Drorbn - User contributions [en]

12-240/The Final Exam

2012-12-18T21:07:58Z

Failure: /* Office Hours */

{{12-240/Navigation}}

==The Results==

101 students took the exam; the average grade was 70, the standard deviation was 26, and the exam itself is at {{pensieve link|Classes/12-240/Final.pdf|Final.pdf}}.

Restricted to the 101 students who took the final, the average course grade was 75 and the standard deviation was 19.

==Announcement==

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===

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

2012-12-18T21:07:18Z

Failure: /* Office Hours */

{{12-240/Navigation}}

==The Results==

101 students took the exam; the average grade was 70, the standard deviation was 26, and the exam itself is at {{pensieve link|Classes/12-240/Final.pdf|Final.pdf}}.

Restricted to the 101 students who took the final, the average course grade was 75 and the standard deviation was 19.

==Announcement==

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, Tenth floor.
* Wednesday December 12, 4PM at least until 6PM, with Dror at Bahen 6178.

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

===Cheat Sheets made by Students===

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/Classnotes for Thursday October 18

2012-10-23T06:05:35Z

Failure:

{{12-240/Navigation}}
===Riddle Along===
The game of 15 is played as follows. Two players alternate choosing cards numbered between 1 and 9, with repetitions forbidden, so that the game ends at most after 9 moves (or four and a half rounds). The first player to have within her/his cards a set of precisely 3 cards that add up to 15 wins.

Does this game have a winning strategy? What is it? Who wins, the first to move or the second? Why am I asking this question at this particular time?

[[Image:12-240-DeckOfCards.png|center]]

See also [https://media.library.utoronto.ca/play.php?DJ6CPFxByy2J&id=8503 a video] and the [https://cmc.math.ca/home/videos/game-of-15-and-isomorphisms/ transcript] of that video.

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

== Theorems ==
1. If G generates, |G| <math>\ge \!\,</math> n and G contains a basis, |G|=n then G is a basis

2. If L is linearly independent, |L| <math>\le \!\,</math> n and L can be extended to be a basis. |L|=n => L is a basis.

3.W <math>\subset \!\,</math> V a subspace then W is finite dimensioned and dim W <math>\le \!\,</math> dim V

If dim W = dim V, then V = W
If dim W < dim V, then any basis of W can be extended to be a basis of V

Proof of W is finite dimensioned:

Let L be a linearly independent subset of W which is of maximal size.

Fact about '''N'''
: Every subset A of '''N''', which is:

1. Non empty

2. Bounded : <math>\exist \!\,</math> N <math>\in \!\,</math> '''N''', <math>\forall \!\,</math> a <math>\in \!\,</math> A, a <math>\le \!\,</math> N

has a maximal element: an element m <math>\in \!\,</math> A, <math>\forall\!\,</math> a <math>\in \!\,</math> A, a <math>\le \!\,</math> m ( m + 1 <math>\notin \!\,</math> A )

== class note ==

<gallery>
Image:12-240-Oct-15-Page-1.jpg |page1
Image:12-240-Oct-15-Page-2.jpg |page2
Image:12-240-Oct-15-Page-3.jpg |page3
</gallery>

== lecture note on oct 18, uploaded by [[User:starash|starash]]==

<gallery>
Image:12-240-1018-1.jpg |page1
Image:12-240-1018-2.jpg |page2
</gallery>

12-240/Classnotes for Thursday October 18

2012-10-23T06:04:21Z

Failure:

12-240/Classnotes for Thursday September 13

2012-10-23T06:01:02Z

Failure:

{{12-240/Navigation}}
In the second day of the class, the professor continues on the definition of a field.

== Definition of a field ==

Combined with a part from the first class, we have a complete definition as follow:

A field is a set "F' with two binary operations +,x defind on it, and two special elements 0 ≠ 1 such that

'''F1:''' commutative law

<math>\forall \!\,</math> a, b <math>\in \!\,</math> F: a+b=b+a and a.b=b.a

'''F2:''' associative law

<math>\forall \!\,</math> a, b, c <math>\in \!\,</math> F: (a+b)+c=a+(b+c) and (a.b).c= a.(b.c)

'''F3:''' the existence of identity elements

<math>\forall \!\,</math> a <math>\in \!\,</math>, a+0=a and a.1=a

'''F4:''' existence of inverses

<math>\forall \!\,</math> a <math>\in \!\,</math> F \0,<math> \exists \!\,</math> c, d <math>\in \!\ </math> F such that a+c=o and a.d=1

'''F5:''' contributive law

<math>\forall \!\,</math> a, b, c <math>\in \!\,</math> F, a.(b+c)=a.b + a.c
== Theorems ==

'''Theorem 1: Cancellation laws'''
<math>\forall \!\,</math> a, b, c <math>\in \!\,</math> F

if a+c=b+c, then a=b

if a.c=b.c and c<math>\ne \!\,</math>0, then a=b

'''Theorem 2: Identity uniqueness'''

Identity elements 0 and 1 mentioned in '''F3''' are unique

<math>\forall \!\,</math> a, b, b' <math>\in \!\,</math> F

if a+b=a and a+b'=a, then b=b'=0

if a.b=a and a.b'=a and a<math>\ne\!\,</math>0, then b=b'=1

'''Theorem 3: Inverse uniqueness'''

Elements c and d mentioned in '''F4''' are unique

<math>\forall \!\,</math> a, b, b' <math>\in \!\,</math> F

if a+b=0 and a+b'=0, then b=b'

if a.b=1 and a.b'=1, then b=b'

'''Theorem 4'''

<math>\forall \!\,</math> a <math>\in \!\,</math> F

-( -a) = a

'''Theorem 4'''

<math>\forall \!\,</math> a <math>\in \!\,</math> F

0.a= 0

== Significance ==

'''inverse uniqueness'''

It makes sense to define an operation
-: F -> F called "negation"

For a <math>\in\!\,</math> F define -a to be equal that b <math>\in\!\,</math> F for which a+b=0, i.e, a+(-a)=0

Ex: F(5)={0,1,2,3,4}, define +,x

Question 1: What is(-3)?

Answer: -3=2 and -3 is unique

Similarly, the inverse uniqueness also makes sense a^(-1)

''' identity uniqueness'''

== Lecture Notes, upload by [[User:Starash|Starash]] ==

<gallery>
Image:Mat240 120913 p1.jpg|Page 1
Image:Mat240 120913 p2.jpg|Page 2
</gallery>

12-240/Classnotes for Thursday September 13

2012-10-23T06:00:50Z

Failure: