Drorbn - User contributions [en]

File:September 29 2009 Lecture Notes Page 6.jpg

2009-10-01T21:51:39Z

Yangjiay:

File:September 29 2009 Lecture Notes Page 5.jpg

2009-10-01T21:51:31Z

Yangjiay:

File:September 29 2009 Lecture Notes Page 4.jpg

2009-10-01T21:51:15Z

Yangjiay:

File:September 29 2009 Lecture Notes Page 3.jpg

2009-10-01T21:51:05Z

Yangjiay:

File:September 29 2009 Lecture Notes Page 2.jpg

2009-10-01T21:50:54Z

Yangjiay:

File:September 29 2009 Lecture Notes Page 1.jpg

2009-10-01T21:50:44Z

Yangjiay:

File:September 22 2009 lecture notes page 5.jpg

2009-09-22T23:43:22Z

Yangjiay:

File:September 22 2009 lecture notes page 4.jpg

2009-09-22T23:43:14Z

Yangjiay:

File:September 22 2009 lecture notes page 3.jpg

2009-09-22T23:43:03Z

Yangjiay:

File:September 22 2009 lecture notes page 2.jpg

2009-09-22T23:42:53Z

Yangjiay:

File:September 22 2009 lecture notes page 1.jpg

2009-09-22T23:42:41Z

Yangjiay:

09-240/Classnotes for Tuesday September 15

2009-09-15T23:28:38Z

Yangjiay:

[[Image:Classnotes For Tuesday, September 15.jpg]]
[[yangjiay:09-240 Classnotes for Tuesday September 15 2009 page 5.jpg]]

The real numbers A set <math>\mathbb R</math> with two binary operators and two special elements <math>0, 1 \in \mathbb R</math> s.t.

: <math>F1.\quad \forall a, b \in \mathbb R, a + b = b + a \mbox{ and } a \cdot b = b \cdot a</math>
: <math>F2.\quad \forall a, b, c, (a + b) + c = a + (b + c) \mbox{ and } (a \cdot b) \cdot c = a \cdot (b \cdot c)</math>
: <math>\mbox{(So for any real numbers } a_1, a_2, ..., a_n, \mbox{ one can sum them in any order and achieve the same result.}</math>
: <math>F3.\quad \forall a, a + 0 = a \mbox{ and } a \cdot 0 = 0 \mbox{ and } a \cdot 1 = a</math>
: <math>F4.\quad \forall a, \exists b, a + b = 0 \mbox{ and } \forall a \ne 0, \exists b, a \cdot b = 1</math>
: <math>\mbox{So } a + (-a) = 0 \mbox{ and } a \cdot a^{-1} = 1</math>
: <math>\mbox{(So } (a + b) \cdot (a - b) = a^2 - b^2)</math>
: <math>\forall a, \exists x, x \cdot x = a \mbox{ or } a + x \cdot x = 0</math>
: Note: '''or''' means '''inclusive or''' in math.
: <math>F5.\quad (a + b) \cdot c = a \cdot c + b \cdot c</math>

Definition: A '''field''' is a set ''F'' with two binary operators <math>\,\!+</math>: ''F''×''F'' → ''F'', <math>\times\,\!</math>: ''F''×''F'' → ''F'' and two elements <math>0, 1 \in \mathbb R</math> s.t.
: <math>F1\quad \mbox{Commutativity } a + b = b + a \mbox{ and } a \cdot b = b \cdot a \forall a, b \in F</math>
: <math>F2\quad \mbox{Associativity } (a + b) + c = a + (b + c) \mbox{ and } (a \cdot b) \cdot c = a \cdot (b \cdot c)</math>
: <math>F3\quad a + 0 = a, a \cdot 1 = a</math>
: <math>F4\quad \forall a, \exists b, a + b = 0 \mbox{ and } \forall a \ne 0, \exists b, a \cdot b = 1</math>
: <math>F5\quad \mbox{Distributivity } (a + b) \cdot c = a \cdot c + b \cdot c</math>

== Examples ==

# <math>F = \mathbb R</math>
# <math>F = \mathbb Q</math>
# <math>\mathbb C = \{ a + bi : a, b \in \mathbb R \}</math>
#: <math>i = \sqrt{-1}</math>
#: <math>\,\!(a + bi) + (c + di) = (a + c) + (b + d)i</math>
#: <math>\,\!0 = 0 + 0i, 1 = 1 + 0i</math>
# <math>\,\!F_2 = \{ 0, 1 \}</math>

...

09-240/Classnotes for Tuesday September 15

2009-09-15T23:28:24Z

Yangjiay:

[[Image:Classnotes For Tuesday, September 15.jpg]]
[[:09-240 Classnotes for Tuesday September 15 2009 page 5.jpg]]

The real numbers A set <math>\mathbb R</math> with two binary operators and two special elements <math>0, 1 \in \mathbb R</math> s.t.

: <math>F1.\quad \forall a, b \in \mathbb R, a + b = b + a \mbox{ and } a \cdot b = b \cdot a</math>
: <math>F2.\quad \forall a, b, c, (a + b) + c = a + (b + c) \mbox{ and } (a \cdot b) \cdot c = a \cdot (b \cdot c)</math>
: <math>\mbox{(So for any real numbers } a_1, a_2, ..., a_n, \mbox{ one can sum them in any order and achieve the same result.}</math>
: <math>F3.\quad \forall a, a + 0 = a \mbox{ and } a \cdot 0 = 0 \mbox{ and } a \cdot 1 = a</math>
: <math>F4.\quad \forall a, \exists b, a + b = 0 \mbox{ and } \forall a \ne 0, \exists b, a \cdot b = 1</math>
: <math>\mbox{So } a + (-a) = 0 \mbox{ and } a \cdot a^{-1} = 1</math>
: <math>\mbox{(So } (a + b) \cdot (a - b) = a^2 - b^2)</math>
: <math>\forall a, \exists x, x \cdot x = a \mbox{ or } a + x \cdot x = 0</math>
: Note: '''or''' means '''inclusive or''' in math.
: <math>F5.\quad (a + b) \cdot c = a \cdot c + b \cdot c</math>

Definition: A '''field''' is a set ''F'' with two binary operators <math>\,\!+</math>: ''F''×''F'' → ''F'', <math>\times\,\!</math>: ''F''×''F'' → ''F'' and two elements <math>0, 1 \in \mathbb R</math> s.t.
: <math>F1\quad \mbox{Commutativity } a + b = b + a \mbox{ and } a \cdot b = b \cdot a \forall a, b \in F</math>
: <math>F2\quad \mbox{Associativity } (a + b) + c = a + (b + c) \mbox{ and } (a \cdot b) \cdot c = a \cdot (b \cdot c)</math>
: <math>F3\quad a + 0 = a, a \cdot 1 = a</math>
: <math>F4\quad \forall a, \exists b, a + b = 0 \mbox{ and } \forall a \ne 0, \exists b, a \cdot b = 1</math>
: <math>F5\quad \mbox{Distributivity } (a + b) \cdot c = a \cdot c + b \cdot c</math>

== Examples ==

# <math>F = \mathbb R</math>
# <math>F = \mathbb Q</math>
# <math>\mathbb C = \{ a + bi : a, b \in \mathbb R \}</math>
#: <math>i = \sqrt{-1}</math>
#: <math>\,\!(a + bi) + (c + di) = (a + c) + (b + d)i</math>
#: <math>\,\!0 = 0 + 0i, 1 = 1 + 0i</math>
# <math>\,\!F_2 = \{ 0, 1 \}</math>

...

09-240/Classnotes for Tuesday September 15

2009-09-15T23:28:04Z

Yangjiay:

[[Image:Classnotes For Tuesday, September 15.jpg]]
09-240 Classnotes for Tuesday September 15 2009 page 5.jpg

The real numbers A set <math>\mathbb R</math> with two binary operators and two special elements <math>0, 1 \in \mathbb R</math> s.t.

: <math>F1.\quad \forall a, b \in \mathbb R, a + b = b + a \mbox{ and } a \cdot b = b \cdot a</math>
: <math>F2.\quad \forall a, b, c, (a + b) + c = a + (b + c) \mbox{ and } (a \cdot b) \cdot c = a \cdot (b \cdot c)</math>
: <math>\mbox{(So for any real numbers } a_1, a_2, ..., a_n, \mbox{ one can sum them in any order and achieve the same result.}</math>
: <math>F3.\quad \forall a, a + 0 = a \mbox{ and } a \cdot 0 = 0 \mbox{ and } a \cdot 1 = a</math>
: <math>F4.\quad \forall a, \exists b, a + b = 0 \mbox{ and } \forall a \ne 0, \exists b, a \cdot b = 1</math>
: <math>\mbox{So } a + (-a) = 0 \mbox{ and } a \cdot a^{-1} = 1</math>
: <math>\mbox{(So } (a + b) \cdot (a - b) = a^2 - b^2)</math>
: <math>\forall a, \exists x, x \cdot x = a \mbox{ or } a + x \cdot x = 0</math>
: Note: '''or''' means '''inclusive or''' in math.
: <math>F5.\quad (a + b) \cdot c = a \cdot c + b \cdot c</math>

Definition: A '''field''' is a set ''F'' with two binary operators <math>\,\!+</math>: ''F''×''F'' → ''F'', <math>\times\,\!</math>: ''F''×''F'' → ''F'' and two elements <math>0, 1 \in \mathbb R</math> s.t.
: <math>F1\quad \mbox{Commutativity } a + b = b + a \mbox{ and } a \cdot b = b \cdot a \forall a, b \in F</math>
: <math>F2\quad \mbox{Associativity } (a + b) + c = a + (b + c) \mbox{ and } (a \cdot b) \cdot c = a \cdot (b \cdot c)</math>
: <math>F3\quad a + 0 = a, a \cdot 1 = a</math>
: <math>F4\quad \forall a, \exists b, a + b = 0 \mbox{ and } \forall a \ne 0, \exists b, a \cdot b = 1</math>
: <math>F5\quad \mbox{Distributivity } (a + b) \cdot c = a \cdot c + b \cdot c</math>

== Examples ==

# <math>F = \mathbb R</math>
# <math>F = \mathbb Q</math>
# <math>\mathbb C = \{ a + bi : a, b \in \mathbb R \}</math>
#: <math>i = \sqrt{-1}</math>
#: <math>\,\!(a + bi) + (c + di) = (a + c) + (b + d)i</math>
#: <math>\,\!0 = 0 + 0i, 1 = 1 + 0i</math>
# <math>\,\!F_2 = \{ 0, 1 \}</math>

...

09-240/Classnotes for Tuesday September 15

2009-09-15T23:26:32Z

Yangjiay:

[[Image:Classnotes For Tuesday, September 15.jpg]]
[[09-240 Classnotes for Tuesday September 15 2009 page 5.jpg]]

The real numbers A set <math>\mathbb R</math> with two binary operators and two special elements <math>0, 1 \in \mathbb R</math> s.t.

: <math>F1.\quad \forall a, b \in \mathbb R, a + b = b + a \mbox{ and } a \cdot b = b \cdot a</math>
: <math>F2.\quad \forall a, b, c, (a + b) + c = a + (b + c) \mbox{ and } (a \cdot b) \cdot c = a \cdot (b \cdot c)</math>
: <math>\mbox{(So for any real numbers } a_1, a_2, ..., a_n, \mbox{ one can sum them in any order and achieve the same result.}</math>
: <math>F3.\quad \forall a, a + 0 = a \mbox{ and } a \cdot 0 = 0 \mbox{ and } a \cdot 1 = a</math>
: <math>F4.\quad \forall a, \exists b, a + b = 0 \mbox{ and } \forall a \ne 0, \exists b, a \cdot b = 1</math>
: <math>\mbox{So } a + (-a) = 0 \mbox{ and } a \cdot a^{-1} = 1</math>
: <math>\mbox{(So } (a + b) \cdot (a - b) = a^2 - b^2)</math>
: <math>\forall a, \exists x, x \cdot x = a \mbox{ or } a + x \cdot x = 0</math>
: Note: '''or''' means '''inclusive or''' in math.
: <math>F5.\quad (a + b) \cdot c = a \cdot c + b \cdot c</math>

Definition: A '''field''' is a set ''F'' with two binary operators <math>\,\!+</math>: ''F''×''F'' → ''F'', <math>\times\,\!</math>: ''F''×''F'' → ''F'' and two elements <math>0, 1 \in \mathbb R</math> s.t.
: <math>F1\quad \mbox{Commutativity } a + b = b + a \mbox{ and } a \cdot b = b \cdot a \forall a, b \in F</math>
: <math>F2\quad \mbox{Associativity } (a + b) + c = a + (b + c) \mbox{ and } (a \cdot b) \cdot c = a \cdot (b \cdot c)</math>
: <math>F3\quad a + 0 = a, a \cdot 1 = a</math>
: <math>F4\quad \forall a, \exists b, a + b = 0 \mbox{ and } \forall a \ne 0, \exists b, a \cdot b = 1</math>
: <math>F5\quad \mbox{Distributivity } (a + b) \cdot c = a \cdot c + b \cdot c</math>

== Examples ==

# <math>F = \mathbb R</math>
# <math>F = \mathbb Q</math>
# <math>\mathbb C = \{ a + bi : a, b \in \mathbb R \}</math>
#: <math>i = \sqrt{-1}</math>
#: <math>\,\!(a + bi) + (c + di) = (a + c) + (b + d)i</math>
#: <math>\,\!0 = 0 + 0i, 1 = 1 + 0i</math>
# <math>\,\!F_2 = \{ 0, 1 \}</math>

...

File:09-240 Classnotes for Tuesday September 15 2009 page 5.jpg

2009-09-15T23:24:49Z

Yangjiay:

File:09-240 Classnotes for Tuesday September 15 2009 page 4.jpg

2009-09-15T23:24:39Z

Yangjiay:

File:09-240 Classnotes for Tuesday September 15 2009 page 3.jpg

2009-09-15T23:24:14Z

Yangjiay:

File:09-240 Classnotes for Tuesday September 15 2009 page 2.jpg

2009-09-15T23:24:01Z

Yangjiay:

File:09-240 Classnotes for Tuesday September 15 2009 page 1.jpg

2009-09-15T23:23:39Z

Yangjiay:

08-401/Class Notes for April 9

2008-04-10T14:41:13Z

Yangjiay:

{{08-401/Navigation}}
{{08-401/Class Notes Warning}}
===Notes===
== '''April 09''' ==
Lecture notes by '''Z.Y'''

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_9_.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_9_1.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_9_2.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_9_3.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_9_4.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_9_5.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_9_6.jpg

Here is one set of [[08-401-Apr09-Notes|notes by LeoDaVinci]]

upload by yangjiay
http://katlas.math.toronto.edu/drorbn/images/f/fb/08_401_April_9_2008_lecture_notes.pdf

08-401/Class Notes for April 9

2008-04-10T14:40:57Z

Yangjiay:

{{08-401/Navigation}}
{{08-401/Class Notes Warning}}
===Notes===
== '''April 09''' ==
Lecture notes by '''Z.Y'''

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_9_.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_9_1.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_9_2.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_9_3.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_9_4.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_9_5.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_9_6.jpg

Here is one set of [[08-401-Apr09-Notes|notes by LeoDaVinci]]

upload by yangjiay

http://katlas.math.toronto.edu/drorbn/images/f/fb/08_401_April_9_2008_lecture_notes.pdf

08-401/Class Notes for April 9

2008-04-10T14:40:44Z

Yangjiay:

08-401

2008-04-10T14:39:57Z

Yangjiay:

__NOEDITSECTION__
{{08-401/Navigation}}
==Polynomial Equations and Fields==
===Department of Mathematics, University of Toronto, Spring 2008===

{{08-401/Crucial Information}}

===Further Resources===

* [http://www.d.umn.edu/~jgallian/ J. Gallian's Algebra web site].
* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]
* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions] at the [http://www.artsci.utoronto.ca/ Faculty of Arts and Science].
* Last year's class: [[07-401]].
* [http://www.math.toronto.edu/gor/mat401.html 06-401] with Julia Gordon.
* [http://www.math.utoronto.ca/shub/mat302S.html 05-302] with Mike Shub.
* [http://ccnet.utoronto.ca/20069/mat301h1f/ 06-301] with Lindsey Shorser.

{| align=center
|- align=center
|[[Image:07-401-Solving The Quartic With Mathematica.png|640px]]
|- align=center
|Solving The Quartic With Mathematica. [http://en.wikipedia.org/wiki/Quartic_equation Read more!]
|}

upload by yangjiay
http://katlas.math.toronto.edu/drorbn/images/f/fb/08_401_April_9_2008_lecture_notes.pdf

File:08 401 April 9 2008 lecture notes.pdf

2008-04-10T14:38:51Z

Yangjiay:

08-401/Class Notes for March 12

2008-04-08T02:26:35Z

Yangjiay:

{{08-401/Navigation}}
{{08-401/Class Notes Warning}}

===Notes===
Here is one set of [[08-401-Mar12-Notes|notes by LeoDaVinci]]

Uploaded by leelorik:

[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:03-12-2008_09%3B09%3B32PM.JPG page 1 front]
[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:03-12-2008_09%3B10%3B04PM.JPG page 1 back]

[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:03-12-2008_09%3B10%3B26PM.JPG page 2 front]
[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:03-12-2008_09%3B10%3B44PM.JPG page 2 back]

[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:03-12-2008_09%3B11%3B03PM.JPG page 3 front]
[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:03-12-2008_09%3B11%3B20PM.JPG page 3 back]

[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:03-12-2008_09%3B11%3B39PM.JPG page 4]

== '''March 12''' ==

Lecture notes by '''Z.Y'''

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_March_12_001.jpg
http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_March_12_002.jpg
http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_March_12_003.jpg
http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_March_12_004.jpg
http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_March_12_005.jpg
http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_March_12_006.jpg
http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_March_12_007.jpg
http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_March_12_008.jpg
http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_March_12_009.jpg
http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_March_12_010.jpg

upload by yangjiay

http://katlas.math.toronto.edu/drorbn/images/d/dc/08-401_Lecture_Notes_March_12_2008.pdf

08-401/Class Notes for March 12

2008-04-08T02:26:21Z

Yangjiay:

File:08-401 Lecture Notes March 12 2008.pdf

2008-04-08T02:25:27Z

Yangjiay:

08-401/Class Notes for March 12

2008-04-08T02:24:45Z

Yangjiay:

File:Media - 08-401 Lecture Notes March 12 2008.pdf

2008-04-08T02:23:41Z

Yangjiay:

08-401/Homework Assignment 8

2008-04-06T01:10:06Z

Yangjiay:

{{08-401/Navigation}}

===Reading===
Read chapter 32 of Gallian's book ('''6th edition''') three times:
* First time as if you were reading a novel - quickly and without too much attention to detail, just to learn what the main keywords and concepts and goals are.
* Second time like you were studying for an exam on the subject - slowly and not skipping anything, verifying every little detail.
* And then a third time, again at a quicker pace, to remind yourself of the bigger picture all those little details are there to paint.

'''Also,''' spend some time reviewing what you've learned in the past about group theory, with special emphasis on normal subgroups, quotients and the isomorphism theorems.

===Doing===
Solve problems 1, '''S2''' (<math>GF(4):=({\mathbb Z}/2)[x]/\langle x^2+x+1\rangle</math> is the field with 4 elements), '''S4''', '''S6''', 7, '''S8''', 9, '''S14''' (<math>D_5</math> is defined on page 34), '''S15''', 16, 17, and 18 in chapter 32 of Gallian's book ('''6th edition'''), but submit only the solutions of the problems marked with the letter "S".

===Due Date===
This assignment is due in class on Wednesday March 26, 2008.

Here is the solution upload by yangjiay
http://katlas.math.toronto.edu/drorbn/images/8/82/Solution_Assignment_8.pdf

===Just for Fun===
Let <math>x</math> be some root of the equation
{{Equation*|<math>x^5+(\sqrt[3]{2}-\sqrt{3})x^4+\frac{1}{\sqrt{\sqrt{3}+\sqrt{5}}}x^3-1=0.</math>}}
We know that <math>x</math> is algebraic. Can you find a polynomial <math>f</math> with rational coefficients whose roots include <math>x</math>? Can you find the minimal polynomial <math>p</math> of <math>x</math>? What is <math>\deg p</math>?

'''Warning.''' "Can you find?" should be interpreted as "How would you find?" and not as "Please find.". The latter is doable, but not by hand!

'''Prize.''' Though if you do actually find such a polynomial <math>f</math> ''through your own efforts'', post it on this site along with your detailed and repeatable computations leading to it (a computer program, I presume, fully documented so that anybody else could run it too), and you will receive up to 60 bonus points for the feat, counted towards your final grade in the same manner as the "good deed" points. The due date for prize claims is the last day of classes.

File:Solution Assignment 8.pdf

2008-04-06T01:09:22Z

Yangjiay:

08-401/Homework Assignment 7

2008-04-06T00:45:58Z

Yangjiay:

{{08-401/Navigation}}

===Reading===
Read chapters 21 (excluding theorem 21.6) and 23 of Gallian's book ('''6th edition''') three times:
* First time as if you were reading a novel - quickly and without too much attention to detail, just to learn what the main keywords and concepts and goals are.
* Second time like you were studying for an exam on the subject - slowly and not skipping anything, verifying every little detail.
* And then a third time, again at a quicker pace, to remind yourself of the bigger picture all those little details are there to paint.

===Doing===
Solve problems 3, 5, '''S6''', 7, '''S8''', 9, 11, '''S12''', '''S13''', 14, '''S16''', '''S18''', 22, 34 in Chapter 21 of Gallian's book ('''6th edition''') and problems 1-5 in Chapter 23 of the same book, but submit only the solutions of the problems marked with the letter "S".

===Due Date===
This assignment is due in class on Wednesday March 19, 2008.

Here is the solution upload by yangjiay
http://katlas.math.toronto.edu/drorbn/images/2/2c/Solution_Assignment_7.pdf
===Just for Fun===
Geometric constructions are related to fields (see it in another way by reading/solving questions 1-4 on pages 394-5 of our text). A related fact is that using "geometrical machines" one can build "analog computers" that are able to "compute" polynomials. For your own entertainment and for that alone, spend some time studying Dori Eldar's java-based interactive site on [http://www.math.toronto.edu/~drorbn/People/Eldar/thesis/comutational_linkages.htm Linkages as Functions] and in particular play with gigantic [http://www.math.toronto.edu/~drorbn/People/Eldar/thesis/squaring.htm squaring machine] (drag the green dot, read the output in red). While you are at it, spend some time pondering the relationship between Eldar's "machines" and our class material. The subjects are not exactly the same, yet not entirely different either.

<center>http://www.math.toronto.edu/~drorbn/classes/0304/157AnalysisI/HW25/SquaringMachine.gif</center>

File:Solution Assignment 7.pdf

2008-04-06T00:45:13Z

Yangjiay:

08-401/Homework Assignment 6

2008-04-06T00:13:33Z

Yangjiay:

{{08-401/Navigation}}

===Reading===
Read chapters 17 and 20 (to page 360) of Gallian's book ('''6th edition''') three times:
* First time as if you were reading a novel - quickly and without too much attention to detail, just to learn what the main keywords and concepts and goals are.
* Second time like you were studying for an exam on the subject - slowly and not skipping anything, verifying every little detail.
* And then a third time, again at a quicker pace, to remind yourself of the bigger picture all those little details are there to paint.

===Doing===
Solve problems '''S10''', 14, 25, 27 and '''S32''' in Chapter 17 of Gallian's book ('''6th edition''') and problems '''S2''', 3, 6, '''S7''', 8, 12 and '''S16''' in Chapter 20 of the same book, but submit only the solutions of the problems marked with the letter "S".

===Due Date===
This assignment is due in class on Wednesday March 12, 2008.

solution to Assignment 6 upload by yangjiay
http://katlas.math.toronto.edu/drorbn/images/c/c2/Solution_Assignment_6.pdf

File:Solution Assignment 6.pdf

2008-04-06T00:12:09Z

Yangjiay:

08-401/Class Notes for April 2

2008-04-05T23:39:13Z

Yangjiay:

{{08-401/Navigation}}
{{08-401/Class Notes Warning}}
===Notes===
Here is one set of [[08-401-Apr02-Notes|notes by LeoDaVinci]]
== '''April 02''' ==

lecture notes by Z.Y

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_001.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_003.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_005.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_007.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_009.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_011.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_012.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_013.jpg

Lecture Notes for Apr 02 by gopsury83[http://katlas.math.toronto.edu/drorbn/images/1/15/080402.pdf]

upload by yangjiay

http://katlas.math.toronto.edu/drorbn/images/e/e1/Media-_08-401_April_2_2008_lecture_Notes.pdf

08-401/Class Notes for April 2

2008-04-05T23:38:57Z

Yangjiay:

{{08-401/Navigation}}
{{08-401/Class Notes Warning}}
===Notes===
Here is one set of [[08-401-Apr02-Notes|notes by LeoDaVinci]]
== '''April 02''' ==

lecture notes by Z.Y

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_001.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_003.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_005.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_007.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_009.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_011.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_012.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_013.jpg

Lecture Notes for Apr 02 by gopsury83[http://katlas.math.toronto.edu/drorbn/images/1/15/080402.pdf]

upload by yangjiay

[http://katlas.math.toronto.edu/drorbn/images/e/e1/Media-_08-401_April_2_2008_lecture_Notes.pdf]

08-401/Class Notes for April 2

2008-04-05T23:38:39Z

Yangjiay:

{{08-401/Navigation}}
{{08-401/Class Notes Warning}}
===Notes===
Here is one set of [[08-401-Apr02-Notes|notes by LeoDaVinci]]
== '''April 02''' ==

lecture notes by Z.Y

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_001.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_003.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_005.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_007.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_009.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_011.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_012.jpg

http://katlas.math.toronto.edu/drorbn/index.php?title=Image:08-401_april_2_013.jpg

Lecture Notes for Apr 02 by gopsury83[http://katlas.math.toronto.edu/drorbn/images/1/15/080402.pdf]

upload by yangjiay

[[http://katlas.math.toronto.edu/drorbn/images/e/e1/Media-_08-401_April_2_2008_lecture_Notes.pdf]]

08-401/Class Notes for April 2

2008-04-05T23:38:10Z

Yangjiay:

08-401/Class Notes for April 2

2008-04-05T23:37:38Z

Yangjiay:

08-401/Class Notes for April 2

2008-04-05T23:36:27Z

Yangjiay:

08-401/Class Notes for April 2

2008-04-05T23:33:48Z

Yangjiay:

08-401/Class Notes for April 2

2008-04-05T23:32:51Z

Yangjiay:

08-401/Class Notes for April 2

2008-04-05T23:30:23Z

Yangjiay:

File:Media- 08-401 April 2 2008 lecture Notes.pdf

2008-04-05T23:29:53Z

Yangjiay:

08-401/Class Notes for April 2

2008-04-05T23:28:48Z

Yangjiay:

08-401/Class Notes for April 2

2008-04-05T23:23:50Z

Yangjiay:

08-401/Class Notes for April 2

2008-04-05T23:21:59Z

Yangjiay:

File:Media 08-401 April 2 2008 lecture Notes.pdf

2008-04-05T23:19:33Z

Yangjiay: