https://drorbn.net/api.php?action=feedcontributions&feedformat=atom&user=ZapyreDrorbn - User contributions [en]2026-05-01T19:34:15ZUser contributionsMediaWiki 1.39.6https://drorbn.net/index.php?title=06-240/Final_Exam_Preparation_Forum&diff=321806-240/Final Exam Preparation Forum2006-12-11T23:13:42Z<p>Zapyre: transfered my question to solved</p>
<hr />
<div>{{06-240/Navigation}}<br />
<br />
If you have questions, ask them here and hopefully someone else will know the answer. (Answering questions will probably help you understand it more).<br />
<br />
Since many of us (including I) don't really know how to use Wiki's, I suggest that we keep the formatting simple: I will post a template at the top of this page, and if you want to add something just click on the "edit", copy the template, and insert your question. Order the questions according to section (solved/unsolved, as judge by whoever created the question), with the newest at the top, except for the template question. In general, I wouldn't retype the question if it's from the book because that's tedious and we all have the book.<br />
<br />
(By the way, I think you leave a space between lines in the code to make a new line; that is, simply pressing enter once will not make a new line. Also, you can press a button at the top of the editing textbox that lets you put in simple equations.)<br />
<br />
==Unsolved Questions==<br />
<br />
===Question Template===<br />
Q: Can someone help me prove: "If an [[integer]] <math>n</math> is greater than 2, then <math>a^n + b^n = c^n</math> has no solutions in non-zero integers <math>a</math>, <math>b</math>, and <math>c</math>."? I had the answer in my head at one point, but the margins of the piece of paper I was working with was too small to fit it.<br />
<br />
===Exam April/May 2006 #7===<br />
Q: Let T : V -> V and U : V -> V be linear operators on a finite-dimensional vector space V. Assume that U is invertible and T is diagonalizable. Prove that the linear operator UTU<Sup>-1</Sup> = U o T o U<Sup>-1</Sup>.<br />
<br />
I dont know where or how to start this question ><.<br />
<br />
A: I think we need to prove that UTU<Sup>-1</Sup> is diagonalizable instead of proving UTU<Sup>-1</Sup> = U o T o U<Sup>-1</Sup>. <br />
<br />
I started by letting A = UTU<Sup>-1</Sup>, then multiplying U<Sup>-1</Sup> and U to the both sides, we get U<Sup>-1</Sup>AU = U<Sup>-1</Sup>UTU<Sup>-1</Sup>U iff U<Sup>-1</Sup>AU = T. Since T is diagonalizable and U is invertible, therefore A and T is similar, thus A is diagonalizable. Please comment. Thanks.<br />
<br />
===Sec. 2.4 Lemma p. 101===<br />
Q: In the proof of the lemma, the second line, we have "T(beta) spans R(T) = W". How do we know that R(T) = W? This would be true if dim V = dim W, because then T would be onto, but we can't assume what we're trying to prove.<br />
<br />
===Exam April/May 2006 #4===<br />
Q: Suppose that A, B Є M<sub>mxn</sub>(F), and rank(A) = rank (B). Prove that there exist invertible matrices P Є M<sub>mxm</sub>(F) and Q Є M<sub>nxn</sub>(F) such that B = PAQ.<br />
<br />
A(partial): Here is a sketch. If you rref A and B by applying a series of elementary row operation matricies, they will both look similar. That is, they will have a section of 1's and 0's (each 1 is the only number in its column) and then a section of "remaining stuff", and these sections will be the same "size" because their ranks are the same. Then, using the elementary column matrix operations, you can essentially modify the "remaining stuff" as much as you like, by adding multiples of the "nice" columns (with single-1's). These row and column operations can then be grouped nicely and set to be equal to P and Q, which are invertible because products of elementary matricies are invertible. <br />
<br />
I know this is very rough, but even if I did have a full answer I wouldn't now how to typeset it.<br />
<br />
===Complex Numbers===<br />
Q: If 'C' is used in the context of a vector space (as in "define T:C->C"), then should we consider C to be the vector space of C over the field C, or instead C over the field R?<br />
<br />
===Readings?===<br />
Q: Are we expected to section 5.2 of the textbook? Although the Assignments tell us to read it, we didn't do any questions, or cover it in class.<br />
<br />
==Solved Questions==<br />
<br />
===Question Template===<br />
Q: How many ways are there to get to the nth stair, if at each step you can move either one or two squares up?<br />
<br />
A: This question can be easily modeled by the Fibonacci numbers, with the nth number being the ways to get to the nth stair. This is because, to get to the nth stair, you can come only from the n-1th or the n-2th. This is exactly how the Fibonacci numbers are defined; the proof is simple by induction.<br />
<br />
===Sec. 1.3 Thm 1.3 Proof===<br />
Q: In the first paragraph of the proof, it says "But also x + 0 = x , and thus 0'=0." How do we know 0 (that is 0 of V) even exists in W? I understand that we know ''some'' zero exists (0'), but not why ''the'' zero (0) exists.<br />
<br />
A: x is in W as well in V. Thus, x + 0 = x (VS 3).<br />
<br />
Reply: Oh I see... now it looks so obvious =/. Thanks.<br />
<br />
===Exam April/May 2006 #3(b)===<br />
Q: Let T : M<Sub>3x2</Sub>(C) -> M<Sub>2x3</Sub>(C) be defined as follows. Given A Є M<Sub>3x2</Sub>(C), let B be the matrix obtained from A by adding i times the second row of A to the third row of A. Let T(A) = B<Sup>t</Sup>, where B<Sup>t</Sup> is the transpose of B. (Note: Here, i is a complex number such that i<Sup>2</Sup> = -1.) Determine whether the linear transformation T is invertible.<br />
<br />
Totally lost on this question :/ Please show some example matrix and how it is transformed as the question asks if possible. I want to see what actually happens to the elements in the matrix rather than the answer (think that would be more important)<br />
<br />
A(Matrix Elements): <br />
This is my interpretation:<br />
<br />
A = <math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}</math> where <math>a_{ij} \in C</math>, then B = <math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\ia_{21}+a_{31}&ia_{22}+a_{32}\end{pmatrix}</math>.<br />
<br />
Therefore, T(A) = B<Sup>t</Sup> is T<math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}</math>=<math>\begin{pmatrix}a_{11}&a_{21}&ia_{21}+a_{31}\\ia_{12}&a_{22}&ia_{22}+a_{32}\end{pmatrix}</math><br />
<br />
R: Thx alot, the matricies are really helpful :)</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Final_Exam_Preparation_Forum&diff=321306-240/Final Exam Preparation Forum2006-12-11T21:03:47Z<p>Zapyre: </p>
<hr />
<div>{{06-240/Navigation}}<br />
<br />
If you have questions, ask them here and hopefully someone else will know the answer. (Answering questions will probably help you understand it more).<br />
<br />
Since many of us (including I) don't really know how to use Wiki's, I suggest that we keep the formatting simple: I will post a template at the top of this page, and if you want to add something just click on the "edit", copy the template, and insert your question. Order the questions according to section (solved/unsolved, as judge by whoever created the question), with the newest at the top, except for the template question. In general, I wouldn't retype the question if it's from the book because that's tedious and we all have the book.<br />
<br />
(By the way, I think you leave a space between lines in the code to make a new line; that is, simply pressing enter once will not make a new line. Also, you can press a button at the top of the editing textbox that lets you put in simple equations.)<br />
<br />
==Unsolved Questions==<br />
<br />
===Question Template===<br />
Q: Can someone help me prove: "If an [[integer]] <math>n</math> is greater than 2, then <math>a^n + b^n = c^n</math> has no solutions in non-zero integers <math>a</math>, <math>b</math>, and <math>c</math>."? I had the answer in my head at one point, but the margins of the piece of paper I was working with was too small to fit it.<br />
<br />
===Exam April/May 2006 #3(b)===<br />
Q: Let T : M<Sub>3x2</Sub>(C) -> M<Sub>2x3</Sub>(C) be defined as follows. Given A Є M<Sub>3x2</Sub>(C), let B be the matrix obtained from A by adding i times the second row of A to the third row of A. Let T(A) = B<Sup>t</Sup>, where B<Sup>t</Sup> is the transpose of B. (Note: Here, i is a complex number such that i<Sup>2</Sup> = -1.) Determine whether the linear transformation T is invertible.<br />
<br />
Totally lost on this question :/ Please show some example matrix and how it is transformed as the question asks if possible. I want to see what actually happens to the elements in the matrix rather than the answer (think that would be more important)<br />
<br />
===Exam April/May 2006 #7===<br />
Q: Let T : V -> V and U : V -> V be linear operators on a finite-dimensional vector space V. Assume that U is invertible and T is diagonalizable. Prove that the linear operator UTU<Sup>-1</Sup> = U o T o U<Sup>-1</Sup>.<br />
<br />
I dont know where or how to start this question ><.<br />
<br />
===Sec. 2.4 Lemma p. 101===<br />
Q: In the proof of the lemma, the second line, we have "T(beta) spans R(T) = W". How do we know that R(T) = W? This would be true if dim V = dim W, because then T would be onto, but we can't assume what we're trying to prove.<br />
<br />
===Exam April/May 2006 #4===<br />
Q: Suppose that A, B Є M<sub>mxn</sub>(F), and rank(A) = rank (B). Prove that there exist invertible matrices P Є M<sub>mxm</sub>(F) and Q Є M<sub>nxn</sub>(F) such that B = PAQ.<br />
<br />
A(partial): Here is a sketch. If you rref A and B by applying a series of elementary row operation matricies, they will both look similar. That is, they will have a section of 1's and 0's (each 1 is the only number in its column) and then a section of "remaining stuff", and these sections will be the same "size" because their ranks are the same. Then, using the elementary column matrix operations, you can essentially modify the "remaining stuff" as much as you like, by adding multiples of the "nice" columns (with single-1's). These row and column operations can then be grouped nicely and set to be equal to P and Q, which are invertible because products of elementary matricies are invertible. <br />
<br />
I know this is very rough, but even if I did have a full answer I wouldn't now how to typeset it.<br />
<br />
===Complex Numbers===<br />
Q: If 'C' is used in the context of a vector space (as in "define T:C->C"), then should we consider C to be the vector space of C over the field C, or instead C over the field R?<br />
<br />
==Solved Questions==<br />
<br />
===Question Template===<br />
Q: How many ways are there to get to the nth stair, if at each step you can move either one or two squares up?<br />
<br />
A: This question can be easily modeled by the Fibonacci numbers, with the nth number being the ways to get to the nth stair. This is because, to get to the nth stair, you can come only from the n-1th or the n-2th. This is exactly how the Fibonacci numbers are defined; the proof is simple by induction.<br />
<br />
===Sec. 1.3 Thm 1.3 Proof===<br />
Q: In the first paragraph of the proof, it says "But also x + 0 = x , and thus 0'=0." How do we know 0 (that is 0 of V) even exists in W? I understand that we know ''some'' zero exists (0'), but not why ''the'' zero (0) exists.<br />
<br />
A: x is in W as well in V. Thus, x + 0 = x (VS 3).<br />
<br />
Reply: Oh I see... now it looks so obvious =/. Thanks.</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Final_Exam_Preparation_Forum&diff=321206-240/Final Exam Preparation Forum2006-12-11T21:03:13Z<p>Zapyre: </p>
<hr />
<div>{{06-240/Navigation}}<br />
<br />
If you have questions, ask them here and hopefully someone else will know the answer. (Answering questions will probably help you understand it more).<br />
<br />
Since many of us (including I) don't really know how to use Wiki's, I suggest that we keep the formatting simple: I will post a template at the top of this page, and if you want to add something just click on the "edit", copy the template, and insert your question. Order the questions according to section (solved/unsolved, as judge by whoever created the question), with the newest at the top, except for the template question. In general, I wouldn't retype the question if it's from the book because that's tedious and we all have the book.<br />
<br />
(By the way, I think you leave a space between lines in the code to make a new line; that is, simply pressing enter once will not make a new line. Also, you can press a button at the top of the editing textbox that lets you put in simple equations.)<br />
<br />
==Unsolved Questions==<br />
<br />
===Question Template===<br />
Q: Can someone help me prove: "If an [[integer]] <math>n</math> is greater than 2, then <math>a^n + b^n = c^n</math> has no solutions in non-zero integers <math>a</math>, <math>b</math>, and <math>c</math>."? I had the answer in my head at one point, but the margins of the piece of paper I was working with was too small to fit it.<br />
<br />
===Exam April/May 2006 #3(b)===<br />
Q: Let T : M<Sub>3x2</Sub>(C) -> M<Sub>2x3</Sub>(C) be defined as follows. Given A Є M<Sub>3x2</Sub>(C), let B be the matrix obtained from A by adding i times the second row of A to the third row of A. Let T(A) = B<Sup>t</Sup>, where B<Sup>t</Sup> is the transpose of B. (Note: Here, i is a complex number such that i<Sup>2</Sup> = -1.) Determine whether the linear transformation T is invertible.<br />
<br />
----<br />
Totally lost on this question :/ Please show some example matrix and how it is transformed as the question asks if possible. I want to see what actually happens to the elements in the matrix rather than the answer (think that would be more important)<br />
<br />
===Exam April/May 2006 #7===<br />
Q: Let T : V -> V and U : V -> V be linear operators on a finite-dimensional vector space V. Assume that U is invertible and T is diagonalizable. Prove that the linear operator UTU<Sup>-1</Sup> = U o T o U<Sup>-1</Sup>.<br />
----<br />
I dont know where or how to start this question ><.<br />
<br />
===Sec. 2.4 Lemma p. 101===<br />
Q: In the proof of the lemma, the second line, we have "T(beta) spans R(T) = W". How do we know that R(T) = W? This would be true if dim V = dim W, because then T would be onto, but we can't assume what we're trying to prove.<br />
<br />
===Exam April/May 2006 #4===<br />
Q: Suppose that A, B Є M<sub>mxn</sub>(F), and rank(A) = rank (B). Prove that there exist invertible matrices P Є M<sub>mxm</sub>(F) and Q Є M<sub>nxn</sub>(F) such that B = PAQ.<br />
<br />
A(partial): Here is a sketch. If you rref A and B by applying a series of elementary row operation matricies, they will both look similar. That is, they will have a section of 1's and 0's (each 1 is the only number in its column) and then a section of "remaining stuff", and these sections will be the same "size" because their ranks are the same. Then, using the elementary column matrix operations, you can essentially modify the "remaining stuff" as much as you like, by adding multiples of the "nice" columns (with single-1's). These row and column operations can then be grouped nicely and set to be equal to P and Q, which are invertible because products of elementary matricies are invertible. <br />
<br />
I know this is very rough, but even if I did have a full answer I wouldn't now how to typeset it.<br />
<br />
===Complex Numbers===<br />
Q: If 'C' is used in the context of a vector space (as in "define T:C->C"), then should we consider C to be the vector space of C over the field C, or instead C over the field R?<br />
<br />
==Solved Questions==<br />
<br />
===Question Template===<br />
Q: How many ways are there to get to the nth stair, if at each step you can move either one or two squares up?<br />
<br />
A: This question can be easily modeled by the Fibonacci numbers, with the nth number being the ways to get to the nth stair. This is because, to get to the nth stair, you can come only from the n-1th or the n-2th. This is exactly how the Fibonacci numbers are defined; the proof is simple by induction.<br />
<br />
===Sec. 1.3 Thm 1.3 Proof===<br />
Q: In the first paragraph of the proof, it says "But also x + 0 = x , and thus 0'=0." How do we know 0 (that is 0 of V) even exists in W? I understand that we know ''some'' zero exists (0'), but not why ''the'' zero (0) exists.<br />
<br />
A: x is in W as well in V. Thus, x + 0 = x (VS 3).<br />
<br />
Reply: Oh I see... now it looks so obvious =/. Thanks.</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Final_Exam_Preparation_Forum&diff=321106-240/Final Exam Preparation Forum2006-12-11T21:00:27Z<p>Zapyre: quick comment</p>
<hr />
<div>{{06-240/Navigation}}<br />
<br />
If you have questions, ask them here and hopefully someone else will know the answer. (Answering questions will probably help you understand it more).<br />
<br />
Since many of us (including I) don't really know how to use Wiki's, I suggest that we keep the formatting simple: I will post a template at the top of this page, and if you want to add something just click on the "edit", copy the template, and insert your question. Order the questions according to section (solved/unsolved, as judge by whoever created the question), with the newest at the top, except for the template question. In general, I wouldn't retype the question if it's from the book because that's tedious and we all have the book.<br />
<br />
(By the way, I think you leave a space between lines in the code to make a new line; that is, simply pressing enter once will not make a new line. Also, you can press a button at the top of the editing textbox that lets you put in simple equations.)<br />
<br />
==Unsolved Questions==<br />
<br />
===Question Template===<br />
Q: Can someone help me prove: "If an [[integer]] <math>n</math> is greater than 2, then <math>a^n + b^n = c^n</math> has no solutions in non-zero integers <math>a</math>, <math>b</math>, and <math>c</math>."? I had the answer in my head at one point, but the margins of the piece of paper I was working with was too small to fit it.<br />
<br />
===Exam April/May 2006 #3(b)===<br />
Q: Let T : M<Sub>3x2</Sub>(C) -> M<Sub>2x3</Sub>(C) be defined as follows. Given A Є M<Sub>3x2</Sub>(C), let B be the matrix obtained from A by adding i times the second row of A to the third row of A. Let T(A) = B<Sup>t</Sup>, where B<Sup>t</Sup> is the transpose of B. (Note: Here, i is a complex number such that i<Sup>2</Sup> = -1.)<br />
Totally lost on this question :/ Please show some example matrix and how it is transformed as the question asks if possible. I want to see what actually happens to the elements in the matrix rather than the answer (think that would be more important)<br />
<br />
===Exam April/May 2006 #7===<br />
Q: Let T : V -> V and U : V -> V be linear operators on a finite-dimensional vector space V. Assume that U is invertible and T is diagonalizable. Prove that the linear operator UTU<Sup>-1</Sup> = U o T o U<Sup>-1</Sup>.<br />
I dont know where or how to start this question ><.<br />
<br />
===Sec. 2.4 Lemma p. 101===<br />
Q: In the proof of the lemma, the second line, we have "T(beta) spans R(T) = W". How do we know that R(T) = W? This would be true if dim V = dim W, because then T would be onto, but we can't assume what we're trying to prove.<br />
<br />
===Exam April/May 2006 #4===<br />
Q: Suppose that A, B Є M<sub>mxn</sub>(F), and rank(A) = rank (B). Prove that there exist invertible matrices P Є M<sub>mxm</sub>(F) and Q Є M<sub>nxn</sub>(F) such that B = PAQ.<br />
<br />
A(partial): Here is a sketch. If you rref A and B by applying a series of elementary row operation matricies, they will both look similar. That is, they will have a section of 1's and 0's (each 1 is the only number in its column) and then a section of "remaining stuff", and these sections will be the same "size" because their ranks are the same. Then, using the elementary column matrix operations, you can essentially modify the "remaining stuff" as much as you like, by adding multiples of the "nice" columns (with single-1's). These row and column operations can then be grouped nicely and set to be equal to P and Q, which are invertible because products of elementary matricies are invertible. <br />
<br />
I know this is very rough, but even if I did have a full answer I wouldn't now how to typeset it.<br />
<br />
===Complex Numbers===<br />
Q: If 'C' is used in the context of a vector space (as in "define T:C->C"), then should we consider C to be the vector space of C over the field C, or instead C over the field R?<br />
<br />
==Solved Questions==<br />
<br />
===Question Template===<br />
Q: How many ways are there to get to the nth stair, if at each step you can move either one or two squares up?<br />
<br />
A: This question can be easily modeled by the Fibonacci numbers, with the nth number being the ways to get to the nth stair. This is because, to get to the nth stair, you can come only from the n-1th or the n-2th. This is exactly how the Fibonacci numbers are defined; the proof is simple by induction.<br />
<br />
===Sec. 1.3 Thm 1.3 Proof===<br />
Q: In the first paragraph of the proof, it says "But also x + 0 = x , and thus 0'=0." How do we know 0 (that is 0 of V) even exists in W? I understand that we know ''some'' zero exists (0'), but not why ''the'' zero (0) exists.<br />
<br />
A: x is in W as well in V. Thus, x + 0 = x (VS 3).<br />
<br />
Reply: Oh I see... now it looks so obvious =/. Thanks.</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Final_Exam_Preparation_Forum&diff=321006-240/Final Exam Preparation Forum2006-12-11T20:57:36Z<p>Zapyre: ===Exam April/May 2006 #3(b)=== (forgot to log in for the previous exam questions) please answer them if possible too =)</p>
<hr />
<div>{{06-240/Navigation}}<br />
<br />
If you have questions, ask them here and hopefully someone else will know the answer. (Answering questions will probably help you understand it more).<br />
<br />
Since many of us (including I) don't really know how to use Wiki's, I suggest that we keep the formatting simple: I will post a template at the top of this page, and if you want to add something just click on the "edit", copy the template, and insert your question. Order the questions according to section (solved/unsolved, as judge by whoever created the question), with the newest at the top, except for the template question. In general, I wouldn't retype the question if it's from the book because that's tedious and we all have the book.<br />
<br />
(By the way, I think you leave a space between lines in the code to make a new line; that is, simply pressing enter once will not make a new line. Also, you can press a button at the top of the editing textbox that lets you put in simple equations.)<br />
<br />
==Unsolved Questions==<br />
<br />
===Question Template===<br />
Q: Can someone help me prove: "If an [[integer]] <math>n</math> is greater than 2, then <math>a^n + b^n = c^n</math> has no solutions in non-zero integers <math>a</math>, <math>b</math>, and <math>c</math>."? I had the answer in my head at one point, but the margins of the piece of paper I was working with was too small to fit it.<br />
<br />
===Exam April/May 2006 #3(b)===<br />
Q: Let T : M<Sub>3x2</Sub>(C) -> M<Sub>2x3</Sub>(C) be defined as follows. Given A Є M<Sub>3x2</Sub>(C), let B be the matrix obtained from A by adding i times the second row of A to the third row of A. Let T(A) = B<Sup>t</Sup>, where B<Sup>t</Sup> is the transpose of B. (Note: Here, i is a complex number such that i<Sup>2</Sup> = -1.)<br />
Totally lost on this question :/<br />
<br />
===Exam April/May 2006 #7===<br />
Q: Let T : V -> V and U : V -> V be linear operators on a finite-dimensional vector space V. Assume that U is invertible and T is diagonalizable. Prove that the linear operator UTU<Sup>-1</Sup> = U o T o U<Sup>-1</Sup>.<br />
I dont know where or how to start this question ><.<br />
<br />
===Sec. 2.4 Lemma p. 101===<br />
Q: In the proof of the lemma, the second line, we have "T(beta) spans R(T) = W". How do we know that R(T) = W? This would be true if dim V = dim W, because then T would be onto, but we can't assume what we're trying to prove.<br />
<br />
===Exam April/May 2006 #4===<br />
Q: Suppose that A, B Є M<sub>mxn</sub>(F), and rank(A) = rank (B). Prove that there exist invertible matrices P Є M<sub>mxm</sub>(F) and Q Є M<sub>nxn</sub>(F) such that B = PAQ.<br />
<br />
A(partial): Here is a sketch. If you rref A and B by applying a series of elementary row operation matricies, they will both look similar. That is, they will have a section of 1's and 0's (each 1 is the only number in its column) and then a section of "remaining stuff", and these sections will be the same "size" because their ranks are the same. Then, using the elementary column matrix operations, you can essentially modify the "remaining stuff" as much as you like, by adding multiples of the "nice" columns (with single-1's). These row and column operations can then be grouped nicely and set to be equal to P and Q, which are invertible because products of elementary matricies are invertible. <br />
<br />
I know this is very rough, but even if I did have a full answer I wouldn't now how to typeset it.<br />
<br />
===Complex Numbers===<br />
Q: If 'C' is used in the context of a vector space (as in "define T:C->C"), then should we consider C to be the vector space of C over the field C, or instead C over the field R?<br />
<br />
==Solved Questions==<br />
<br />
===Question Template===<br />
Q: How many ways are there to get to the nth stair, if at each step you can move either one or two squares up?<br />
<br />
A: This question can be easily modeled by the Fibonacci numbers, with the nth number being the ways to get to the nth stair. This is because, to get to the nth stair, you can come only from the n-1th or the n-2th. This is exactly how the Fibonacci numbers are defined; the proof is simple by induction.<br />
<br />
===Sec. 1.3 Thm 1.3 Proof===<br />
Q: In the first paragraph of the proof, it says "But also x + 0 = x , and thus 0'=0." How do we know 0 (that is 0 of V) even exists in W? I understand that we know ''some'' zero exists (0'), but not why ''the'' zero (0) exists.<br />
<br />
A: x is in W as well in V. Thus, x + 0 = x (VS 3).<br />
<br />
Reply: Oh I see... now it looks so obvious =/. Thanks.</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Class_Photo&diff=239206-240/Class Photo2006-10-17T17:28:00Z<p>Zapyre: Quick reorder of names</p>
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<div>Our class on September 28, 2006:<br />
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[[Image:06-240-ClassPhoto.jpg|thumb|centre|500px|Class Photo: click to enlarge]]<br />
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===Who We Are...===<br />
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{| align=center border=1 <br />
|-<br />
!First name<br />
!Last name<br />
!UserID<br />
!Email<br />
!In the photo<br />
!Comments<br />
{{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}}<br />
{{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!}}<br />
{{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= }}<br />
{{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 }}<br />
{{Photo Entry|last=Carberry|first=Mick|userid=MC|email=Mick.Carberry@ utoronto.ca|location=long haired, bearded old guy in back|comments= }}<br />
{{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= }}<br />
{{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 }}<br />
{{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 ;) }}<br />
{{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=}}<br />
{{Photo Entry|last=Halacheva|first=Iva|userid=Haliv|email=iva.halacheva@ utoronto.ca|location=Third row from the front, light brown jacket|comments= }}<br />
{{Photo Entry|last=Kaifosh|first=Patrick|userid=Pat|email=patrick.kaifosh@ utoronto.ca|location=Front row, rightmost.|comments= }}<br />
{{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= }}<br />
{{Photo Entry|last=Koziar|first=John|userid=John.koziar|email=|location=Front row, centre.|comments= }}<br />
{{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= }}<br />
{{Photo Entry|last=McIntyre|first=Sean|userid=Smcintyre|email=s.mcintyre@ utoronto.ca|location=mini-picture, fourth from the right|comments= }}<br />
{{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= }}<br />
{{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=}}<br />
{{Photo Entry|last=Qiao|first=Li|userid=joy9999|email=li.qiao@ utoronto.ca|location=behind the guy in black, last row|comments= }}<br />
{{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= }}<br />
{{Photo Entry|last=Soreanu|first=Alla|userid=Alla|email=alla.soreanu@ utoronto.ca|location=mini-picture, first from the left|comments= }}<br />
{{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}}<br />
{{Photo Entry|last=Tai|first=Johnathan|userid=Zapyre|email=zapyre_1@ hotmail.com|location= At the very back, left hand corner (all blue) |comments= >.<}}<br />
{{Photo Entry|last=Veytsman|first=Maxim|userid=Mveytsman|email=|location=small picture in the bottom left corner. Second one from the right.|comments=}}<br />
{{Photo Entry|last=Wong|first=Pak|userid=wongpak|email=plwong@ utoronto.ca|location=Third row from the back, left most, black shirt|comments= }}</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Class_Photo&diff=222606-240/Class Photo2006-10-04T20:05:43Z<p>Zapyre: </p>
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Our class on September 28, 2006:<br />
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[[Image:06-240-ClassPhoto.jpg|thumb|center|500px|Class Photo: click to enlarge]]<br />
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!Email<br />
!In the photo<br />
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{{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}}<br />
{{Photo Entry|last=Carberry|first=Mick|userid=MC|email=Mick.Carberry@utoronto.ca|location=long haired, bearded old guy in back|comments= }}<br />
{{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= }}<br />
{{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 }}<br />
{{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=}}<br />
{{Photo Entry|last=Halacheva|first=Iva|userid=Haliv|email=iva.halacheva@utoronto.ca|location=Third row from the front, light brown jacket|comments= }}<br />
{{Photo Entry|last=Kaifosh|first=Patrick|userid=Pat|email=patrick.kaifosh@utoronto.ca|location=Front row, rightmost.|comments= }}<br />
{{Photo Entry|last=Koziar|first=John|userid=John.koziar|email=|location=Front row, centre.|comments= }}<br />
{{Photo Entry|last=McIntyre|first=Sean|userid=Smcintyre|email=s.mcintyre@utoronto.ca|location=mini-picture, fourth from the right|comments= }}<br />
{{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= }}<br />
{{Photo Entry|last=Qiao|first=Li|userid=joy9999|email=li.qiao@ utoronto.ca|location=behind the guy in black, last row|comments= }}<br />
{{Photo Entry|last=Soreanu|first=Alla|userid=Alla|email=alla.soreanu@utoronto.ca|location=mini-picture, first from the left|comments= }}<br />
{{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}}<br />
{{Photo Entry|last=Tai|first=Johnathan|userid=Zapyre|email=zapyre_1@hotmail.com|location= At the very back, left hand corner (all blue) |comments= >.<}}<br />
{{Photo Entry|last=Veytsman|first=Maxim|userid=Mveytsman|email=|location=small picture in the bottom left corner. Second one from the right.|comments=}}<br />
{{Photo Entry|last=Wong|first=Pak|userid=wongpak|email=plwong@utoronto.ca|location=Third row from the back, left most, black shirt|comments= }}</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Class_Photo&diff=222506-240/Class Photo2006-10-04T20:04:52Z<p>Zapyre: put pplz in alpha order (and corrected my own name :/)</p>
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Our class on September 28, 2006:<br />
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{{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}}<br />
{{Photo Entry|last=Carberry|first=Mick|userid=MC|email=Mick.Carberry@utoronto.ca|location=long haired, bearded old guy in back|comments= }}<br />
{{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= }}<br />
{{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 }}<br />
{{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=}}<br />
{{Photo Entry|last=Halacheva|first=Iva|userid=Haliv|email=iva.halacheva@utoronto.ca|location=Third row from the front, light brown jacket|comments= }}<br />
{{Photo Entry|last=Kaifosh|first=Patrick|userid=Pat|email=patrick.kaifosh@utoronto.ca|location=Front row, rightmost.|comments= }}<br />
{{Photo Entry|last=Koziar|first=John|userid=John.koziar|email=|location=Front row, centre.|comments= }}<br />
{{Photo Entry|last=McIntyre|first=Sean|userid=Smcintyre|email=s.mcintyre@utoronto.ca|location=mini-picture, fourth from the right|comments= }}<br />
{{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= }}<br />
{{Photo Entry|last=Qiao|first=Li|userid=joy9999|email=li.qiao@ utoronto.ca|location=behind the guy in black, last row|comments= }}<br />
{{Photo Entry|last=Soreanu|first=Alla|userid=Alla|email=alla.soreanu@utoronto.ca|location=mini-picture, first from the left|comments= }}<br />
{{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}}<br />
{{Photo Entry|last=Tai|first=Johnathan|userid=Zapyre|email=zapyre_1@hotmail.com|location= At the very back, right hand corner (all blue) |comments= >.<}}<br />
{{Photo Entry|last=Veytsman|first=Maxim|userid=Mveytsman|email=|location=small picture in the bottom left corner. Second one from the right.|comments=}}<br />
{{Photo Entry|last=Wong|first=Pak|userid=wongpak|email=plwong@utoronto.ca|location=Third row from the back, left most, black shirt|comments= }}</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Class_Photo&diff=222406-240/Class Photo2006-10-04T20:02:27Z<p>Zapyre: </p>
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Our class on September 28, 2006:<br />
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[[Image:06-240-ClassPhoto.jpg|thumb|center|500px|Class Photo: click to enlarge]]<br />
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{{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}}<br />
{{Photo Entry|last=Carberry|first=Mick|userid=MC|email=Mick.Carberry@utoronto.ca|location=long haired, bearded old guy in back|comments= }}<br />
{{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= }}<br />
{{Photo Entry|last=Kaifosh|first=Patrick|userid=Pat|email=patrick.kaifosh@utoronto.ca|location=Front row, rightmost.|comments= }}<br />
{{Photo Entry|last=Koziar|first=John|userid=John.koziar|email=|location=Front row, centre.|comments= }}<br />
{{Photo Entry|last=McIntyre|first=Sean|userid=Smcintyre|email=s.mcintyre@utoronto.ca|location=mini-picture, fourth from the right|comments= }}<br />
{{Photo Entry|last=Soreanu|first=Alla|userid=Alla|email=alla.soreanu@utoronto.ca|location=mini-picture, first from the left|comments= }}<br />
{{Photo Entry|last=Tai|first=Johnathan|userid=Zapyre|email=zapyre_1@hotmail.com|location= At the very back, right hand corner (all blue) |comments= >.<}}<br />
{{Photo Entry|last=Veytsman|first=Maxim|userid=Mveytsman|email=|location=small picture in the bottom left corner. Second one from the right.|comments=}}<br />
{{Photo Entry|last=Wong|first=Pak|userid=wongpak|email=plwong@utoronto.ca|location=Third row from the back, left most, black shirt|comments= }}<br />
{{Photo Entry|last=Halacheva|first=Iva|userid=Haliv|email=iva.halacheva@utoronto.ca|location=Third row from the front, light brown jacket|comments= }}<br />
{{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= }}<br />
{{Photo Entry|last=Qiao|first=Li|userid=joy9999|email=li.qiao@ utoronto.ca|location=behind the guy in black, last row|comments= }}<br />
{{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}}<br />
{{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=}}<br />
{{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 }}</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Class_Photo&diff=222306-240/Class Photo2006-10-04T20:01:08Z<p>Zapyre: </p>
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Our class on September 28, 2006:<br />
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|-<br />
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!In the photo<br />
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{{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}}<br />
{{Photo Entry|last=Carberry|first=Mick|userid=MC|email=Mick.Carberry@utoronto.ca|location=long haired, bearded old guy in back|comments= }}<br />
{{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= }}<br />
{{Photo Entry|last=Kaifosh|first=Patrick|userid=Pat|email=patrick.kaifosh@utoronto.ca|location=Front row, rightmost.|comments= }}<br />
{{Photo Entry|last=Koziar|first=John|userid=John.koziar|email=|location=Front row, centre.|comments= }}<br />
{{Photo Entry|last=McIntyre|first=Sean|userid=Smcintyre|email=s.mcintyre@utoronto.ca|location=mini-picture, fourth from the right|comments= }}<br />
{{Photo Entry|last=Soreanu|first=Alla|userid=Alla|email=alla.soreanu@utoronto.ca|location=mini-picture, first from the left|comments= }}<br />
{{Photo Entry|last=Veytsman|first=Maxim|userid=Mveytsman|email=|location=small picture in the bottom left corner. Second one from the right.|comments=}}<br />
{{Photo Entry|last=Wong|first=Pak|userid=wongpak|email=plwong@utoronto.ca|location=Third row from the back, left most, black shirt|comments= }}<br />
{{Photo Entry|last=Halacheva|first=Iva|userid=Haliv|email=iva.halacheva@utoronto.ca|location=Third row from the front, light brown jacket|comments= }}<br />
{{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= }}<br />
{{Photo Entry|last=Qiao|first=Li|userid=joy9999|email=li.qiao@ utoronto.ca|location=behind the guy in black, last row|comments= }}<br />
{{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}}<br />
{{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=}}<br />
{{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 }}<br />
{{Photo Entry|last=Tai|first=Johnathan|userid=Zapyre|email=zapyre_1@hotmail.com|location= At the very back, right hand corner (all blue) |comments= :/}}</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Class_Photo&diff=222206-240/Class Photo2006-10-04T20:00:50Z<p>Zapyre: </p>
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Our class on September 28, 2006:<br />
<br />
[[Image:06-240-ClassPhoto.jpg|thumb|center|500px|Class Photo: click to enlarge]]<br />
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Please identify yourself in this photo! There are two ways to do that:<br />
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!In the photo<br />
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{{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}}<br />
{{Photo Entry|last=Carberry|first=Mick|userid=MC|email=Mick.Carberry@utoronto.ca|location=long haired, bearded old guy in back|comments= }}<br />
{{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= }}<br />
{{Photo Entry|last=Kaifosh|first=Patrick|userid=Pat|email=patrick.kaifosh@utoronto.ca|location=Front row, rightmost.|comments= }}<br />
{{Photo Entry|last=Koziar|first=John|userid=John.koziar|email=|location=Front row, centre.|comments= }}<br />
{{Photo Entry|last=McIntyre|first=Sean|userid=Smcintyre|email=s.mcintyre@utoronto.ca|location=mini-picture, fourth from the right|comments= }}<br />
{{Photo Entry|last=Soreanu|first=Alla|userid=Alla|email=alla.soreanu@utoronto.ca|location=mini-picture, first from the left|comments= }}<br />
{{Photo Entry|last=Veytsman|first=Maxim|userid=Mveytsman|email=|location=small picture in the bottom left corner. Second one from the right.|comments=}}<br />
{{Photo Entry|last=Wong|first=Pak|userid=wongpak|email=plwong@utoronto.ca|location=Third row from the back, left most, black shirt|comments= }}<br />
{{Photo Entry|last=Halacheva|first=Iva|userid=Haliv|email=iva.halacheva@utoronto.ca|location=Third row from the front, light brown jacket|comments= }}<br />
{{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= }}<br />
{{Photo Entry|last=Qiao|first=Li|userid=joy9999|email=li.qiao@ utoronto.ca|location=behind the guy in black, last row|comments= }}<br />
{{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}}<br />
{{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=}}<br />
{{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 }}<br />
{{Photo Entry|last=Tai|first=Johnathan (Tom)|userid=Zapyre|email=zapyre_1@hotmail.com|location= At the very back, right hand corner (all blue) |comments= :/}}</div>Zapyrehttps://drorbn.net/index.php?title=06-240&diff=191406-2402006-09-15T02:03:08Z<p>Zapyre: </p>
<hr />
<div>__NOEDITSECTION__<br />
{{06-240/Navigation}}<br />
==Algebra I==<br />
===Department of Mathematics, University of Toronto, Fall 2006===<br />
<br />
{{06-240/Crucial Information}}<br />
<br />
===06-240 Sep 12 class=== Below are a couple of lemmata critical to the derivation we did in class - the Professor left this little work to the students:<br />
* [[06-240: Edit1.jpg]]<br />
* [[06-240: Edit2.jpg]]<br />
<br />
===Links To Notes===<br />
* [http://www.yousendit.com/transfer.php?action=download&ufid=38FF36BF7ED1E1BA September 12 Notes] for re-uploading, please email at jeff.matskin@utoronto.ca<br />
<br />
===Famous Quotes About Math and Mind===<br />
* "The problems that exist in the world today cannot be solved by the level of thinking that created them." by Albert Einstein<br />
* "Do not worry about your difficulties in mathematics, I can assure you mine are still greater." by Albert Einstein<br />
* "The mind, once expanded to the dimensions of larger ideas, never returns to its original size." by Oliver Wendell Holmes, Sr.<br />
* "I don't think necessity is the mother of invention. Invention, in my opinion, arises directly from idleness, possibly also from laziness - to save oneself trouble." by Dame Agatha Christie (1890-1976)<br />
<br />
===Further Resources===<br />
<br />
* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]<br />
* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].<br />
* [http://www.math.toronto.edu/murnaghan/courses/mat240/index.html Last year's Math 240 web site].<br />
* [http://www.maths.leeds.ac.uk/~khouston/httlam.html "How to Think Like a Mathematician" by Kevin Houston].</div>Zapyrehttps://drorbn.net/index.php?title=06-240&diff=187406-2402006-09-13T16:07:32Z<p>Zapyre: </p>
<hr />
<div>__NOEDITSECTION__<br />
{{06-240/Navigation}}<br />
==Algebra I==<br />
===Department of Mathematics, University of Toronto, Fall 2006===<br />
<br />
{{06-240/Crucial Information}}<br />
<br />
===Class Notes=== 06-240 ... ?<br />
* [[Lecture 01, September 12, 2006]] <br />
<br />
===Links To Notes===<br />
* [http://www.yousendit.com/transfer.php?action=download&ufid=38FF36BF7ED1E1BA/ September 12 Notes] for re-uploading, please email at jeff.matskin@utoronto.ca<br />
===Further Resources===<br />
<br />
* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]<br />
* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].<br />
* [http://www.math.toronto.edu/murnaghan/courses/mat240/index.html Last year's Math 240 web site].</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Lecture_01,_September_12,_2006&diff=187306-240/Lecture 01, September 12, 20062006-09-13T16:06:16Z<p>Zapyre: Lecture 1 notes</p>
<hr />
<div>Hi its John.<br />
Okay, This time... I hope I put the notes in the right place and such... :/ <br><br />
I usually post on forums anyway so i'll start off here =). <br><br />
Any questions about handwriting (i.e. chickenscratch... yes i get that alot ><) email me at zapyre_1@hotmail.com.<br />
<br />
<br />
Anyway heres the notes from Lecture 1. <br><br />
<br />
http://img96.imageshack.us/img96/739/mat240l1p1go8.jpg<br />
http://img246.imageshack.us/img246/6005/mat240l1p2db6.jpg<br />
http://img147.imageshack.us/img147/7478/mat240l1p3hj4.jpg<br />
<br />
And to Give credit where credit is due,<br><br />
Images scanned here hosted by ImageShack,<br><br />
Visit our upload file link or imageshack to scan pages<br><br />
http://www.imageshack.us (you might want to register)<br><br />
Johnathan Tai out :/</div>Zapyrehttps://drorbn.net/index.php?title=06-240&diff=187206-2402006-09-13T16:02:09Z<p>Zapyre: </p>
<hr />
<div>__NOEDITSECTION__<br />
{{06-240/Navigation}}<br />
==Algebra I==<br />
===Department of Mathematics, University of Toronto, Fall 2006===<br />
<br />
{{06-240/Crucial Information}}<br />
<br />
===Class Notes=== 06-260 ... ?<br />
* [[Lecture 01, September 12, 2006]] <br />
<br />
===Links To Notes===<br />
* [http://www.yousendit.com/transfer.php?action=download&ufid=38FF36BF7ED1E1BA/ September 12 Notes] for re-uploading, please email at jeff.matskin@utoronto.ca<br />
===Further Resources===<br />
<br />
* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]<br />
* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].<br />
* [http://www.math.toronto.edu/murnaghan/courses/mat240/index.html Last year's Math 240 web site].</div>Zapyrehttps://drorbn.net/index.php?title=06-240&diff=187106-2402006-09-13T15:59:15Z<p>Zapyre: </p>
<hr />
<div>__NOEDITSECTION__<br />
{{06-240/Navigation}}<br />
==Algebra I==<br />
===Department of Mathematics, University of Toronto, Fall 2006===<br />
<br />
{{06-240/Crucial Information}}<br />
<br />
===Class Notes=== 06-260 ... ?<br />
* [[Lecture 01, September 13, 2006]] <br />
<br />
===Links To Notes===<br />
* [http://www.yousendit.com/transfer.php?action=download&ufid=38FF36BF7ED1E1BA/ September 12 Notes] for re-uploading, please email at jeff.matskin@utoronto.ca<br />
===Further Resources===<br />
<br />
* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]<br />
* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].<br />
* [http://www.math.toronto.edu/murnaghan/courses/mat240/index.html Last year's Math 240 web site].</div>Zapyrehttps://drorbn.net/index.php?title=06-240&diff=187006-2402006-09-13T15:58:28Z<p>Zapyre: </p>
<hr />
<div>__NOEDITSECTION__<br />
{{06-240/Navigation}}<br />
==Algebra I==<br />
===Department of Mathematics, University of Toronto, Fall 2006===<br />
<br />
{{06-240/Crucial Information}}<br />
<br />
===Class Notes=== 06-260 ... ?<br />
* [[Lecture 01, September 13, 2006]] <br />
hmm, guess i put it in the wrong place the first time ><!<br />
<br />
===Links To Notes===<br />
* [http://www.yousendit.com/transfer.php?action=download&ufid=38FF36BF7ED1E1BA/ September 12 Notes] for re-uploading, please email at jeff.matskin@utoronto.ca<br />
===Further Resources===<br />
<br />
* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]<br />
* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].<br />
* [http://www.math.toronto.edu/murnaghan/courses/mat240/index.html Last year's Math 240 web site].</div>Zapyrehttps://drorbn.net/index.php?title=06-240&diff=186906-2402006-09-13T15:57:46Z<p>Zapyre: </p>
<hr />
<div>__NOEDITSECTION__<br />
{{06-240/Navigation}}<br />
==Algebra I==<br />
===Department of Mathematics, University of Toronto, Fall 2006===<br />
<br />
{{06-240/Crucial Information}}<br />
<br />
===Class Notes===<br />
* [[Lecture 01, September 13, 2006]] {{06-260/??}}<br />
hmm, guess i put it in the wrong place the first time ><!<br />
<br />
===Links To Notes===<br />
* [http://www.yousendit.com/transfer.php?action=download&ufid=38FF36BF7ED1E1BA/ September 12 Notes] for re-uploading, please email at jeff.matskin@utoronto.ca<br />
===Further Resources===<br />
<br />
* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]<br />
* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].<br />
* [http://www.math.toronto.edu/murnaghan/courses/mat240/index.html Last year's Math 240 web site].</div>Zapyrehttps://drorbn.net/index.php?title=06-240&diff=186806-2402006-09-13T15:55:24Z<p>Zapyre: </p>
<hr />
<div>__NOEDITSECTION__<br />
{{06-240/Navigation}}<br />
==Algebra I==<br />
===Department of Mathematics, University of Toronto, Fall 2006===<br />
<br />
{{06-240/Crucial Information}}<br />
<br />
===Class Notes===<br />
* [[Lecture 01, September 13, 2006]]<br />
hmm, guess i put it in the wrong place the first time ><!<br />
<br />
===Links To Notes===<br />
* [http://www.yousendit.com/transfer.php?action=download&ufid=38FF36BF7ED1E1BA/ September 12 Notes] for re-uploading, please email at jeff.matskin@utoronto.ca<br />
===Further Resources===<br />
<br />
* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]<br />
* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].<br />
* [http://www.math.toronto.edu/murnaghan/courses/mat240/index.html Last year's Math 240 web site].</div>Zapyrehttps://drorbn.net/index.php?title=06-240&diff=186706-2402006-09-13T15:54:58Z<p>Zapyre: </p>
<hr />
<div>__NOEDITSECTION__<br />
{{06-240/Navigation}}<br />
==Algebra I==<br />
===Department of Mathematics, University of Toronto, Fall 2006===<br />
<br />
{{06-240/Crucial Information}}<br />
<br />
===Class Notes===<br />
hmm, guess i put it in the wrong place ><!<br />
* [[Lecture 01, September 13, 2006]]<br />
<br />
===Links To Notes===<br />
* [http://www.yousendit.com/transfer.php?action=download&ufid=38FF36BF7ED1E1BA/ September 12 Notes] for re-uploading, please email at jeff.matskin@utoronto.ca<br />
===Further Resources===<br />
<br />
* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]<br />
* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].<br />
* [http://www.math.toronto.edu/murnaghan/courses/mat240/index.html Last year's Math 240 web site].</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Lecture_01,_September_13,_2006&diff=186606-240/Lecture 01, September 13, 20062006-09-13T15:52:26Z<p>Zapyre: </p>
<hr />
<div>Hi its John.<br />
Okay, This time... I hope I put the notes in the right place and such... :/ <br><br />
I usually post on forums anyway so i'll start off here =). <br><br />
Any questions about handwriting (i.e. chickenscratch... yes i get that alot ><) email me at zapyre_1@hotmail.com.<br />
<br />
<br />
Anyway heres the notes from Lecture 1. <br><br />
<br />
http://img96.imageshack.us/img96/739/mat240l1p1go8.jpg<br />
http://img246.imageshack.us/img246/6005/mat240l1p2db6.jpg<br />
http://img147.imageshack.us/img147/7478/mat240l1p3hj4.jpg<br />
<br />
And to Give credit where credit is due,<br><br />
Images scanned here hosted by ImageShack,<br><br />
Visit our upload file link or imageshack to scan pages<br><br />
http://www.imageshack.us (you might want to register)<br><br />
Johnathan Tai out :/</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Lecture_01,_September_13,_2006&diff=186506-240/Lecture 01, September 13, 20062006-09-13T15:52:02Z<p>Zapyre: </p>
<hr />
<div>Hi its John.<br />
Okay, This time... I hope I put the notes in the right place and such... :/ <br><br />
I usually post on forums anyway so i'll start off here =). <br><br />
Any questions about handwriting (i.e. chickenscratch... yes i get that alot ><) email me at zapyre_1@hotmail.com.<br><br />
<br />
Anyway heres the notes from Lecture 1. <br><br />
<br />
http://img96.imageshack.us/img96/739/mat240l1p1go8.jpg<br />
http://img246.imageshack.us/img246/6005/mat240l1p2db6.jpg<br />
http://img147.imageshack.us/img147/7478/mat240l1p3hj4.jpg<br />
<br />
And to Give credit where credit is due,<br><br />
Images scanned here hosted by ImageShack,<br><br />
Visit our upload file link or imageshack to scan pages<br><br />
http://www.imageshack.us (you might want to register)<br><br />
Johnathan Tai out :/</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Lecture_01,_September_13,_2006&diff=186406-240/Lecture 01, September 13, 20062006-09-13T15:50:56Z<p>Zapyre: </p>
<hr />
<div>Hi its John.<br />
Okay, This time... I hope I put the notes in the right place and such... :/<br />
I usually post on forums anyway so i'll start off here =).<br />
Any questions about handwriting (i.e. chickenscratch... yes i get that alot ><) email me at zapyre_1@hotmail.com.<br />
<br />
Anyway heres the notes from Lecture 1.<br />
<br />
http://img96.imageshack.us/img96/739/mat240l1p1go8.jpg<br />
http://img246.imageshack.us/img246/6005/mat240l1p2db6.jpg<br />
http://img147.imageshack.us/img147/7478/mat240l1p3hj4.jpg<br />
<br />
And to Give credit where credit is due,<br><br />
Images scanned here hosted by ImageShack,<br><br />
Visit our upload file link or imageshack to scan pages<br><br />
http://www.imageshack.us <br><br />
Johnathan Tai out :/</div>Zapyrehttps://drorbn.net/index.php?title=06-240&diff=186306-2402006-09-13T15:45:19Z<p>Zapyre: </p>
<hr />
<div>__NOEDITSECTION__<br />
{{06-240/Navigation}}<br />
==Algebra I==<br />
===Department of Mathematics, University of Toronto, Fall 2006===<br />
<br />
{{06-240/Crucial Information}}<br />
<br />
{{06-240/Class notes}}<br />
===Class Notes===<br />
hmm, guess i put it in the wrong place ><!<br />
* [[Lecture 01, September 13, 2006]]<br />
<br />
===Links To Notes===<br />
* [http://www.yousendit.com/transfer.php?action=download&ufid=38FF36BF7ED1E1BA/ September 12 Notes] for re-uploading, please email at jeff.matskin@utoronto.ca<br />
===Further Resources===<br />
<br />
* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]<br />
* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].<br />
* [http://www.math.toronto.edu/murnaghan/courses/mat240/index.html Last year's Math 240 web site].</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Lecture_1_(Tuesday_12th_of_Sept)&diff=185406-240/Lecture 1 (Tuesday 12th of Sept)2006-09-13T05:06:43Z<p>Zapyre: </p>
<hr />
<div>Hi this is John, I hope this works ^^,<br />
I usually Post in Forums anyway so i'll start things out :P<br />
Any Questions about unreadable handwriting (i.e. chickenscratch, yes I get that alot) or anything at all as long as its not about explaining material... (obv u ask the prof for that =)) e-mail me at zapyre_1@hotmail.com<br />
<br />
Anyway, here are the first set of notes from lecture 1.<br />
<br />
<br />
http://img96.imageshack.us/img96/739/mat240l1p1go8.jpg<br />
http://img246.imageshack.us/img246/6005/mat240l1p2db6.jpg<br />
http://img147.imageshack.us/img147/7478/mat240l1p3hj4.jpg<br />
<br />
<br />
Oh and to give credit where credit is due, these notes hosted by ImageShack <br><br />
Visit ImageShack to upload scans at:<br><br />
http://www.imageshack.us (you will need to register and such)<br><br />
<br />
Omg I didnt see the upload file button ><! lol w/e <br><br />
Johnathan Tai out :/</div>Zapyrehttps://drorbn.net/index.php?title=Main_Page&diff=1853Main Page2006-09-13T04:24:27Z<p>Zapyre: </p>
<hr />
<div>This is the wiki portion of my [http://www.math.toronto.edu/~drorbn web site]. It is not a standalone page - it has no overall tree structure and there is no specific entry point. Rather it is just the common location for those of for my pages that could benefit from being editable by others and/or from having their history preserved and/or by having an associated "[[Talk:Main_Page|discussion]]" page.<br />
<br />
Ok. Here are some of the pages here anyway:<br />
<br />
===[[Paperlets]]===<br />
* Finite Type Invariants:<br />
** [[The Envelope of a Finite Type Invariant]].<br />
** [[Knotted Trivalent Graphs are Finitely Presented]].<br />
** [[Some Quotients of A]].<br />
** [[Associators with Frozen Feet]].<br />
** [[Coloured Chord Diagrams]].<br />
* Khovanov Homology:<br />
** [[Mutation Invariance of Khovanov Homology]].<br />
* Other:<br />
** [[KTG-Definable Properties of Knots and Links]].<br />
<br />
===Projects===<br />
* [[VasCalc - A Vassiliev Invariants Calculator]].<br />
* [[HoriAsso - Computing a Horizontal Associator]].<br />
<br />
=== Classes ===<br />
* [[06-240]].<br />
* [[06-1350]].<br />
<br />
=== Class Notes === (New! possibly :/)<br />
* [[Lecture 1 (Tuesday 12th of Sept)]].<br />
<br />
<br />
===Other===<br />
* [[Survey of Finite Type Invariants]].<br />
* [[CMS Winter 2006 Session on Knot Homologies]].<br />
* [[2009 Knot Theory Program at the Fields Institute]].</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Lecture_1_(Tuesday_12th_of_Sept)&diff=185206-240/Lecture 1 (Tuesday 12th of Sept)2006-09-13T04:22:46Z<p>Zapyre: </p>
<hr />
<div>Hi this is John, I hope this works ^^,<br />
I usually Post in Forums anyway so i'll start things out :P<br />
Any Questions about unreadable handwriting (i.e. chickenscratch, yes I get that alot) or anything at all as long as its not about explaining material... (obv u ask the prof for that =)) e-mail me at zapyre_1@hotmail.com<br />
<br />
Anyway, here are the first set of notes from lecture 1.<br />
<br />
<br />
http://img96.imageshack.us/img96/739/mat240l1p1go8.jpg<br />
http://img246.imageshack.us/img246/6005/mat240l1p2db6.jpg<br />
http://img147.imageshack.us/img147/7478/mat240l1p3hj4.jpg<br />
<br />
<br />
Oh and to give credit where credit is due, these notes hosted by ImageShack <br><br />
Visit ImageShack to upload scans at:<br><br />
http://www.imageshack.us (you will need to register and such)<br><br />
Johnathan Tai out :/</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Lecture_1_(Tuesday_12th_of_Sept)&diff=185106-240/Lecture 1 (Tuesday 12th of Sept)2006-09-13T04:21:41Z<p>Zapyre: </p>
<hr />
<div>Hi this is John, I hope this works ^^,<br />
I usually Post in Forums anyway so i'll start things out :P<br />
Any Questions about unreadable handwriting (i.e. chickenscratch, yes I get that alot) or anything at all as long as its not about explaining material... (obv u ask the prof for that =)) e-mail me at zapyre_1@hotmail.com<br />
<br />
Anyway, here are the first set of notes from lecture 1.<br />
<br />
<br />
http://img96.imageshack.us/img96/739/mat240l1p1go8.jpg<br />
http://img246.imageshack.us/img246/6005/mat240l1p2db6.jpg<br />
http://img147.imageshack.us/img147/7478/mat240l1p3hj4.jpg<br />
<br />
<br />
Oh and to give credit where credit is due, these notes hosted by ImageShack<br />
Visit ImageShack to upload scans at:<br><br />
http://www.imageshack.us (you will need to register and such)<br><br />
Johnathan Tai out :/</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Lecture_1_(Tuesday_12th_of_Sept)&diff=185006-240/Lecture 1 (Tuesday 12th of Sept)2006-09-13T04:19:44Z<p>Zapyre: </p>
<hr />
<div>Hi this is John, I hope this works ^^,<br />
I usually Post in Forums anyway so i'll start things out :P<br />
Any Questions about unreadable handwriting (i.e. chickenscratch, yes I get that alot) or anything at all as long as its not about explaining material... (obv u ask the prof for that =)) e-mail me at zapyre_1@hotmail.com<br />
<br />
Anyway, here are the first set of notes from lecture 1.<br />
<br />
<br />
http://img96.imageshack.us/img96/739/mat240l1p1go8.jpg<br />
http://img246.imageshack.us/img246/6005/mat240l1p2db6.jpg<br />
http://img147.imageshack.us/img147/7478/mat240l1p3hj4.jpg<br />
<br />
<br />
Oh and to give credit where credit is due, these notes hosted by ImageShack<br />
Visit ImageShack to upload scans at:<br><br />
http://www.imageshack.us (you will need to register and such)<br />
Johnathan Tai out :/</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Lecture_1_(Tuesday_12th_of_Sept)&diff=184906-240/Lecture 1 (Tuesday 12th of Sept)2006-09-13T04:10:18Z<p>Zapyre: </p>
<hr />
<div>Hi this is Johnathan, I hope this works ^^,<br />
I usually Post in Forums anyway so i'll start things out :P<br />
Any Questions about unreadable handwriting (i.e. chickenscratch, yes I get that alot) or anything at all as long as its not about explaining material... (obv u ask the prof for that =)) e-mail me at zapyre_1@hotmail.com<br />
<br />
Anyway, here are the first set of notes from lecture 1.<br />
<br />
<br />
http://img174.imageshack.us/img174/5263/mat240l1p1es5.jpg<br />
http://img246.imageshack.us/img246/6005/mat240l1p2db6.jpg<br />
http://img147.imageshack.us/img147/7478/mat240l1p3hj4.jpg<br />
<br />
<br />
Oh and to give credit where credit is due, these notes hosted by ImageShack<br />
Visit ImageShack to upload scans at:<br><br />
http://www.imageshack.us (you will need to register and such)</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Lecture_1_(Tuesday_12th_of_Sept)&diff=184806-240/Lecture 1 (Tuesday 12th of Sept)2006-09-13T04:10:04Z<p>Zapyre: </p>
<hr />
<div>Hi this is Johnathan, I hope this works ^^,<br />
I usually Post in Forums anyway so i'll start things out :P<br />
Any Questions about unreadable handwriting (i.e. chickenscratch, yes I get that alot) or anything at all<br><br />
as long as its not about explaining material... (obv u ask the prof for that =)) e-mail me at zapyre_1@hotmail.com<br />
<br />
Anyway, here are the first set of notes from lecture 1.<br />
<br />
<br />
http://img174.imageshack.us/img174/5263/mat240l1p1es5.jpg<br />
http://img246.imageshack.us/img246/6005/mat240l1p2db6.jpg<br />
http://img147.imageshack.us/img147/7478/mat240l1p3hj4.jpg<br />
<br />
<br />
Oh and to give credit where credit is due, these notes hosted by ImageShack<br />
Visit ImageShack to upload scans at:<br><br />
http://www.imageshack.us (you will need to register and such)</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Lecture_1_(Tuesday_12th_of_Sept)&diff=184706-240/Lecture 1 (Tuesday 12th of Sept)2006-09-13T04:05:45Z<p>Zapyre: </p>
<hr />
<div>Hi this is Johnathan, I hope this works ^^,<br />
I usually Post in Forums anyway so i'll start things out :P<br />
Any Questions about unreadable handwriting (i.e. chickenscratch, yes I get that alot) or anything at all, as long as its not about explaining material... (obv u ask the prof for that =)) e-mail me at zapyre_1@hotmail.com<br />
<br />
Anyway, here are the first set of notes from lecture 1.<br />
<br />
<br />
http://img154.imageshack.us/img154/6534/mat240l1p1fj2.jpg<br />
http://img246.imageshack.us/img246/6005/mat240l1p2db6.jpg<br />
http://img147.imageshack.us/img147/7478/mat240l1p3hj4.jpg<br />
<br />
<br />
Oh and to give credit where credit is due, these notes hosted by ImageShack<br />
Visit ImageShack to upload scans at:<br><br />
http://www.imageshack.us (you will need to register and such)</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Lecture_1_(Tuesday_12th_of_Sept)&diff=184606-240/Lecture 1 (Tuesday 12th of Sept)2006-09-13T04:04:58Z<p>Zapyre: </p>
<hr />
<div>Hi this is Johnathan, I hope this works ^^,<br />
I usually Post in Forums anyway so i'll start things out :P<br />
Any Questions about unreadable handwriting (i.e. chickenscratch, yes I get that alot) or anything at all, as long as its not about explaining material... (obv u ask the prof for that =)) e-mail me at zapyre_1@hotmail.com<br />
<br />
Anyway, here are the first set of notes from lecture 1.<br />
<br />
<br />
http://img154.imageshack.us/img154/6534/mat240l1p1fj2.jpg<br />
http://img246.imageshack.us/img246/6005/mat240l1p2db6.jpg<br />
http://img147.imageshack.us/img147/7478/mat240l1p3hj4.jpg<br />
<br />
<br />
Oh and to give credit where credit is due, these notes hosted by ImageShack<br />
Visit ImageShack to upload scans at:<br />
http://www.imageshack.us</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Lecture_1_(Tuesday_12th_of_Sept)&diff=184506-240/Lecture 1 (Tuesday 12th of Sept)2006-09-13T04:04:30Z<p>Zapyre: </p>
<hr />
<div>Hi this is Johnathan, I hope this works ^^,<br />
I usually Post in Forums anyway so i'll start things out :P<br />
Any Questions about unreadable handwriting (i.e. chickenscratch, yes I get that alot) or anything at all, as long as its not about explaining material... (obv u ask the prof for that =)) e-mail me at zapyre_1@hotmail.com<br />
<br />
Anyway, here are the first set of notes from lecture 1.<br />
<br />
<br />
http://img154.imageshack.us/img154/6534/mat240l1p1fj2.jpg<br />
http://img246.imageshack.us/img246/6005/mat240l1p2db6.jpg<br />
http://img147.imageshack.us/img147/7478/mat240l1p3hj4.jpg<br />
<br />
<br />
Oh and to give credit where credit is due, these notes hosted by ImageShack<br />
Visit ImageShack to upload scans at [[www.imageshack.us]]</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Lecture_1_(Tuesday_12th_of_Sept)&diff=184406-240/Lecture 1 (Tuesday 12th of Sept)2006-09-13T04:03:47Z<p>Zapyre: </p>
<hr />
<div>Hi this is Johnathan, I hope this works ^^,<br />
I usually Post in Forums anyway so i'll start things out :P<br />
Any Questions about unreadable handwriting (i.e. chickenscratch, yes I get that alot) or anything at all, as long as its not about explaining material... (obv u ask the prof for that =)) e-mail me at zapyre_1@hotmail.com<br />
<br />
Anyway, here are the first set of notes from lecture 1.<br />
<br />
<br />
http://img154.imageshack.us/img154/6534/mat240l1p1fj2.jpg<br />
http://img246.imageshack.us/img246/6005/mat240l1p2db6.jpg<br />
http://img147.imageshack.us/img147/7478/mat240l1p3hj4.jpg<br />
<br />
Oh and to give credit where credit is due, these notes hosted by ImageShack<br />
Visit ImageShack to upload scans at www.imageshack.us</div>Zapyrehttps://drorbn.net/index.php?title=06-240/Lecture_1_(Tuesday_12th_of_Sept)&diff=184306-240/Lecture 1 (Tuesday 12th of Sept)2006-09-13T04:02:11Z<p>Zapyre: </p>
<hr />
<div>Hi this is Johnathan, I hope this works ^^,<br />
I usually Post in Forums anyway so i'll start things out :P<br />
Any Questions about unreadable handwriting (i.e. chickenscratch, yes I get that alot) or anything at all, as long as its not about explaining material... (obv u ask the prof for that =)) e-mail me at zapyre_1@hotmail.com<br />
<br />
Anyway, here are the first set of notes from lecture 1.<br />
<br />
<br />
<img src="http://img154.imageshack.us/img154/6534/mat240l1p1fj2.jpg" alt="Image Hosted by ImageShack.us" /><br />
<img src="http://img246.imageshack.us/img246/6005/mat240l1p2db6.jpg" alt="Image Hosted by ImageShack.us" /><br />
<img src="http://img147.imageshack.us/img147/7478/mat240l1p3hj4.jpg" alt="Image Hosted by ImageShack.us" /></div>Zapyrehttps://drorbn.net/index.php?title=06-240/Lecture_1_(Tuesday_12th_of_Sept)&diff=184206-240/Lecture 1 (Tuesday 12th of Sept)2006-09-13T04:01:10Z<p>Zapyre: </p>
<hr />
<div>Hi this is Johnathan, I hope this works ^^,<br />
I usually Post in Forums anyway so i'll start things out :P<br />
Any Questions about unreadable handwriting (i.e. chickenscratch, yes I get that alot) or anything at all, as long as its not about explaining material... (obv u ask the prof for that :)) e-mail me at zapyre_1@hotmail.com<br />
<br />
Anyway, here are the first set of notes from lecture 1.<br />
<br />
<br />
<img src="http://img154.imageshack.us/img154/6534/mat240l1p1fj2.jpg" alt="Image Hosted by ImageShack.us" /><br />
<img src="http://img246.imageshack.us/img246/6005/mat240l1p2db6.jpg" alt="Image Hosted by ImageShack.us" /><br />
<img src="http://img147.imageshack.us/img147/7478/mat240l1p3hj4.jpg" alt="Image Hosted by ImageShack.us" /></div>Zapyrehttps://drorbn.net/index.php?title=06-240/Lecture_1_(Tuesday_12th_of_Sept)&diff=184106-240/Lecture 1 (Tuesday 12th of Sept)2006-09-13T03:58:57Z<p>Zapyre: </p>
<hr />
<div>Hi this is Johnathan, I hope this works ^^,<br />
I usually Post in Forums anyway so i'll start things out :P<br />
Any Questions about unreadable handwriting (i.e. chickenscratch, yes I get that alot) or anything at all, as long as its not about explaining material... (obv u ask the prof for that :)) e-mail me at zapyre_1@hotmail.com<br />
<br />
Anyway, here are the first set of notes from lecture 1.<br />
<br />
[IMG]http://img154.imageshack.us/img154/6534/mat240l1p1fj2.jpg[/IMG]<br />
[IMG]http://img246.imageshack.us/img246/6005/mat240l1p2db6.jpg[/IMG]<br />
[IMG]http://img147.imageshack.us/img147/7478/mat240l1p3hj4.jpg[/IMG]</div>Zapyrehttps://drorbn.net/index.php?title=Main_Page&diff=1840Main Page2006-09-13T03:52:36Z<p>Zapyre: Gonna try and upload notes :/</p>
<hr />
<div>This is the wiki portion of my [http://www.math.toronto.edu/~drorbn web site]. It is not a standalone page - it has no overall tree structure and there is no specific entry point. Rather it is just the common location for those of for my pages that could benefit from being editable by others and/or from having their history preserved and/or by having an associated "[[Talk:Main_Page|discussion]]" page.<br />
<br />
Ok. Here are some of the pages here anyway:<br />
<br />
===[[Paperlets]]===<br />
* Finite Type Invariants:<br />
** [[The Envelope of a Finite Type Invariant]].<br />
** [[Knotted Trivalent Graphs are Finitely Presented]].<br />
** [[Some Quotients of A]].<br />
** [[Associators with Frozen Feet]].<br />
** [[Coloured Chord Diagrams]].<br />
* Khovanov Homology:<br />
** [[Mutation Invariance of Khovanov Homology]].<br />
* Other:<br />
** [[KTG-Definable Properties of Knots and Links]].<br />
<br />
===Projects===<br />
* [[VasCalc - A Vassiliev Invariants Calculator]].<br />
* [[HoriAsso - Computing a Horizontal Associator]].<br />
<br />
=== Classes ===<br />
* [[06-240]].<br />
* [[06-1350]].<br />
<br />
=== Notes === (possibly :/)<br />
* [[Lecture 1 (Tuesday 12th of Sept)]].<br />
<br />
<br />
===Other===<br />
* [[Survey of Finite Type Invariants]].<br />
* [[CMS Winter 2006 Session on Knot Homologies]].<br />
* [[2009 Knot Theory Program at the Fields Institute]].</div>Zapyrehttps://drorbn.net/index.php?title=Main_Page&diff=1839Main Page2006-09-13T03:52:21Z<p>Zapyre: Gonna try and upload notes :/</p>
<hr />
<div>This is the wiki portion of my [http://www.math.toronto.edu/~drorbn web site]. It is not a standalone page - it has no overall tree structure and there is no specific entry point. Rather it is just the common location for those of for my pages that could benefit from being editable by others and/or from having their history preserved and/or by having an associated "[[Talk:Main_Page|discussion]]" page.<br />
<br />
Ok. Here are some of the pages here anyway:<br />
<br />
===[[Paperlets]]===<br />
* Finite Type Invariants:<br />
** [[The Envelope of a Finite Type Invariant]].<br />
** [[Knotted Trivalent Graphs are Finitely Presented]].<br />
** [[Some Quotients of A]].<br />
** [[Associators with Frozen Feet]].<br />
** [[Coloured Chord Diagrams]].<br />
* Khovanov Homology:<br />
** [[Mutation Invariance of Khovanov Homology]].<br />
* Other:<br />
** [[KTG-Definable Properties of Knots and Links]].<br />
<br />
===Projects===<br />
* [[VasCalc - A Vassiliev Invariants Calculator]].<br />
* [[HoriAsso - Computing a Horizontal Associator]].<br />
<br />
=== Classes ===<br />
* [[06-240]].<br />
* [[06-1350]].<br />
<br />
=== Notes === (possibly :/)<br />
** [[Lecture 1 (Tuesday 12th of Sept)]].<br />
<br />
<br />
===Other===<br />
* [[Survey of Finite Type Invariants]].<br />
* [[CMS Winter 2006 Session on Knot Homologies]].<br />
* [[2009 Knot Theory Program at the Fields Institute]].</div>Zapyre