Welcome to Math 240! (additions to this web site no longer count towards good deed points)

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Week of...

Notes and Links

1

Sep 8

About This Class, What is this class about? (PDF, HTML), Monday, Wednesday

2

Sep 15

HW1, Monday, Wednesday, TheComplexField.pdf,HW1_solutions.pdf

3

Sep 22

HW2, Class Photo, Monday, Wednesday, HW2_solutions.pdf

4

Sep 29

HW3, Wednesday, Tutorial, HW3_solutions.pdf

5

Oct 6

HW4, Monday, Wednesday, Tutorial, HW4_solutions.pdf

6

Oct 13

No Monday class (Thanksgiving), Wednesday, Tutorial

7

Oct 20

HW5, Term Test at tutorials on Tuesday, Wednesday

8

Oct 27

HW6, Monday, Why LinAlg?, Wednesday, Tutorial

9

Nov 3

Monday is the last day to drop this class, HW7, Monday, Wednesday, Tutorial

10

Nov 10

HW8, Monday, Tutorial

11

Nov 17

MondayTuesday is UofT November break

12

Nov 24

HW9

13

Dec 1

Wednesday is a "makeup Monday"! EndofCourse Schedule, Tutorial

F

Dec 8

The Final Exam

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Definition of Subtraction and Division
 Subtraction: if $a,b\in F,ab=a+(b)$.
 Division: if $a,b\in F,a/b=a\times b^{1}$.
Basic Properties of a Field (cont'd)
8. $\forall a\in F$, $a\times 0=0$.
 Proof of 8
 By F3 , $a\times 0=a\times (0+0)$
 By F5 , $a\times (0+0)=a\times 0+a\times 0$;
 By F3 , $a\times 0=0+a\times 0$;
 By Thm P1, $0=a\times 0$.
9. $\nexists b\in F$ s.t. $0\times b=1$;
 $\forall b\in F$ s.t. $0\times b\neq 1$.
 Proof of 9
 By F3 , $\times b=0\neq 1$.
10. $(a)\times b=a\times (b)=(a\times b)$.
11. $(a)\times (b)=a\times b$.
12. $a\times b=0\iff a=0$ or $b=0$.
 Proof of 12
 <= :
 By P8 , if $a=0$ , then $a\times b=0\times b=0$;
 By P8 , if $b=0$ , then $a\times b=a\times 0=0$.
 => : Assume $a\times b=0$ , if a = 0 we are done;
 Otherwise , by P8 , $a\neq 0$ and we have $a\times b=0=a\times 0$;
 by cancellation (P2) , $b=0$.
$(a+b)\times (ab)=a^{2}b^{2}$.
 Proof
 By F5 , $(a+b)\times (ab)=a\times (a+(b))+b\times (a+(b))$
 $=a^{2}b^{2}$
Theorem
$\exists !\iota :\mathbb {Z} \rightarrow F$ s.t.
 1. $\iota (0)=0,\iota (1)=1$;
 2. $\forall m,n\in \mathbb {Z} ,\iota (m+n)=\iota (m)+\iota (n)$;
 3. $\forall m,n\in \mathbb {Z} ,\iota (m\times n)=\iota (m)\times \iota (n)$.
 Examples
$\iota (2)=\iota (1+1)=\iota (1)+\iota (1)=1+1;$
$\iota (3)=\iota (2+1)=\iota (2)+\iota (1)=\iota (2)+1;$
......
In F2:
${\begin{aligned}27>\iota (27)&=\iota (26+1)\\&=\iota (26)+\iota (1)\\&=\iota (26)+1\\&=\iota (13\times 2)+1\\&=\iota (2)\times \iota (13)+1\\&=(1+1)\times \iota (13)+1\\&=0\times \iota (13)+1\\&=1\end{aligned}}$
Scanned Lecture Notes by AM
Scanned Lecture Notes by Boyang.wu
File:W21.pdf