Drorbn - User contributions [en]

Talk:06-240/Homework Assignment 5

2006-10-14T23:35:28Z

70.24.245.27:

For the test, do we have to know the LaGrange formula? Although not covered in class, it is in Section 1.6, which we've been asked to read.

The test material will only be announced on Tuesday. --[[User:Drorbn|Drorbn]] 13:02, 14 October 2006 (EDT)

For question 28: "Let V be a finite-dimensional vector space over C with dimension n. Prove that if V is now ''regarded as a vector space over R'', then dim V = 2n"...
Is this a formally defined concept? (that is, while it is obvious what they mean, how could you state it rigorously)

06-240/Classnotes For Tuesday, September 12

2006-10-08T19:45:22Z

70.24.245.27:

{{06-240/Navigation}}

* PDF notes by [[User:Harbansb]]: [[Media:06-240-0912.pdf|September 12 Notes]].
* If I have made an error in my notes, or you would like the editable OpenOffice file, feel free to e-mail me at harbansb@msn.com.
* [http://www.yousendit.com/transfer.php?action=download&ufid=38FF36BF7ED1E1BA September 12 Notes] for re-uploading, please email at jeff.matskin@utoronto.ca
* PDF notes by [[User:Alla]]: [[Media:MAT_Lect001.pdf|Week 1 Lecture 1 notes]]
* 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]]

=Notes=

==The Real Numbers==
The Real Numbers are a set (denoted by <math>\mathbb{R}</math>) along with two binary operations: + (plus) and · (times) and two special elements: 0 (zero) and 1 (one), such that the following laws hold true: 
<math>\mathbb{R}1:\forall a, b\in \mathbb{R}\mbox{ s.th.} \quad a+b=b+a \quad \mbox{and} \quad a\cdot b=b\cdot a</math> (The Commutative Laws) 
<math>\mathbb{R}2:\forall a, b, c\in \mathbb{R}\mbox{ s.th.} \quad (a+b)+c=a+(b+c) \quad \mbox{and} \quad (a\cdot b)\cdot c=a\cdot (b\cdot c) </math> (The Associative Laws) 
<math>\mathbb{R}3:0\mbox{ is an additive unit} \quad \mbox{and} \quad 1\mbox{ is a multiplicative unit}</math> (The Existence of Units/Identities) 
<math>\mathbb{R}4:\forall a\in \mathbb{R} \ \exists b\in \mathbb(R) \mbox{ s.th.} \ a+b=0</math>

This is incomplete.

06-240

2006-09-24T01:40:28Z

70.24.245.27:

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

{{06-240/Crucial Information}}

===Text===
Freidberg, Insel, Spence. Linear Algebra, 4e. New Jersy: Pearson Education Inc, 2003. 

===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 Math 240 web 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.

Talk:06-240

2006-09-23T16:08:49Z

70.24.245.27:

I think this page needs to list the course textbook. I do not have it on me right now, however, so someone will have to add it.

Also, if anyone is interested in typesetting the lectures I think they should follow these Wikipedia guidelines: http://en.wikipedia.org/wiki/WP:MSM

I attempted to typeset the Sept 12 notes, but it takes way too long to do. If we really want to have typesetted notes, it would probably be much more efficient to make a Word document and then upload it, although it's not as "official" looking.

I doubt it is eaiser to type with Word; typing is typing; you just have to get used to the different interface. And if you type in word others can view and edit it only if they have word too, and even then, fixing a typo becomes way harder. --[[User:Drorbn|Drorbn]] 03:27, 23 September 2006 (EDT)

The big differnce is formatting. In Word, it is easier to format text and MUCH easier to type equations. In Wikipedia, you have to use tags that are tedious and hard to keep track of: see the Sept 12 notes for an example. Perhpas there is an easier way of formatting text in Wikipedia that I am not aware of? (For example, maybe you can convert a Word file?)

== title of the Textbook ==

the title of the textbook is:

LINEAR ALGEBRA
by Friedberf, Insel and Spence.
4th edition.
Publisher: Prentice Hall.

-nicole =)

== Modular Arithmetic ==

This was particularly interesting after being introduced to Modular Multiplication tables and seeing some visual patterns with the numbers, such as the in the '1' column where the elements go from 1 to n-1 in Zn and backwards in the 'n-1' column.

After searching around, it seems that people had been able to discover other, more interesting patterns!

Make sure to analyze the tables since they begin from the bottom left corner instead of top left which we saw in class.

http://whistleralley.com/mod/mod25.htm

The following site allows you to see tables up to mod 30.

http://www.cut-the-knot.org/blue/Modulo.shtml

-Richard

Also, notice how in modular multiplication tables for prime numbers, in specific for modulo 5 in the columns and rows for 0 and 5 only 0s appear. The 0s create a sort of frame around a 4x4 square of elements. Specifically all elements within the frame of 0s are between 1 and n-1 and all are non-zero. In the case of the mod 4 table there was a 0 which, as proved in class causes Z4 to fail as a field. There must be something deeper about all those 0s.

06-240/Classnotes For Tuesday, September 12

2006-09-23T02:46:24Z

70.24.245.27:

Talk:06-240

2006-09-15T00:22:43Z

70.24.245.27:

Talk:06-240

2006-09-15T00:21:26Z

70.24.245.27:

Talk:06-240

2006-09-14T00:25:27Z

70.24.245.27:

I think this page needs to list the course textbook. I do not have it on me right now, however, so someone will have to add it.