Drorbn - User contributions [en]

12-240/Classnotes for Thursday November 8

2012-12-04T06:15:46Z

Grace.zhu:

{{12-240/Navigation}}
== Riddle Along ==
Four cars drive in the Sahara desert at constant speeds and with constant directions. A meets B, C, D; B meets C, D. Do C and D meet?

== Goals ==
1. compute Rank T over A 
2. Compute <math>T^{-1}</math> over <math>A^{-1}</math> 
3. Solve systems of linear equations

== Theorems ==
1. Given V' -> V -> W -> W' (where the linear transformations are Q, T, P respectively) 
such that P and Q are invertible (i.e. Q is surjective and P is injective)
then rank T = rank PTQ 
 
2. if T: V -> W, V with basis <math>\beta</math> and W with basis <math>\gamma</math>
rank <math>[T]_\beta^\gamma</math> = rank T 
 
3. if P and Q are invertible matrices, A is some other matrix, rank A = rank PAQ

== Definitions ==
if A = <math>M_(m \times n)</math>, then it is linear transformation <math>T_A : F^n -> F^m</math>

== Lecture notes upload by [[User:yaaleni.vijay|yaaleni.vijay]] ==

<gallery>
Image:12-240 Nov 8 Page 1.jpg|Page 1
Image:12-240 Nov 8 Page 2.jpg|Page 2

</gallery>

12-240/Classnotes for Thursday November 8

2012-12-03T21:51:16Z

Grace.zhu: /* Goals */

== Riddle Along ==
Four cars drive in the Sahara desert at constant speeds and with constant directions. A meets B, C, D; B meets C, D. Do C and D meet?

== Goals ==
1. compute Rank T over A 
2. Compute <math>T^{-1}</math> over <math>A^{-1}</math> 
3. Solve systems of linear equations

== Theorems ==
1. Given V' -> V -> W -> W' (where the linear transformations are Q, T, P respectively) 
such that P and Q are invertible (i.e. Q is surjective and P is injective)
then rank T = rank PTQ 
 
2. if T: V -> W, V with basis <math>\beta</math> and W with basis <math>\gamma</math>
rank <math>[T]_\beta^\gamma</math> = rank T 
 
3. if P and Q are invertible matrices, A is some other matrix, rank A = rank PAQ

== Definitions ==
if A = <math>M_(m \times n)</math>, then it is linear transformation <math>T_A : F^n -> F^m</math>

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

<gallery>
Image:12-240 Nov 8 Page 1.jpg|Page 1
Image:12-240 Nov 8 Page 2.jpg|Page 2

</gallery>

12-240/Classnotes for Thursday November 8

2012-12-03T21:50:02Z

Grace.zhu:

== Riddle Along ==
Four cars drive in the Sahara desert at constant speeds and with constant directions. A meets B, C, D; B meets C, D. Do C and D meet?

== Goals ==
1. compute Rank T over A
2. Compute <math>T^(-1)</math> over <math>A^(-1)</math>
3. Solve systems of linear equations

== Theorems ==
1. Given V' -> V -> W -> W' (where the linear transformations are Q, T, P respectively) 
such that P and Q are invertible (i.e. Q is surjective and P is injective)
then rank T = rank PTQ 
 
2. if T: V -> W, V with basis <math>\beta</math> and W with basis <math>\gamma</math>
rank <math>[T]_\beta^\gamma</math> = rank T 
 
3. if P and Q are invertible matrices, A is some other matrix, rank A = rank PAQ

== Definitions ==
if A = <math>M_(m \times n)</math>, then it is linear transformation <math>T_A : F^n -> F^m</math>

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

<gallery>
Image:12-240 Nov 8 Page 1.jpg|Page 1
Image:12-240 Nov 8 Page 2.jpg|Page 2

</gallery>

12-240/Classnotes for Tuesday November 6

2012-12-03T21:39:05Z

Grace.zhu:

{{12-240/Navigation}}

==Riddle==

Find A and B such that AB - BA = I

==Theorems==
1. Given U with basis <math>\alpha</math>, V with basis <math>\beta</math>, W with basis <math>\gamma,</math>
<math>[T \circ S]_\alpha^\beta = [T]_\beta^\gamma \times [S]_\alpha^\beta</math>

2. For A <math>\in M_(m \times n)</math> and B <math>\in M_(n \times p)</math> and C <math>\in M_(p \times q)</math>,
(AB)C = a(BC)

== Lecture notes scanned by [[User:Zetalda|Zetalda]] ==
<gallery>
Image:12-240-Nov6-1.jpeg|Page 1
Image:12-240-Nov6-2.jpeg|Page 2
Image:12-240-Nov6-3.jpeg|Page 3
</gallery>

12-240/Classnotes for Tuesday November 20

2012-11-22T06:04:07Z

Grace.zhu:

{{12-240/Navigation}}

== Lecture notes scanned by [[User:Zetalda|Zetalda]] ==
<gallery>
Image:12-240-Nov20-1.jpeg|Page 1
Image:12-240-Nov20-2.jpeg|Page 2
Image:12-240-Nov20-3.jpeg|Page 3
Image:12-240-Nov20-4.jpeg|Page 4
</gallery>

12-240/Classnotes for Thursday November 1

2012-11-05T19:23:29Z

Grace.zhu:

{{12-240/Navigation}}
== Lecture notes scanned by [[User:Zetalda|Zetalda]] ==
<gallery>
Image:12-240-Nov1-1.jpeg|Page 1
Image:12-240-Nov1-2.jpeg|Page 2
</gallery>

== Lecture notes uploaded by [[User:Grace.zhu|gracez]] ==
<gallery>
Image:12-240-N1.jpg|Page 1
</gallery>

12-240/Classnotes for Tuesday October 30

2012-11-05T19:23:20Z

Grace.zhu:

{{12-240/Navigation}}
== Lecture notes scanned by [[User:KJMorenz|KJMorenz]] ==
<gallery>
Image:12-240-Oct30.jpg|Oct 30 Page 1
Image:12-240-Oct30-2.jpg|Oct 30 Page 2
</gallery>

== Lecture notes uploaded by [[User:Grace.zhu|gracez]] ==
<gallery>
Image:12-240-O30.jpg|Page 1
</gallery>

12-240/Class Photo

2012-11-04T18:22:17Z

Grace.zhu: /* 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=Wu|first=Lin-Liang|userid=okmijn22|email=wlinliang@gmail.com|location= U of T Shirt standing on the left. |comments=}}
{{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=}}
{{Photo Entry|last=Zhu|first=Grace|userid=gracez|email=grace.zhu@mail.utoronto.ca|location=Third row, second from right |comments=}}


|}

12-240/Classnotes for Thursday November 1

2012-11-04T18:19:50Z

Grace.zhu: /* Lecture notes scanned by Zetalda */

File:12-240-N1.jpg

2012-11-04T18:19:40Z

Grace.zhu:

12-240/Classnotes for Tuesday October 30

2012-11-04T18:17:55Z

Grace.zhu: /* Lecture notes scanned by KJMorenz */

== Lecture notes scanned by [[User:KJMorenz|KJMorenz]] ==
<gallery>
Image:12-240-Oct30.jpg|Oct 30 Page 1
Image:12-240-Oct30-2.jpg|Oct 30 Page 2
</gallery>

== Lecture notes uploaded by [[User:Grace.zhu|gracez]] ==
<gallery>
Image:12-240-O30.jpg|Page 1
</gallery>

File:12-240-O30.jpg

2012-11-04T18:17:43Z

Grace.zhu:

12-240/Classnotes for Thursday October 18

2012-11-04T18:16:27Z

Grace.zhu: /* lecture note on oct 18, uploaded by starash */

{{12-240/Navigation}}
'''== Linear transformation =='''
'''Definition:''' A function L: V-> W is called a linear transformation if it preserve following structures:

1) L(x + y)= L(x) + L(y)

2) L(cx)= c.L(x)

3) L(0 of V) = 0 of W

'''Proposition:'''

1) property 2 => property 3

2) L: V -> W is a linear transformation iff <math>\forall\,\!</math> c <math>\in\,\!</math> F, <math>\forall\,\!</math> x, y <math>\in\,\!</math> V: L(cx + y)= cL(x) + L(y)

'''Proof:
'''
1) take c= 0 in F and x=0 in V. Then L(cx)=cL(x) -> L(0 of F * 0 of V)=(0 of F)*L(0 of V)=0 of W

2)(=>)Assume L is linear transformation

L(cx + y)= L(cx) + L(y)= c*L(x) + L(y)

(<=) 1. Follows from L(c*x+y) = c*L(x)+L(y) by taking c=1
2. Follows by taking y=0

'''Examples'''

1. L: '''R'''^2 -> '''R'''^2 by

2. P,Q: P(F)

== 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>

== Lecture notes uploaded by [[User:Grace.zhu|gracez]] ==
<gallery>
Image:12-240-O18-1.jpg|Page 1
Image:12-240-O18-2.jpg|Page 2
</gallery>

File:12-240-O18-2.jpg

2012-11-04T18:15:45Z

Grace.zhu:

File:12-240-O18-1.jpg

2012-11-04T18:15:35Z

Grace.zhu:

12-240/Classnotes for Tuesday October 09

2012-11-04T18:14:08Z

Grace.zhu: /* Lecture notes scanned by gracez */

{{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>

== Lecture notes uploaded by [[User:Grace.zhu|gracez]] ==
<gallery>
Image:12-240-O9-1.jpg|Page 1
Image:12-240-O9-2.jpg|Page 2
</gallery>

12-240/Classnotes for Tuesday October 09

2012-11-04T18:13:04Z

Grace.zhu: /* Lecture notes scanned by 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>

== Lecture notes scanned by [[User:Grace.zhu|gracez]] ==
<gallery>
Image:12-240-O9-1.jpg|Page 1
Image:12-240-O9-2.jpg|Page 2
</gallery>

File:12-240-O9-2.jpg

2012-11-04T18:12:00Z

Grace.zhu:

File:12-240-O9-1.jpg

2012-11-04T18:10:47Z

Grace.zhu: