Drorbn - User contributions [en]

12-240/The Final Exam

2012-12-12T17:42:47Z

Quanle: /* Cheat Sheets made by Students */

{{12-240/Navigation}}

Our final exam is coming up. It will take place on Thursday December 13th, from 9AM until noon, at NF003 - Northrop Frye Hall, Victoria College, 73 Queen's Park Crescent.

===Content and Style===
It will consist of 5-6 questions (each may have several parts) on everything that we have covered in class this semester:
*Fields and vector spaces.
*Spans, independence, replacement and bases.
*Linear transformation, rank, nullity, matrices.
*Row and column reduction and elementary matrices, systems of linear equations.
*Determinants.
*A bit on diagonalization.
*Several other "smaller" topics.

As for the style -

*You can expect to be asked to reproduce some proofs that were given in class.
*You can expect some fresh things to prove, though generally not as hard as the previous type of proofs.
*You can expect questions (or parts of questions) that will be identical or nearly identical to questions that were assigned for homework.
*You can expect some calculations (but nothing that will require a calculator).

It is likely that the exam will be close in spirit to the exams of six and three years ago. See [[06-240/The_Final_Exam]] and [[09-240/The Final Exam]].

Basic calculators (not capable of displaying text or sounding speech) will be allowed but will not be necessary. You may wish to bring one nevertheless, as under pressure <math>5+7</math> often comes out to be <math>13</math>.

'''Remember.''' Neatness counts! Organization counts! Language counts! Proofs are best given as short and readable essays; without the English between the formulas one never knows how to interpret those formulas. When you write, say, "<math>v\in V</math>", does it mean "choose <math>v\in V</math>", or "we've just proven that <math>v\in V</math>", or "assume by contradiction that <math>v\in V</math>", or "for every <math>v\in V</math>" or "there exists <math>v\in V</math>"? If you don't say, your reader has no way of knowing. Also remember that long and roundabout solutions of simple problems, full of detours and irrelevant facts, are often an indication that their author didn't quite get the point, even if they are entirely correct. Avoid those!

==Office Hours==
Brandon and Peter and I will hold pre-exam office hours as follows:
* Friday December 7, 10AM-11AM, with Peter at 215 Huron, 10th floor.
* Monday December 10, 10:30AM-11:30AM, with Dror at Bahen 6178.
* Monday December 10, 2PM-4PM, with Peter at 215 Huron, 10th floor.
* Tuesday December 11, 1PM-3:30PM, with Brandon at 215 Huron, 10th floor.
* Wednesday December 12, 11AM-1PM, with Peter at 215 Huron, 10th floor.
* Wednesday December 12, 1PM-3:30PM, with Brandon at 215 Huron, 10th floor.
* Wednesday December 12, 4PM at least until 6PM, with Dror at Bahen 6178.

{{12-240:Dror/Students Divider}}

==Cheat Sheets made by Students==
Caveat emptor! These notes should not be used in anyway that violates the Code of Academic Integrity. Furthermore, to use these notes effectively, you should be able to prove every bullet points stated.

Materials covered before the midterm: [http://drorbn.net/images/1/12/Midterm1.JPG Part 1] [http://drorbn.net/images/2/23/Midterm2.JPG Part 2] [http://drorbn.net/images/6/62/Midterm3.JPG Part 3]

[http://drorbn.net/images/1/17/DeterminantsforDummies.JPG Determinants for Determined Dummies] (this would especially helps if you are lost in the topic of determinants) (or if you like bad puns)

Errata:
Part 1, 1L says "the characteristic of a finite field is either prime or zero." This is incorrect. The characteristic of a finite field must be nonzero and prime. On an unrelated notes, I believe there exists infinite fields of prime characteristics. Yet, a field with characteristic 0 has to be infinite.
Part 3, 7i, the only "and" should read "or."
Claim 7k is actually by definition, a smarter claim would have been "If there exists a finite set S such that S spans a vector space V, then V has a finite basis."

I don't know where to post this, but if you are doing some last minute study questions like me and you want to check the even answers (as the odd can be checked in the back of the book), here's a good link:
http://www.scribd.com/doc/64217240/Linear-Algebra-Friedberg-4th-Ed-Solutions-Manual

12-240/The Final Exam

2012-12-12T17:26:40Z

Quanle: /* Cheat Sheets made by Students */

{{12-240/Navigation}}

Our final exam is coming up. It will take place on Thursday December 13th, from 9AM until noon, at NF003 - Northrop Frye Hall, Victoria College, 73 Queen's Park Crescent.

===Content and Style===
It will consist of 5-6 questions (each may have several parts) on everything that we have covered in class this semester:
*Fields and vector spaces.
*Spans, independence, replacement and bases.
*Linear transformation, rank, nullity, matrices.
*Row and column reduction and elementary matrices, systems of linear equations.
*Determinants.
*A bit on diagonalization.
*Several other "smaller" topics.

As for the style -

*You can expect to be asked to reproduce some proofs that were given in class.
*You can expect some fresh things to prove, though generally not as hard as the previous type of proofs.
*You can expect questions (or parts of questions) that will be identical or nearly identical to questions that were assigned for homework.
*You can expect some calculations (but nothing that will require a calculator).

It is likely that the exam will be close in spirit to the exams of six and three years ago. See [[06-240/The_Final_Exam]] and [[09-240/The Final Exam]].

Basic calculators (not capable of displaying text or sounding speech) will be allowed but will not be necessary. You may wish to bring one nevertheless, as under pressure <math>5+7</math> often comes out to be <math>13</math>.

'''Remember.''' Neatness counts! Organization counts! Language counts! Proofs are best given as short and readable essays; without the English between the formulas one never knows how to interpret those formulas. When you write, say, "<math>v\in V</math>", does it mean "choose <math>v\in V</math>", or "we've just proven that <math>v\in V</math>", or "assume by contradiction that <math>v\in V</math>", or "for every <math>v\in V</math>" or "there exists <math>v\in V</math>"? If you don't say, your reader has no way of knowing. Also remember that long and roundabout solutions of simple problems, full of detours and irrelevant facts, are often an indication that their author didn't quite get the point, even if they are entirely correct. Avoid those!

==Office Hours==
Brandon and Peter and I will hold pre-exam office hours as follows:
* Friday December 7, 10AM-11AM, with Peter at 215 Huron, 10th floor.
* Monday December 10, 10:30AM-11:30AM, with Dror at Bahen 6178.
* Monday December 10, 2PM-4PM, with Peter at 215 Huron, 10th floor.
* Tuesday December 11, 1PM-3:30PM, with Brandon at 215 Huron, 10th floor.
* Wednesday December 12, 11AM-1PM, with Peter at 215 Huron, 10th floor.
* Wednesday December 12, 1PM-3:30PM, with Brandon at 215 Huron, 10th floor.
* Wednesday December 12, 4PM at least until 6PM, with Dror at Bahen 6178.

{{12-240:Dror/Students Divider}}

==Cheat Sheets made by Students==
Caveat emptor! These notes should not be used in anyway that violates the Code of Academic Integrity. Furthermore, to use these notes effectively, you should be able to prove every bullet points stated.

Materials covered before the midterm: [http://drorbn.net/images/1/12/Midterm1.JPG Part 1] [http://drorbn.net/images/2/23/Midterm2.JPG Part 2] [http://drorbn.net/images/6/62/Midterm3.JPG Part 3]

[http://drorbn.net/images/1/17/DeterminantsforDummies.JPG Determinants for Determined Dummies] (this would especially helps if you are lost in the topic of determinants) (or if you like bad puns)

Errata:
Part 1, 1L says "the characteristic of a finite field is either prime or zero." This is incorrect. The characteristic of a finite field must be nonzero and prime. On an unrelated notes, I believe there exists infinite fields of prime characteristics. Yet, a field with characteristic 0 has to be infinite.
Part 3, 7i, the only "and" should read "or."
Claim 7k is actually by definition.

I don't know where to post this, but if you are doing some last minute study questions like me and you want to check the even answers (as the odd can be checked in the back of the book), here's a good link:
http://www.scribd.com/doc/64217240/Linear-Algebra-Friedberg-4th-Ed-Solutions-Manual

12-240/The Final Exam

2012-12-11T00:57:03Z

Quanle: /* Cheat Sheets made by Students */

{{12-240/Navigation}}

Our final exam is coming up. It will take place on Thursday December 13th, from 9AM until noon, at NF003 - Northrop Frye Hall, Victoria College, 73 Queen's Park Crescent.

===Content and Style===
It will consist of 5-6 questions (each may have several parts) on everything that we have covered in class this semester:
*Fields and vector spaces.
*Spans, independence, replacement and bases.
*Linear transformation, rank, nullity, matrices.
*Row and column reduction and elementary matrices, systems of linear equations.
*Determinants.
*A bit on diagonalization.
*Several other "smaller" topics.

As for the style -

*You can expect to be asked to reproduce some proofs that were given in class.
*You can expect some fresh things to prove, though generally not as hard as the previous type of proofs.
*You can expect questions (or parts of questions) that will be identical or nearly identical to questions that were assigned for homework.
*You can expect some calculations (but nothing that will require a calculator).

It is likely that the exam will be close in spirit to the exams of six and three years ago. See [[06-240/The_Final_Exam]] and [[09-240/The Final Exam]].

Basic calculators (not capable of displaying text or sounding speech) will be allowed but will not be necessary. You may wish to bring one nevertheless, as under pressure <math>5+7</math> often comes out to be <math>13</math>.

'''Remember.''' Neatness counts! Organization counts! Language counts! Proofs are best given as short and readable essays; without the English between the formulas one never knows how to interpret those formulas. When you write, say, "<math>v\in V</math>", does it mean "choose <math>v\in V</math>", or "we've just proven that <math>v\in V</math>", or "assume by contradiction that <math>v\in V</math>", or "for every <math>v\in V</math>" or "there exists <math>v\in V</math>"? If you don't say, your reader has no way of knowing. Also remember that long and roundabout solutions of simple problems, full of detours and irrelevant facts, are often an indication that their author didn't quite get the point, even if they are entirely correct. Avoid those!

==Office Hours==
Brandon and Peter and I will hold pre-exam office hours as follows:
* Friday December 7, 10AM-11AM, with Peter at 215 Huron, 10th floor.
* Monday December 10, 10:30AM-11:30AM, with Dror at Bahen 6178.
* Monday December 10, 2PM-4PM, with Peter at 215 Huron, 10th floor.
* Tuesday December 11, 1PM-3:30PM, with Brandon at 215 Huron, 10th floor.
* Wednesday December 12, 11AM-1PM, with Peter at 215 Huron, 10th floor.
* Wednesday December 12, 1PM-3:30PM, with Brandon at 215 Huron, 10th floor.
* Wednesday December 12, 4PM at least until 6PM, with Dror at Bahen 6178.

==Cheat Sheets made by Students==
Caveat emptor! These notes should not be used in anyway that violates the Code of Academic Integrity. Furthermore, to use these notes effectively, you should be able to prove every bullet points stated.

Materials covered before the midterm: [http://drorbn.net/images/1/12/Midterm1.JPG Part 1] [http://drorbn.net/images/2/23/Midterm2.JPG Part 2] [http://drorbn.net/images/6/62/Midterm3.JPG Part 3]

[http://drorbn.net/images/1/17/DeterminantsforDummies.JPG Determinants for Determined Dummies] (this would especially helps if you are lost in the topic of determinants) (or if you like bad puns)

Errata:
Part 1, 1L says "the characteristic of a finite field is either prime or zero." This is incorrect. The characteristic of a finite field must be nonzero and prime. On an unrelated notes, I believe there exists infinite fields of prime characteristics. Yet, a field with characteristic 0 has to be infinite.

12-240

2012-12-10T19:54:13Z

Quanle:

__NOEDITSECTION__
__NOTOC__
{{12-240/Navigation}}
==Algebra I==
===Department of Mathematics, University of Toronto, Fall 2012===

{{12-240/Crucial Information}}

===Text===
Our main text book will be ''Linear Algebra'' (fourth edition) by Friedberg, Insel and Spence, ISBN 0-13-008451-4; it is a required reading. An errata is at http://www.math.ilstu.edu/linalg/errata.html.

===Further Resources===

* [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].

* [http://wiki.math.toronto.edu/TorontoMathWiki/index.php/2011F_MAT240_Algebra_I Marco Gualtieri's 2011 Math 240 web site].

* [http://wiki.math.toronto.edu/TorontoMathWiki/index.php/10-240 Marco Gualtieri's 2010 Math 240 web site].

* [[09-240|My 2009 Math 240 web site]].

* [http://www.math.toronto.edu/murnaghan/courses/mat240/index.html The 2008 MAT240 site].

* [[06-240|My 2006 Math 240 web site]].

* My {{Pensieve Link|Classes/12-240/|12-240 notebook}}.

{{12-240:Dror/Students Divider}}

* Mathematics at Google publication: [http://research.google.com/pubs/pub38331.html (abstract)] [http://research.google.com/pubs/archive/38331.pdf (slides)].
* [[12-240/Proofs_in_Vector_Spaces|ProofWiki]]
* Pretty much the [http://www.math.uri.edu/~eaton/NotesCh3&4.pdf outline] for the text's chapter 3 and 4, helpful for review.
* A few cheat sheets made by students have been posted in [http://drorbn.net/index.php?title=12-240/The_Final_Exam The Final Exam] section.

===Online Discussion Platform===
* [http://mat240.wordpress.com/ Click to go to the online discussion platform]

12-240/The Final Exam

2012-12-10T19:48:52Z

Quanle: /* Cheat Sheets made by Students */

{{12-240/Navigation}}

Our final exam is coming up. It will take place on Thursday December 13th, from 9AM until noon, at NF003 - Northrop Frye Hall, Victoria College, 73 Queen's Park Crescent.

===Content and Style===
It will consist of 5-6 questions (each may have several parts) on everything that we have covered in class this semester:
*Fields and vector spaces.
*Spans, independence, replacement and bases.
*Linear transformation, rank, nullity, matrices.
*Row and column reduction and elementary matrices, systems of linear equations.
*Determinants.
*A bit on diagonalization.
*Several other "smaller" topics.

As for the style -

*You can expect to be asked to reproduce some proofs that were given in class.
*You can expect some fresh things to prove, though generally not as hard as the previous type of proofs.
*You can expect questions (or parts of questions) that will be identical or nearly identical to questions that were assigned for homework.
*You can expect some calculations (but nothing that will require a calculator).

It is likely that the exam will be close in spirit to the exams of six and three years ago. See [[06-240/The_Final_Exam]] and [[09-240/The Final Exam]].

Basic calculators (not capable of displaying text or sounding speech) will be allowed but will not be necessary. You may wish to bring one nevertheless, as under pressure <math>5+7</math> often comes out to be <math>13</math>.

'''Remember.''' Neatness counts! Organization counts! Language counts! Proofs are best given as short and readable essays; without the English between the formulas one never knows how to interpret those formulas. When you write, say, "<math>v\in V</math>", does it mean "choose <math>v\in V</math>", or "we've just proven that <math>v\in V</math>", or "assume by contradiction that <math>v\in V</math>", or "for every <math>v\in V</math>" or "there exists <math>v\in V</math>"? If you don't say, your reader has no way of knowing. Also remember that long and roundabout solutions of simple problems, full of detours and irrelevant facts, are often an indication that their author didn't quite get the point, even if they are entirely correct. Avoid those!

==Office Hours==
Brandon and Peter and I will hold pre-exam office hours as follows:
* Friday December 7, 10AM-11AM, with Peter at 215 Huron, 10th floor.
* Monday December 10, 10:30AM-11:30AM, with Dror at Bahen 6178.
* Monday December 10, 2PM-4PM, with Peter at 215 Huron, 10th floor.
* Tuesday December 11, 1PM-3:30PM, with Brandon at 215 Huron, 10th floor.
* Wednesday December 12, 11AM-1PM, with Peter at 215 Huron, 10th floor.
* Wednesday December 12, 1PM-3:30PM, with Brandon at 215 Huron, 10th floor.
* Wednesday December 12, 4PM at least until 6PM, with Dror at Bahen 6178.

==Cheat Sheets made by Students==
Caveat emptor! These notes should not be used in anyway that violates the Code of Academic Integrity. Furthermore, to use these notes effectively, you should be able to prove every bullet points stated.

Materials covered before the midterm: [http://drorbn.net/images/1/12/Midterm1.JPG Part 1] [http://drorbn.net/images/2/23/Midterm2.JPG Part 2] [http://drorbn.net/images/6/62/Midterm3.JPG Part 3]

[http://drorbn.net/images/1/17/DeterminantsforDummies.JPG Determinants for Determined Dummies] (this would especially helps if you are lost in the topic of determinants) (or if you like bad puns)

12-240/The Final Exam

2012-12-10T19:48:37Z

Quanle: /* Cheat Sheets made by Students */

{{12-240/Navigation}}

Our final exam is coming up. It will take place on Thursday December 13th, from 9AM until noon, at NF003 - Northrop Frye Hall, Victoria College, 73 Queen's Park Crescent.

===Content and Style===
It will consist of 5-6 questions (each may have several parts) on everything that we have covered in class this semester:
*Fields and vector spaces.
*Spans, independence, replacement and bases.
*Linear transformation, rank, nullity, matrices.
*Row and column reduction and elementary matrices, systems of linear equations.
*Determinants.
*A bit on diagonalization.
*Several other "smaller" topics.

As for the style -

*You can expect to be asked to reproduce some proofs that were given in class.
*You can expect some fresh things to prove, though generally not as hard as the previous type of proofs.
*You can expect questions (or parts of questions) that will be identical or nearly identical to questions that were assigned for homework.
*You can expect some calculations (but nothing that will require a calculator).

It is likely that the exam will be close in spirit to the exams of six and three years ago. See [[06-240/The_Final_Exam]] and [[09-240/The Final Exam]].

Basic calculators (not capable of displaying text or sounding speech) will be allowed but will not be necessary. You may wish to bring one nevertheless, as under pressure <math>5+7</math> often comes out to be <math>13</math>.

'''Remember.''' Neatness counts! Organization counts! Language counts! Proofs are best given as short and readable essays; without the English between the formulas one never knows how to interpret those formulas. When you write, say, "<math>v\in V</math>", does it mean "choose <math>v\in V</math>", or "we've just proven that <math>v\in V</math>", or "assume by contradiction that <math>v\in V</math>", or "for every <math>v\in V</math>" or "there exists <math>v\in V</math>"? If you don't say, your reader has no way of knowing. Also remember that long and roundabout solutions of simple problems, full of detours and irrelevant facts, are often an indication that their author didn't quite get the point, even if they are entirely correct. Avoid those!

==Office Hours==
Brandon and Peter and I will hold pre-exam office hours as follows:
* Friday December 7, 10AM-11AM, with Peter at 215 Huron, 10th floor.
* Monday December 10, 10:30AM-11:30AM, with Dror at Bahen 6178.
* Monday December 10, 2PM-4PM, with Peter at 215 Huron, 10th floor.
* Tuesday December 11, 1PM-3:30PM, with Brandon at 215 Huron, 10th floor.
* Wednesday December 12, 11AM-1PM, with Peter at 215 Huron, 10th floor.
* Wednesday December 12, 1PM-3:30PM, with Brandon at 215 Huron, 10th floor.
* Wednesday December 12, 4PM at least until 6PM, with Dror at Bahen 6178.

==Cheat Sheets made by Students==
Caveat emptor! These notes should not be used in anyway that violate the Code of Academic Integrity. Furthermore, to use these notes effectively, you should be able to prove every bullet points stated.

Materials covered before the midterm: [http://drorbn.net/images/1/12/Midterm1.JPG Part 1] [http://drorbn.net/images/2/23/Midterm2.JPG Part 2] [http://drorbn.net/images/6/62/Midterm3.JPG Part 3]

[http://drorbn.net/images/1/17/DeterminantsforDummies.JPG Determinants for Determined Dummies] (this would especially helps if you are lost in the topic of determinants) (or if you like bad puns)

12-240/The Final Exam

2012-12-10T19:47:55Z

Quanle:

{{12-240/Navigation}}

Our final exam is coming up. It will take place on Thursday December 13th, from 9AM until noon, at NF003 - Northrop Frye Hall, Victoria College, 73 Queen's Park Crescent.

===Content and Style===
It will consist of 5-6 questions (each may have several parts) on everything that we have covered in class this semester:
*Fields and vector spaces.
*Spans, independence, replacement and bases.
*Linear transformation, rank, nullity, matrices.
*Row and column reduction and elementary matrices, systems of linear equations.
*Determinants.
*A bit on diagonalization.
*Several other "smaller" topics.

As for the style -

*You can expect to be asked to reproduce some proofs that were given in class.
*You can expect some fresh things to prove, though generally not as hard as the previous type of proofs.
*You can expect questions (or parts of questions) that will be identical or nearly identical to questions that were assigned for homework.
*You can expect some calculations (but nothing that will require a calculator).

It is likely that the exam will be close in spirit to the exams of six and three years ago. See [[06-240/The_Final_Exam]] and [[09-240/The Final Exam]].

Basic calculators (not capable of displaying text or sounding speech) will be allowed but will not be necessary. You may wish to bring one nevertheless, as under pressure <math>5+7</math> often comes out to be <math>13</math>.

'''Remember.''' Neatness counts! Organization counts! Language counts! Proofs are best given as short and readable essays; without the English between the formulas one never knows how to interpret those formulas. When you write, say, "<math>v\in V</math>", does it mean "choose <math>v\in V</math>", or "we've just proven that <math>v\in V</math>", or "assume by contradiction that <math>v\in V</math>", or "for every <math>v\in V</math>" or "there exists <math>v\in V</math>"? If you don't say, your reader has no way of knowing. Also remember that long and roundabout solutions of simple problems, full of detours and irrelevant facts, are often an indication that their author didn't quite get the point, even if they are entirely correct. Avoid those!

==Office Hours==
Brandon and Peter and I will hold pre-exam office hours as follows:
* Friday December 7, 10AM-11AM, with Peter at 215 Huron, 10th floor.
* Monday December 10, 10:30AM-11:30AM, with Dror at Bahen 6178.
* Monday December 10, 2PM-4PM, with Peter at 215 Huron, 10th floor.
* Tuesday December 11, 1PM-3:30PM, with Brandon at 215 Huron, 10th floor.
* Wednesday December 12, 11AM-1PM, with Peter at 215 Huron, 10th floor.
* Wednesday December 12, 1PM-3:30PM, with Brandon at 215 Huron, 10th floor.
* Wednesday December 12, 4PM at least until 6PM, with Dror at Bahen 6178.

==Cheat Sheets made by Students==
Caveat emptor! These notes should not be used in anyway that violate the Code of Academic Integrity. Furthermore, to use these notes effectively, you should be able to prove every bullet points stated.

Materials covered before the midterm: [http://drorbn.net/images/1/12/Midterm1.JPG Part 1] [http://drorbn.net/images/2/23/Midterm2.JPG Part 2] [http://drorbn.net/images/6/62/Midterm3.JPG Part 3]

[http://drorbn.net/images/1/17/DeterminantsforDummies.JPG Determinants for Determined Dummies] (this would especially helps if you are lost in the topic of determinants)

12-240

2012-12-09T00:21:37Z

Quanle:

__NOEDITSECTION__
__NOTOC__
{{12-240/Navigation}}
==Algebra I==
===Department of Mathematics, University of Toronto, Fall 2012===

{{12-240/Crucial Information}}

===Text===
Our main text book will be ''Linear Algebra'' (fourth edition) by Friedberg, Insel and Spence, ISBN 0-13-008451-4; it is a required reading. An errata is at http://www.math.ilstu.edu/linalg/errata.html.

===Further Resources===

* [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].

* [http://wiki.math.toronto.edu/TorontoMathWiki/index.php/2011F_MAT240_Algebra_I Marco Gualtieri's 2011 Math 240 web site].

* [http://wiki.math.toronto.edu/TorontoMathWiki/index.php/10-240 Marco Gualtieri's 2010 Math 240 web site].

* [[09-240|My 2009 Math 240 web site]].

* [http://www.math.toronto.edu/murnaghan/courses/mat240/index.html The 2008 MAT240 site].

* [[06-240|My 2006 Math 240 web site]].

* My {{Pensieve Link|Classes/12-240/|12-240 notebook}}.

{{12-240:Dror/Students Divider}}

* Mathematics at Google publication: [http://research.google.com/pubs/pub38331.html (abstract)] [http://research.google.com/pubs/archive/38331.pdf (slides)].
* [http://drorbn.net/index.php?title=12-240/Proofs_in_Vector_Spaces ProofWiki]
* Pretty much the [http://www.math.uri.edu/~eaton/NotesCh3&4.pdf outline] for the text's chapter 3 and 4, helpful for review.
===Online Discussion Platform===
* [http://mat240.wordpress.com/ Click to go to the online discussion platform]

12-240/Term Test

2012-10-30T22:49:14Z

Quanle:

{{12-240/Navigation}}
Our term test took place on Thursday October 25 at 3-5PM. The results are available on the UofT Portal.

'''The Test:''' {{pensieve link|Classes/12-240/TT.pdf|TT.pdf}}.

==The Results==
Excluding some exceptions, 115 students took the test. Before appeals, the average grade was approximately 75 and the standard deviation was approximately 20.

The results are quite similar to what I expected them to be. The easiest questions (on average) were the computational ones, the hardest were the ones involving proofs.

How should you read your grade?

* If you got 100 you should pat yourself on your shoulder and feel good.
* If you got something like 95, you're doing great. You made a few relatively minor mistakes; find out what they are and try to avoid them next time.
* If you got something like 85 you're doing fine but you did miss something significant, probably more than just a minor thing. Figure out what it was and make a plan to fix the problem for next time.
* If you got something like 65 you should be concerned. You are still in position to improve greatly and get an excellent grade at the end, but what you missed is quite significant and you are at the risk of finding yourself far behind. You must analyze what happened - perhaps it was a minor mishap, but more likely you misunderstood something major or something major is missing in your background. Find out what it is and try to come up with a realistic strategy to overcome the difficulty!
* If you got something like 45, most likely you are not gaining much from this class and you should consider dropping it, unless you are convinced that you fully understand the cause of your difficulty (you were very sick, you really couldn't study at all for the two weeks before the exam because of some unusual circumstances, something like that) and you feel confident you have a fix for next time. If you do decide to drop the class, don't feel too bad about it. It is the hardest first year algebra class at UofT and of the thousands of students taking math here, very few come with sufficient preparation to do well in it.

Note that problems with writing are problems, period. Perhaps you got a low grade but you feel you know the material enough for a high grade only you didn't write everything you know or you didn't it write well enough or the silly graders simply didn't get what you wrote (and it isn't a simple misunderstanding - see "appeals" below). If this describes you, don't underestimate your problem. If you don't process and resolve it, it is likely to recur.

====Appeals.====
Remember! Grading is a difficult process and mistakes '''always''' happen - solutions get misread, parts are forgotten, grades are not added up correctly. You '''must''' read your exam and make sure that you understand how it was graded. If you disagree with anything, don't hesitate to complain! Your first stop should be the person who graded the problem in question, and only if you can't agree with him you should appeal to {{Dror}}.

Problems 1 and 3 were graded by Peter Crooks and problems 2 and 4 by Brandon Hanson. {{Dror}} graded problem number 5 and did the data entry.

The deadline to start the appeal process is Tuesday November 6 at 1PM. Once you've started the process by talking to {{Dror}} or to Brandon or to Peter, it ends when a final decision is made, with no deadline.

12-240/Classnotes for Thursday October 4

2012-10-21T19:17:28Z

Quanle: /* Pre - Basis */

{{12-240/Navigation}}
== Reminders ==
Web Fact: No link, doesn't exist!

Life Fact: Dror doesn't do email math!

Riddle: Professor and lion in a ring with <math>V_p = V_l</math>, help the professor live as long as possible.

== Recap ==

Base - what were doing today

Linear combination (lc) - We say <math>v</math> is a linear combination of a set <math>S = \{u_1, \dots, u_n\}</math> if <math>v = a_1u_1, \dots, a_nu_n</math> for scalars from a field <math>F</math>.

Span - <math>\operatorname{span}(S)</math> is the set of all linear combination of set <math>S</math>.

Generate - We say <math>S</math> generates a vector space <math>V</math> is <math>\operatorname{span}(S) = V</math>.

== Pre - Basis ==

'''Linear dependence'''

'''Definition''' A set S ⊂ V is called linearly dependent if you can express the zero vector as a linear combination of distinct vectors from S, excluding the non-trivial linear combination where all of the scalars are 0.

Otherwise, we call S '''linearly independent.'''

=== Examples ===

1. In '''R'''^3, take S = {u1 = (1,4,7), u2 = (2,5,8), u3 = (3,6,9)}

u1 - 2u2 + u3 = 0

S is linearly dependent.

2. In R^n, take {e_i} = {0, 0, ... 1, 0, 0, 0} where 1 is in the ith position, and (ei) is a vector with n entries.

Claim: This set is linearly independent.

Proof: Suppose (∑ ai*ei) = 0 ({ei} is linearly dependent.)

(∑ ai*ei) = 0 ⇔ a1(1, 0, ..., 0) + ... + an(0, ..., 0, 1) = 0 ⇔ (a1, 0, ... , 0) + ... + (0, ... , an) = 0

⇒ a1 = a2 = ... = an = 0!

===Comments ===

1. {u} is linearly independent.
Proof:
⇐ If u≠0, suppose au =0
By property (a*u = 0, a = 0 or u = 0), a = 0. Thus, {u} is linearly independent.

⇒ By definition, au = 0 for {u} only when a = 0.

2. ∅ is linearly independent.

Exercise: Prove: '''Theorem''' Suppose S1 ⊂ S2 ⊂ V.

-> If S1 is linearly dependent, then S2 is dependent.

-> If S2 is linearly independent, then S1 is linearly independent. (Hint: contrapositive)

== Basis ==

'''Definition''': A subset β is called a basis if 1. β generates V → span(β) = V and 2. β is linearly independent.

===Examples===

1. V = {0}, β = {}

2. {ei} for F^n, this is what we call the '''standard basis'''

3. B = {(1,1),(1, -1)} is a basis for R^2

4. P_n(F) β = {x^n, x^n-1, ... , x^1, x^0}

5. P(F), β = (x^0, x^1 ... and on} ('''Infinite basis'''!)

== Interesting inequality ==

This holds is true if the field does not have characteristic 2. Can you see why?

(a,b) = (a+b)/2 * (1, 1) + (a-b)/2 * (1, -1)

== Lecture notes scanned by [[User:starash|starash]] ==
<gallery>
Image:12-240-1004-1.jpg|Page 1
Image:12-240-1004-2.jpg|Page 2
</gallery></math>

12-240/Class Photo

2012-09-29T05:17:17Z

Quanle:

Our class on September 25, 2012:

[[Image:12-240-ClassPhoto.jpg|thumb|centre|800px|Class Photo: click to enlarge]]
{{12-240/Navigation}}

Please identify yourself in this photo! There are two ways to do that:

* [[Special:Userlogin|Log in]] to this Wiki and edit this page. Put your name, userid, email address and location in the picture in the alphabetical list below.
* Send [[User:Drorbn|Dror]] an email message with this information.

The first option is more fun but less private.

===Who We Are...===

{| align=center border=1 cellspacing=0
|-
!First name
!Last name
!User ID
!Email
!Place in photo
!Comments

{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn@ math. toronto. edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank. For better line-breaking, leave a space next to the "@" in email addresses.}}
{{Photo Entry|last=Frailich|first=Rebecca|userid=Rebecca.frailich|email=rebecca.frailich@ mail.utoronto.ca|location=Last row, in between two guys standing at the back (one in red, one in black) |comments=}}
{{Photo Entry|last=Hoover|first=Ken|userid=Khoover|email=ken.hoover@ mail.utoronto.ca|location=First row, fourth from the right.|comments=}}
{{Photo Entry|last=Klingspor|first=Josefine|userid=Josefine|email=josefine. klingspor@ mail. utoronto. ca|location=First row, second from left.|comments=}}
{{Photo Entry|last=Le|first=Quan|userid=Quanle|email=quan.le@mail.utoronto.ca|location=Start bottom right corner, third from right. Go three steps north-west. Directly north-east from there, in blue collar shirt|comments=}}
{{Photo Entry|last=Liu|first=Zhaowei|userid=tod|email=tod. liu@ mail. utoronto .ca|location=First row, third from the right|comments=}}
{{Photo Entry|last=Millson|first=Richard|userid=Richardm|email=r.millson@ mail. utoronto. ca|location=Seventh row from the front, fourth from the right, blue sweater|comments=}}
{{Photo Entry|last=McGrath|first=Celton|userid=CeltonMcGrath|email=celton. mcgrath@ mail. utoronto. ca|location=4th row front from, centre right, brown sweater|comments=}}
{{Photo Entry|last=Morenz|first=Karen|userid=KJMorenz|email=kjmorenz@ gmail.com|location=3rd-ish row from the back, centre right, purple shirt|comments=}}
{{Photo Entry|last=Vicencio-Heap|first=Felipe|userid=Heapfeli|email=felipe. vicencio. heap@ mail. utoronto. ca|location=Second row from the front, furthest to the right.|comments=}}
{{Photo Entry|last=Wamer|first=Kyle|userid=kylewamer|email=kyle. wamer @ mail. utoronto. ca|location=Second row, fifth from the left in the red shirt.|comments=}}
{{Photo Entry|last=Yang|first=Chen|userid=chen|email=neochen. yang@ mail. utoronto. ca|location=sixth row, first from the right in the black pull-over.|comments=}}
{{Photo Entry|last=Zhang|first=BingZhen|userid=Zetalda|email=bingzhen.zhang@ mail. utoronto. ca|location=Second last row, third from left.|comments=}}
{{Photo Entry|last=Zhao|first=TianChen|userid=Ericolony|email=zhao_tianchen@ hotmail. com|location=fourth row, the guy in green shirt.|comments=}}
{{Photo Entry|last=Zibert|first=Vincent|userid=vincezibert|email=vincent. zibert@ mail. utoronto. ca|location=Directly beneath the white notice posted on the door on the right-hand side.|comments=}}
{{Photo Entry|last=Zoghi|first=Sina|userid=sina.zoghi|email=sina.zoghi@ utoronto .ca|location=First row, leftest left.|comments=}}
{{Photo Entry|last=Léger|first=Zacharie|userid=zach.leger8|email=zacharie. leger@ mail. utronto. ca|location= 5th row in a black T-shirt.|comments=}}

|}