12-240/Classnotes for Tuesday September 18: Difference between revisions

Revision as of 22:03, 18 September 2012

#	Week of...	Notes and Links
Additions to this web site no longer count towards good deed points.
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)
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Thrm: In a field F: 1. a+b = c+b ⇒ a=c

2. b≠0, a∙b=c∙b ⇒ a=c

3. 0 is unique.

4. 1 is unique.

5. -a is unique.

6. a^-1 is unique (a≠0)

7. -(-a)=a

8. (a^-1)^-1 =a

9. a∙0=0 **Surprisingly difficult, required distributivity.

10. ∄ 0^-1, aka, ∄ b∈F s.t 0∙b=1

11. (-a)∙(-b)=a∙b

12. a∙b=0 iff a=0 or b=0

. . .

16. (a+b)∙(a-b)= a^2 - b^2 [Define a^2 = a∙a] Hint: Use distributive law

Thrm 2: Given a field F, there exists a map Ɩ: Z → F with the properties:

1) Ɩ(0) =0, Ɩ(1)=1

2) Ɩ(m+n) = Ɩ(m) +Ɩ(n)

3) Ɩ(mn) = Ɩ(m)∙Ɩ(n)

Furthermore, Ɩ is unique.

@@ Line 1: / Line 1: @@
 {{12-240/Navigation}}
+== Various properties of fields ==
-== Recap: ==
 Thrm: In a field F:
-. a+b = c+b => a=c
+. a+b = c+b ⇒ a=c
-. b≠0, a∙b=c∙b => a=c
+. b≠0, a∙b=c∙b ⇒ a=c
 . 0 is unique.
@@ Line 24: / Line 24: @@
 . (-a)∙(-b)=a∙b
+. a∙b=0 ''iff'' a=0 or b=0
+.
+.
+.
+. (a+b)∙(a-b)= a^2 - b^2
+[Define a^2 = a∙a]
+Hint: Use distributive law
+Thrm 2: Given a field F, there exists a map Ɩ: Z → F with the properties:
+) Ɩ(0) =0, Ɩ(1)=1
+) Ɩ(m+n) = Ɩ(m) +Ɩ(n)
+) Ɩ(mn) = Ɩ(m)∙Ɩ(n)
+Furthermore, Ɩ is unique.