Drorbn - User contributions [en]

06-240

2006-10-25T02:21:18Z

Adamgolding:

__NOEDITSECTION__
{{06-240/Navigation}}
==Algebra I==
===Department of Mathematics, University of Toronto, Fall 2006===

{{06-240/Crucial Information}}

===Text===
Freidberg, Insel, Spence. <u>Linear Algebra, 4e</u>. New Jersy: Pearson Education Inc, 2003.<br>

===Famous Quotes About Math and Mind===
* "The problems that exist in the world today cannot be solved by the level of thinking that created them." by Albert Einstein
* "Do not worry about your difficulties in mathematics, I can assure you mine are still greater." by Albert Einstein
* "The mind, once expanded to the dimensions of larger ideas, never returns to its original size." by Oliver Wendell Holmes, Sr.
* "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)
* "A mathematician is a machine for turning coffee into theorems." by Alfréd Rényi, colleague of Paul Erdős.

===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://www.math.toronto.edu/murnaghan/courses/mat240/index.html Last year's MAT240 web site].
* [http://www.math.toronto.edu/~megumi/MAT240/240.html Two years ago's MAT240 site].
* [http://www.maths.leeds.ac.uk/~khouston/httlam.html "How to Think Like a Mathematician"] by Kevin Houston.
* [http://www.stonehill.edu/compsci/History_Math/math-read.htm "How to Read Mathematics"] by Shai Simonson and Fernando Gouvea.
* A wide variety of free online mathematics texts is [http://www.math.gatech.edu/~cain/textbooks/onlinebooks.html here]. Want more examples? Get them here. Also, for those in 157 I'd suggest looking at Elias Zakon's Analysis 1 text as a supplement for Spivak.

06-240

2006-10-25T02:20:14Z

Adamgolding:

__NOEDITSECTION__
{{06-240/Navigation}}
==Algebra I==
===Department of Mathematics, University of Toronto, Fall 2006===

{{06-240/Crucial Information}}

===Text===
Freidberg, Insel, Spence. <u>Linear Algebra, 4e</u>. New Jersy: Pearson Education Inc, 2003.<br>

===Famous Quotes About Math and Mind===
* "The problems that exist in the world today cannot be solved by the level of thinking that created them." by Albert Einstein
* "Do not worry about your difficulties in mathematics, I can assure you mine are still greater." by Albert Einstein
* "The mind, once expanded to the dimensions of larger ideas, never returns to its original size." by Oliver Wendell Holmes, Sr.
* "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)
* "A mathematician is a machine for turning coffee into theorems." by Alfréd Rényi, colleague of Paul Erdős.

===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://www.math.toronto.edu/murnaghan/courses/mat240/index.html Last year's MAT240 web site].
* [http://www.math.toronto.edu/~megumi/MAT240/240.html Two years ago's MAT240 site].
* [http://www.maths.leeds.ac.uk/~khouston/httlam.html "How to Think Like a Mathematician"] by Kevin Houston.
* A wide variety of free online mathematics texts is [http://www.math.gatech.edu/~cain/textbooks/onlinebooks.html here]. Want more examples? Get them here. Also, for those in 157 I'd suggest looking at Elias Zakon's Analysis 1 text as a supplement for Spivak.

06-240/Classnotes For Tuesday October 10

2006-10-22T08:34:39Z

Adamgolding: /* A Quick Summary by {{Dror}} */

{{06-240/Navigation}}

====Scan of Lecture Notes====

* PDF file by [[User:Alla]]: [[Media:MAT_Lect009.pdf|Week 5 Lecture 1 notes]]

==A Quick Summary by {{Dror}}==
(Intentionally terse. A sea of details appears in the book and already appeared on the blackboard. But these are useless without some '''organizing principles'''; in some sense, "understanding" is precisely being able to see those principles within the sea of details. Yet don't fool yourself into thinking that the principles are enough even without the details!)

'''Theorem.''' A finite generating set <math>G</math> has a subset which is a basis.

'''Proof Sketch.''' Grab more and more elements of <math>G</math> so long as they are linearly independent. When you can't any more, you have a basis.

'''Lemma.''' (The Replacement Lemma) If <math>G</math> generates and <math>L</math> is linearly independent, then <math>|L|\leq|G|</math> and you can replace <math>|L|</math> of the elements of <math>G</math> by the elements of <math>L</math>, and still have a generating set.

'''Proof Sketch.''' Insert the elements of <math>L</math> one by one, and for each one that comes in, take one out of <math>G</math>. Which one? One used in expressing the newcomer in terms of the vectors already in <math>G</math>. Such a vector must exist or else the newcomer is a linear combination of some of the elements of <math>L</math>, but <math>L</math> is linearly independent.

'''Theorem.''' If a vector space <math>V</math> has a finite basis, all bases thereof are finite and have the same number of elements, the "dimension of <math>V</math>".

'''Proof Sketch.''' By replacement, <math>|\alpha|\leq|\beta|</math> and <math>|\beta|\leq|\alpha|</math>.

'''Theorem.''' Assume <math>\dim V=n</math>.
# If <math>G</math> generates, <math>|G|\geq n</math>. In case of equality, <math>G</math> is a basis.
# If <math>L</math> is linearly independent, <math>|L|\leq n</math>. In case of equality, <math>L</math> is a basis.

'''Proof Sketch.'''
# Find a basis within <math>G</math>; it has <math>n</math> elements.
# Use replacement to place the elements of <math>L</math> within some basis.

06-240/Classnotes For Tuesday October 10

2006-10-22T08:31:20Z

Adamgolding: /* A Quick Summary by {{Dror}} */

{{06-240/Navigation}}

====Scan of Lecture Notes====

* PDF file by [[User:Alla]]: [[Media:MAT_Lect009.pdf|Week 5 Lecture 1 notes]]

==A Quick Summary by {{Dror}}==
(Intentionally terse. A sea of details appears in the book and already appeared on the blackboard. But these are useless without some '''organizing principles'''; in some sense, "understanding" is precisely being able to see those principles within the sea of details. Yet don't fool yourself into thinking that the principles are enough even without the details!)

'''Theorem.''' A finite generating set <math>G</math> has a subset which is a basis.

'''Proof Sketch.''' Grab more and more elements of <math>G</math> so long as they are linearly independent. When you can't any more, you have a basis.

'''Lemma.''' (The Replacement Lemma) If <math>G</math> generates and <math>L</math> is linearly independent, then <math>|L|\leq|G|</math> and you can replace <math>|L|</math> of the elements of <math>G</math> by the elements of <math>L</math>, and still have a generating set.

'''Proof Sketch.''' Insert the elements of <math>L</math> one by one, and for each one that comes in, take one out of <math>G</math>. Which one? One used in expressing the newcomer in terms of the vectors already in the set. Such a vector must exist or else the newcomer is a linear combination of some of the elements of <math>L</math>, but <math>L</math> is linearly independent.

'''Theorem.''' If a vector space <math>V</math> has a finite basis, all bases thereof are finite and have the same number of elements, the "dimension of <math>V</math>".

'''Proof Sketch.''' By replacement, <math>|\alpha|\leq|\beta|</math> and <math>|\beta|\leq|\alpha|</math>.

'''Theorem.''' Assume <math>\dim V=n</math>.
# If <math>G</math> generates, <math>|G|\geq n</math>. In case of equality, <math>G</math> is a basis.
# If <math>L</math> is linearly independent, <math>|L|\leq n</math>. In case of equality, <math>L</math> is a basis.

'''Proof Sketch.'''
# Find a basis within <math>G</math>; it has <math>n</math> elements.
# Use replacement to place the elements of <math>L</math> within some basis.

06-240

2006-09-14T06:43:18Z

Adamgolding:

__NOEDITSECTION__
{{06-240/Navigation}}
==Algebra I==
===Department of Mathematics, University of Toronto, Fall 2006===

{{06-240/Crucial Information}}

===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:
* [[06-240: Edit1.jpg]]
* [[06-240: Edit2.jpg]]

===Class Notes=== 06-240 ... ?
* [[Lecture 01, September 12, 2006]]

===Links To Notes===
* [http://www.yousendit.com/transfer.php?action=download&ufid=38FF36BF7ED1E1BA September 12 Notes] for re-uploading, please email at jeff.matskin@utoronto.ca
===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://www.math.toronto.edu/murnaghan/courses/mat240/index.html Last year's Math 240 web site].
* [http://www.maths.leeds.ac.uk/~khouston/httlam.html "How to Think Like a Mathematician" by Kevin Houston].

06-240/About This Class

2006-09-06T04:16:30Z

Adamgolding: /* Text Book(s) */

{{06-240/Navigation}}

===Crucial Information===
{{06-240/Crucial Information}}

'''URL:''' {{SERVER}}/drorbn/index.php?title=06-240.

===Abstract===
Taken from the Faculty of Arts and Science [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Calendar]:
<blockquote>
A theoretical approach to: vector spaces over arbitrary fields including <math>{\mathbb C}</math>, <math>{\mathbb Z}_p</math>. Subspaces, bases and dimension. Linear transformations, matrices, change of basis, similarity, determinants. Polynomials over a field (including unique factorization, resultants). Eigenvalues, eigenvectors, characteristic polynomial, diagonalization. Minimal polynomial, Cayley-Hamilton theorem.
</blockquote>

*Prerequisite: MCB4U, MGA4U
*Co-requisite: MAT157Y1

[[Image:Friedberg_Insel_Spence_Cover.jpg|right|200px]]
===Text Book(s)===
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. I am told that Schaum’s ''Outline of Linear Algebra'', ISBN 0-07-136200-2, may contain useful examples; it is not a required reading.

Please note that the eratta for various printings of the 4th edition of the text is available from the publisher here:
http://www.math.ilstu.edu/linalg/errata.html

===Wiki===
The class web site is a wiki, as in [http://www.wikipedia.org Wikipedia] - meaning that anyone can and is welcome to edit almost anything and in particular, students can post notes, comments, pictures, whatever. Some rules, though -
* This wiki is a part of my ([[User:Drorbn|Dror's]]) academic web page. All postings on it must be class-related (or related to one of the other projects I'm involved with).
* If there's no specific reason for your edit to be anonymous, please log in and don't have it anonymous.
* Criticism is fine, but no insults or foul language, please.
* I ([[User:Drorbn|Dror]]) will allow myself to exercise editorial control, when necessary.
* The titles of all pages related to this class should begin with "06-240/", just like the title of this page.
Some further editing help is available at [[Help:Contents]].

===Marking Scheme===
There will be one term test (25% of the total grade) and a final exam (50%), as well as about 10 homework assignments (25%).

====The Term Test====
The term test will take place in class on Tuesday October 24th. A student who misses the term test without providing a valid reason (for example, a doctor’s note) within one week of the test will receive a mark of 0 on the term test. There will be no make-up term test. If a student misses the term test for a valid reason, the weight of the problem sets will increase to 35% and the weight of the final exam to 65%.

====Homework====
Assignments will be posted on the course web page and distributed in class (usually on Thursdays) on the weeks shown in the class timeline. They will be due a week later at the tutorials and they will be (at least partially) marked by the TAs. All students (including those who join the course late) will receive a mark of 0 on each assignment not handed in; though in computing the homework grade, your worst two assignments will not count. I encourage you to discuss the assignments with other students or even browse the web, so long as you do at least some of the thinking on your own and you write up your own solutions. Remember that cheating is always possible and may increase your homework grade a bit. But it will hurt your exam grades a lot more.

===Good Deeds===
Students will be able to earn up to 25 "good deeds" points throughout the year for doing services to the class as a whole. There is no pre-set system for awarding these points, but the following will definitely count:
* Drawing a beautiful picture to illustrate a point discussed in class and posting it on this site.
* Taking class notes in nice handwriting, scanning them and posting them here.
* Typing up or formatting somebody else's class notes, correcting them or expanding them in any way.
* Writing an essay on expanding on anything mentioned in class and posting it here; correcting or expanding somebody else's article.
* Doing anything on our [[06-240/To do]] list.
* Any other service to the class as a whole.

Good deed points will count towards your final grade! If you got <math>n</math> of those, they are solidly your and the formula for the final grade below will only be applied to the remaining <math>100-n</math> points. So if you got 25 good deed points (say) and your final grade is 80, I will report your grade as <math>25+80(100-25)/100=85</math>. Yet you can get an overall 100 even without doing a single good deed.

'''Important.''' For your good deeds to count, you '''must''' do them under your own name. So you must set up an account for yourself on this wiki and you must use it whenever you edit something. I will periodically check [[Special:Recentchanges|Recent changes]] to assign good deeds credits.

===Class Photo===
To help me learn your names, I will take a class photo on Thursday of the third week of classes. I will post the picture on the class' web site and you will be ''required'' to send me an email and identify yourself in the picture or to identify yourself on the [[06-240/Class Photo|Class Photo]] page of this wiki.

===How to Succeed in this Class===
* '''Keep up!''' Don't fall behind on reading, listening, and doing assignments! University goes at a different pace than high school. New material is covered once and just once. There will be no going over the same thing again and again - if you fall behind, you stay behind. Unless you are an Einstein, there is ''no way'' to do well in this class merely by attending lectures - you '''must''' think about the material more than 3 or 5 hours a week if you want it to sink in. And if you are planning on not attending lectures, well, think again. Most people find it very hard to pace their own studies without a human contact; if you'll try, you are likely to discover the hard way that you belong to the majority.
* If in high school you were the best in your class in math, now remember that everybody around you was the same. You may find that what was enough then simply doesn't cut it any more. Try to catch that early in the year!
* Math is about '''understanding''', not about memorizing. To understand is to internalize; it is to come to the point where whatever the professor does on the blackboard or whatever is printed in the books becomes '''yours'''; it is to come to the point where you appreciate why everything is done the way it is done, what does it mean, what are the reasons and motivations and what is it all good for. Don't settle for less!
* Keep asking yourself questions; many of them will be answered in class, but not all. Remember the old Chinese proverb:

<div style="text-align: center; font-size: 18px">'''"Teachers open the door, but you must enter by yourself"'''</div>

The saddest that can happen to you in this class is if you won't notice the door being opened.