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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Knowledge about Fields:
During this lecture, we first talked about the properties of the real numbers. Then we applied these properties to the "Field". At the end of the lecture, we learned how to prove basic properties of fields.
The Real Numbers
Properties of Real Numbers
The real numbers are a set $\mathbb {R}$ with two binary operations:
$+:\mathbb {R} \times \mathbb {R} \rightarrow \mathbb {R}$
$*:\mathbb {R} \times \mathbb {R} \rightarrow \mathbb {R}$
such that the following properties hold.
 R1 : $\forall a,b\in \mathbb {R} ,a+b=b+a~\&~a*b=b*a$ (the commutative law)
 R2 : $\forall a,b,c\in \mathbb {R} ,(a+b)+c=a+(b+c)~\&~(a*b)*c=a*(b*c)$ (the associative law)
 R3 : $\forall a\in \mathbb {R} ,a+0=a~\&~a*1=a$ (existence of units: 0 is known as the
 "additive unit" and 1 as the "multiplicative unit")
 R4 : $\forall a\in \mathbb {R} ,\exists b\in \mathbb {R} ,a+b=0$;
 $\forall a\in \mathbb {R} ,a\neq 0\Rightarrow \exists b\in \mathbb {R} ,a*b=1$ (existence of inverses)
 R5 : $\forall a,b,c\in \mathbb {R} ,(a+b)*c=(a*c)+(b*c)$ (the distributive law)
Properties That Do Not Follow from R1R5
There are properties which are true for $\mathbb {R}$, but do not follow from R1 to R5. For example (note that OR in mathematics denotes an "inclusive or"):
$\forall a\in \mathbb {R} ,\exists x\in \mathbb {R} ,a=x^{2}$ OR $a=x^{2}$ (the existence of square roots)
Consider another set that satisfies all the properties R1 to R5. In $\mathbb {Q}$ (the rational numbers), let us take </math>a = 2</math>. There is no $x\in \mathbb {Q}$ such that $x^{2}=a=2$, so the statement above is not true for the rational numbers!
Fields
Definition
A "field" is a set $\mathbb {F}$ along with a pair of binary operations:
$+:\mathbb {F} \times \mathbb {F} \rightarrow \mathbb {F}$
$*:\mathbb {F} \times \mathbb {F} \rightarrow \mathbb {F}$
and along with a pair $(0,1)\in \mathbb {F} ,0\neq 1$, such that the following properties hold.
 F1 : $\forall a,b\in \mathbb {F} ,a+b=b+a~\&~a*b=b*a$ (the commutative law)
 F2 : $\forall a,b,c\in \mathbb {F} ,(a+b)+c=a+(b+c)~\&~(a*b)*c=a*(b*c)$ (the associative law)
 F3 : $\forall a\in \mathbb {F} ,a+0=a~\&~a*1=a$ (existence of units)
 F4 : $\forall a\in \mathbb {F} ,\exists b\in \mathbb {F} ,a+b=0$;
 $\forall a\in \mathbb {F} ,a\neq 0\Rightarrow \exists b\in \mathbb {F} ,a*b=1$ (existence of inverses)
 F5 : $\forall a,b,c\in \mathbb {F} ,(a+b)*c=(a*c)+(b*c)$ (the distributive law)
Examples
 $\mathbb {R}$ is a field.
 $\mathbb {Q}$ (the rational numbers) is a field.
 $\mathbb {C}$ (the complex numbers) is a field.
 $\mathbb {F} =\{0,1\}$ with operations defined as follows (known as $\mathbb {F} _{2}$ or $\mathbb {Z} /2$) is a field:
More generally, for every prime number $P$, $\mathbb {F} _{p}=\{0,1,2,3,\cdots ,p1\}$ is a field, with operations defined by
$(a,b)\rightarrow a+b\mod P$.
An example: $\mathbb {F} _{7}=\{0,1,2,3,4,5,6\}$, the operations are like remainders when divided by 7, or "like remainders mod 7". For example, $4+6=4+6\mod 7$ and $3*5=3*5\mod 7$.
Basic Properties of Fields
Theorem:
Let $\mathbb {F}$ be a field, and let $a,b,c$ denote elements of $\mathbb {F}$. Then:
 $a+b=c+b\Rightarrow a=c$ (cancellation law)
 $b\neq 0~\&~a*b=c*b\Rightarrow a=c$
Proof of 1:
1. By F4, $\exists b'\in \mathbb {F} ,b+b'=0$.
We know that $a+b=c+b$;
Therefore $(a+b)+b'=(c+b)+b'$.
2. By F2, $a+(b+b')=c+(b+b')$,
so by the choice of $b'$, we know that $a+0=c+0$.
3. Therefore, by F3, $a=c$.
＾_＾
Proof of 2: more or less the same.
3. If $0'\in \mathbb {F}$ is "like 0", then it is 0:
If $0'\in \mathbb {F}$ satisfies $\forall a\in \mathbb {F} ,a+0'=a$, then 0' = 0.
4. If $1'\in \mathbb {F}$ is "like 1", then it is 1:
If $1'\in \mathbb {F}$ satisfies that $\forall a\in \mathbb {F} ,a*1'=a$, then 1' = 1.
Proof of 3 :
1. By F3 , 0' = 0' + 0.
2. By F1 , 0' + 0 = 0 + 0'.
3. By assumption on 0', 0' = 0 + 0' = 0.
＾_＾
5. $\forall a,b,b'\in \mathbb {F} ,a+b=0~\&~a+b'=0\Rightarrow b=b'$:
In any field "$a$" makes sense because it is unique  it has an unambiguous meaning.
$(a):$ the $b$ such that $a+b=0$.
6. $\forall a,b,b'\in \mathbb {F} ,a\neq 0~\&~a*b=1=a*b'\Rightarrow b=b'$:
In any field, if $a\neq 0$, "$a^{1}$" makes sense.
Proof of 5 :
1. $a+b=0=a+b'$.
2. By F1, $b+a=b'+a$.
3. By cancellation, $b=b'$.
＾_＾
7. $(a)=a$ and when $a\neq 0$, $(a^{1})^{1}=a$.
Proof of 7 :
1. By definition, $a+(a)=0$. (*)
2. By definition, $(a)+((a)=0$.
3. By (*) and F1, $(a)+a=0$.
4. By property 5, $(a)=a$.
＾_＾
Scanned Lecture Notes by AM
Scanned Lecture Notes by Boyang.wu
File:W121.pdf
File:W122.pdf