ B^t=B \ </math> 

<math>(A+B)^t=A^t+B^t=A+B\ so\ A+B\in W. </math> 

<math>\mbox{If } A\in W,\ c\in F \mbox{ need to show } cA\in W {}_{}^{} </math> 

<math>(cA)^t=cA^t=cA\ \Rightarrow cA\in W </math> 

3.<math>0_M=\begin{pmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0\end{pmatrix} \Rightarrow 0^t=0 \ so \ 0\in W</math> 

'''Example 2.''' 

<math>V=M_{n\times n}(F) </math> 

<math>A=A_{ij}\ \ trA=\sum_{i=1}^n A_{ii}\ \mbox{(the trace of A)} </math> 

<math> \mbox{Properties of tr:}{}_{}^{} </math> 

#<math>tr0_M=0 \ </math> 
#<math>tr(A+B)=tr(A)+tr(B) \ </math> 
#<math> tr(cA)=c.trA \ </math> 

<math>A=\begin{pmatrix} 1 & 0 \\ 0 & 0\end{pmatrix}\ \ B=\begin{pmatrix} 0 & 0 \\ 0 & 1\end{pmatrix} \ </math> 
<math>trA=1\ \ trB=1 \ </math> 

<math>Set\ \ W=\big\{A\in V: trA=0\big\}=\bigg\{\begin{pmatrix} 1 & 7 \\ \pi\ & -1\end{pmatrix},...\bigg\} \ </math> 

Claim

<math> \mbox{W is a subspace.}{}_{}^{} </math> 

<math> \mbox{Indeed,}{}_{}^{} </math> 
#<math>A,B\in W \Rightarrow trA=0=trB\ \ tr(A+B)=tr(A)+tr(B)=0+0=0\ so\ A+B\in W\ </math>
#<math>A\in W\ \ trA=0\ \ tr(cA)=c(trA)=c.0=0\ so\ cA\in W\ </math>
#<math>tr0_M=0\ \ 0_M\in W \ </math>

'''Example 3.''' 

<math> W_3=\big\{ A\in M_{n\times n}(F): trA=1\big\} \mbox{ Not a subspace.} {}_{}^{} </math> 
<math> A,B\in W_3 \Rightarrow tr(A+B)=trA+trB=1+1=2\ so\ A+B\ \not\in W_3\ </math> 

'''Theorem''' 
<math> \mbox{The intersection of two subspaces of the same space is always a subspace.}{}_{}^{}</math> 
<math> \mbox{Assume }W_1\subset V \mbox{ is a subspace of V, } W_2\subset V \mbox{ is a subspace of V, then }W_1\cap W_2=\big\{ x: x\in W_1 \ and\ x\in W_2\big\} \mbox{ is a subspace.}{}_{}^{} </math> 
<math>\mbox{However, }W_1\cup W_2=\big\{x: x\in W_1\ or\ W_2\big\} \mbox{ is most often not a subspace.} {}_{}^{}</math> 

'''Proof''' 

1.<math> \mbox{Assume }x,y \in W_1\cap W_2 \mbox{ , that is, } x\in W_1, x\in W_2, y\in W_1, y\in W_2. \ {}_{}^{} </math> 
<math> x+y\in W_1 \ as\ x,y\in W_1 \mbox{ and } W_1 \mbox{ is a subspace}{}_{}^{} </math> 
<math> x+y\in W_2 \ as\ x,y\in W_2 \mbox{ and } W_2 \mbox{ is a subspace}{}_{}^{} </math> 
<math> \mbox{So }x+y \in W_1\cap W_2. \ {}_{}^{} </math> 

2.<math>\mbox{If} \ x\in W_1\cap W_2\ then\ x\in W_1 \Rightarrow cx\in W_1\ ,\ x\in W_2 \Rightarrow cx\in W_2\ \Rightarrow cx\in W_1\cap W_2. \ {}_{}^{} </math> 

3.<math>0 \in W_1\ ,\ 0\in W_2 \Rightarrow 0\in W_1\cap W_2. \ </math>

06-240/Classnotes For Thursday, September 21

2006-09-24T15:53:12Z

Haliv:

A force has a direction & a magnitude.

==<center>'''Force Vectors'''</center>==
#There is a special force vector called 0.
#They can be added.
#They can be multiplied by any scalar.

====''Properties''==== (convention: x,y,z-vectors; a,b,c-scalars)
# <math> x+y=y+x \ </math>
#<math> x+(y+z)=(x+y)+z \ </math>
#<math> x+0=x \ </math>
#<math> \forall x\; \exists\ y \ s.t.\ x+y=0 \ </math>
#<math> 1.x=x \ </math>
#<math> a(bx=(ab)x \ </math>
#<math> a(x+y)=ax+ay \ </math>
#<math> (a+b)x=ax+bx \ </math>

=====Definition===== Let F be a field "of scalars". A vector space over F is a set V (of "vectors") along with two operations:
: <math> +: V \times V \to V </math>
: <math> \cdot: F \times V \to V </math>, so that
#<math> \forall x,y \in V\ x+y=y+x </math>
#<math> \forall x,y \in V\ x+(y+z)=(x+y)+z </math>
#<math> \exists\ 0 \in V s.t.\; \forall x \in V\ x+0=x </math>
#<math> \forall x \in V\; \exists\ y \in V\ s.t.\ x+y=0</math>
#<math> 1.x=x\ </math>
#<math> a(bx)=(ab)x\ </math>
#<math> a(x+y)=ax+ay\ </math>
#<math> \forall x \in V\ ,\forall a,b \in F\ (a+b)x=ax+bx </math>
-----
9. <math> x \mapsto |x| \in \mathbb{R} \ \ |x+y| \le |x|+|y| </math>
====''Examples''====
'''Ex.1.'''
<math> F^n= \big\{ (a_1,a_2,a_3,...,a_{n-1},a_n):\forall i\ a_i \in F \big\} </math> 
<math> n \in \mathbb{Z}\ , n \ge 0 </math> 
<math> x=(a_1,...,a_2)\ y=(b_1,...,b_2)\ </math> 
<math> x+y:=(a_1+b_1,a_2+b_2,...,a_n+b_n)\ </math> 
<math> 0_{F^n}=(0,...,0) </math> 
<math> a\in F\ ax=(aa_1,aa_2,...,aa_n) </math> 
<math> In \ \mathbb{Q}^3 \ ( \frac{3}{2},-2,7)+( \frac{-3}{2}, \frac{1}{3},240)=(0, \frac{-5}{3},247) </math> 
<math> 7( \frac{1}{5},\frac{1}{7},\frac{1}{9})=( \frac{7}{5},1,\frac{7}{9}) </math> 
'''Ex.2.'''
<math> V=M_{m \times n}(F)=\Bigg\{\begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots &
& \vdots \\ a_{m1} & \cdots & a_{mn}\end{pmatrix}: a_{ij} \in F \Bigg\} </math> 
<math> M_{3\times 2}( \mathbb{R})\ni \begin{pmatrix} 7 & -7 \\ \pi & \mathit{e} \\ -5 & 2 \end{pmatrix} </math> 
Add by adding entry by entry:<math> M_{2\times 2}\ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}+\begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}=\begin{pmatrix} {a_{11}+b_{11}} & {a_{12}+b_{12}} \\ {a_{21}+b_{21}} & {a_{22}+b_{22}} \end{pmatrix}</math> 
Multiplication by a is multiplication of all entries by a. 
<math> 0_{M_{m\times n}}=\begin{pmatrix} 0 & \cdots & 0 \\ \vdots &
& \vdots \\ 0 & \cdots & 0\end{pmatrix} </math> 
'''Ex.3.'''
<math> \mathbb{C}</math> form a vector space over <math> \mathbb{R}</math>. 
'''Ex.4.'''
F is a vector space over itself. 
'''Ex.5.'''
<math> \mathbb{R}</math> is a vector space over <math> \mathbb{Q}</math>. 
'''Ex.6.'''
Let S be a set. Let 
<math> \mathcal{F}(S,\mathbb{R})=\big\{f:S\to \mathbb{R} \big\} </math> 
<math> f,g \in \mathcal{F}(S,\mathbb{R}) </math> 
<math> (f+g)(t)=f(t)+g(t)\ for\ any\ t\in S </math> 
<math> (af)(t)=a.f(t)\ </math>

Talk:06-240/Classnotes For Thursday, September 21

2006-09-23T21:20:35Z

Haliv:

For some reason the \textstyle function was not working so I couldn't format the fractions to the text size. I would appreciate it if someone attempts to do that or, of course, corrects any mistakes I might have made.--[[User:Haliv|Haliv]] 17:20, 23 September 2006 (EDT)

Template:06-240/Navigation

2006-09-23T17:36:48Z

Haliv:

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!Week of...
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|align=center|1
|Sep 11
|[[06-240/About This Class|About]], [[06-240/Classnotes For Tuesday, September 12|Tue]], [[06-240/Homework Assignment 1|HW1]], [[06-240/Putnam Competition|Putnam]], [[06-240/Classnotes for Thursday, September 14|Thu]]
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|Sep 18
|[[06-240/Classnotes For Tuesday, September 19|Tue]], [[06-240/Homework Assignment 2|HW2]], [[06-240/Classnotes For Thursday, September 21|Thu]]
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|align=center|3
|Sep 25
|HW3
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|Oct 2
|HW4
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|align=center|5
|Oct 9
|HW5
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|align=center|6
|Oct 16
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|align=center|7
|Oct 23
|Term Test
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|align=center|8
|Oct 30
|HW6
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|align=center|9
|Nov 6
|HW7
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|align=center|10
|Nov 13
|HW8
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|align=center|11
|Nov 20
|HW9
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|align=center|12
|Nov 27
|HW10
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|align=center|13
|Dec 4
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