12-240/Classnotes for Thursday September 13

DBN: Classes: 2012-13: 12-240 / Navigation Additions to this web site no longer count towards good deed points. # Week of... Notes and Links 1 Sep 10 About This Class, Tuesday, Thursday 2 Sep 17 HW1, Tuesday, Thursday, HW1 Solutions 3 Sep 24 HW2, Tuesday, Class Photo, Thursday 4 Oct 1 HW3, Tuesday, Thursday 5 Oct 8 HW4, Tuesday, Thursday 6 Oct 15 Tuesday, Thursday 7 Oct 22 HW5, Tuesday, Term Test was on Thursday. HW5 Solutions 8 Oct 29 Why LinAlg?, HW6, Tuesday, Thursday, Nov 4 is the last day to drop this class 9 Nov 5 Tuesday, Thursday 10 Nov 12 Monday-Tuesday is UofT November break, HW7, Thursday 11 Nov 19 HW8, Tuesday,Thursday 12 Nov 26 HW9, Tuesday , Thursday 13 Dec 3 Tuesday UofT Fall Semester ends Wednesday F Dec 10 The Final Exam (time, place, style, office hours times) Register of Good Deeds Add your name / see who's in!

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

$\forall \!\,$ a, b $\in \!\,$ F: a+b=b+a and a.b=b.a

F2: associative law

$\forall \!\,$ a, b, c $\in \!\,$ F: (a+b)+c=a+(b+c) and (a.b).c= a.(b.c)

F3: the existence of identity elements

$\forall \!\,$ a $\in \!\,$, a+0=a and a.1=a

F4: existence of inverses

$\forall \!\,$ a $\in \!\,$ F \0,$\exists \!\,$ c, d $\in \!\$ F such that a+c=o and a.d=1

F5: contributive law

$\forall \!\,$ a, b, c $\in \!\,$ F, a.(b+c)=a.b + a.c

Theorems

Theorem 1: Cancellation laws $\forall \!\,$ a, b, c $\in \!\,$ F

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

if a.c=b.c and c$\neq \!\,$0, then a=b

Theorem 2: Identity uniqueness

Identity elements 0 and 1 mentioned in F3 are unique

$\forall \!\,$ a, b, b' $\in \!\,$ F

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

if a.b=a and a.b'=a and a$\neq \!\,$0, then b=b'=1

Theorem 3: Inverse uniqueness

Elements c and d mentioned in F4 are unique

$\forall \!\,$ a, b, b' $\in \!\,$ 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

$\forall \!\,$ a $\in \!\,$ F

-( -a) = a

Theorem 4

$\forall \!\,$ a $\in \!\,$ F

0.a= 0

Significance

inverse uniqueness

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

For a $\in \!\,$ F define -a to be equal that b $\in \!\,$ 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 Starash

• 

  Page 1

• 

  Page 2