14-240/Classnotes for Monday September 15: Difference between revisions

Revision as of 10:05, 21 September 2014

DBN: Classes: 2014-15: 14-240 / Navigation

#	Week of...	Notes and Links
Welcome to Math 240! (additions to this web site no longer count towards good deed points)
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	Monday-Tuesday is UofT November break
12	Nov 24	HW9
13	Dec 1	Wednesday is a "makeup Monday"! End-of-Course Schedule, Tutorial
F	Dec 8	The Final Exam
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Definition of Subtraction and Division

Subtraction: if $a,b\in F,a-b=a+(-b)$ .
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) , $b=0$ .

$(a+b)\times (a-b)=a^{2}-b^{2}$ .

Proof: By F5 , $(a+b)\times (a-b)=a\times (a+(-b))+b\times (a+(-b))$; $=a^{2}-b^{2}$

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: ${\begin{aligned}27---->\iota (27)&=\iota (26+1)\\&=\iota (26)+\iota (1)\\&=\iota (26)+1\\&=\iota (13\times 2)+1\\&=\iota (2)\times \iota (13)+1\\&=(1+1)\times \iota (13)+1\\&=0\times \iota (13)+1\\&=1\end{aligned}}$ http://drorbn.net/images/c/cd/MAT_240_lecture_3_%281_of_2%29.pdf (Lecture 3 notes by AM part 1 of 2) http://drorbn.net/images/6/6a/MAT240_lectuire_3_%282_of_2%29.pdf (Lecture 3 notes by AM part 2 of 2)

@@ Line 1: / Line 1: @@
 {{14-240/Navigation}}
-Definition:
+==Definition of Subtraction and Division==
 * Subtraction: if <math>a, b \in F, a - b = a + (-b)</math>.
 * Division: if <math>a, b \in F, a / b = a \times b^{-1}</math>.
+==Basic Properties of a Field (cont'd)==
-Theorem:
-* 8. <math>\forall a \in F</math>, <math>a \times 0 = 0</math>.
+'''8.''' <math>\forall a \in F</math>, <math>a \times 0 = 0</math>.
+;Proof of 8
-            *proof of 8: By F3 , <math>a \times 0 = a \times (0 + 0)</math>
-                               By F5 , <math>a \times (0 + 0) = a \times 0 + a \times 0</math>;
+:By F3 , <math>a \times 0 = a \times (0 + 0)</math>
-                               By F3 , <math>a \times 0 = 0 + a \times 0</math>;
+:By F5 , <math>a \times (0 + 0) = a \times 0 + a \times 0</math>;
-                               By Thm P1,<math>0 = a \times 0</math>.
+:By F3 , <math>a \times 0 = 0 + a \times 0</math>;
+:By Thm P1, <math>0 = a \times 0</math>.
-* 9. <math>\nexists b \in F</math> s.t. <math>0 \times b = 1</math>;
+'''9.''' <math>\nexists b \in F</math> s.t. <math>0 \times b = 1</math>;
-            <math>\forall b \in F</math> s.t. <math>0 \times b \neq 1</math>.
+:<math>\forall b \in F</math> s.t. <math>0 \times b \neq 1</math>.
+;Proof of 9
-                    proof of 9: By F3 , <math>\times b = 0 \neq 1</math>.
+:By F3 , <math>\times b = 0 \neq 1</math>.
-* 10. <math>(-a) \times b = a \times (-b) = -(a \times b)</math>.
+'''10.''' <math>(-a) \times b = a \times (-b) = -(a \times b)</math>.
-* 11. <math>(-a) \times (-b) = a \times b</math>.
+'''11.''' <math>(-a) \times (-b) = a \times b</math>.
-* 12. <math>a \times b = 0 \iff a = 0 or b = 0</math>.
+'''12.''' <math>a \times b = 0 \iff a = 0</math> or <math>b = 0</math>.
+;Proof of 12
-                    proof of 12: <= : By P8 , if <math>a = 0</math> , then <math>a \times b = 0 \times b = 0</math>;
+:'''<= :'''
-                                      By P8 , if <math>b = 0</math> , then <math>a \times b = a \times 0 = 0</math>.
-                                 => : Assume <math>a \times b = 0 </math> , if a = 0 we are done;
+:By P8 , if <math>a = 0</math> , then <math>a \times b = 0 \times b = 0</math>;
-                                      Otherwise , by P8 , <math>a \neq 0 </math> and we have <math>a \times b = 0 = a \times 0</math>;
+:By P8 , if <math>b = 0</math> , then <math>a \times b = a \times 0 = 0</math>.
+:'''=> :''' Assume <math>a \times b = 0 </math> , if a = 0 we are done;
-                                                  by cancellation (P2) , <math>b = 0</math>.
+:Otherwise , by P8 , <math>a \neq 0 </math> and we have <math>a \times b = 0 = a \times 0</math>;
+:by cancellation (P2) , <math>b = 0</math>.
 <math>(a + b) \times (a - b) = a^2 - b^2</math>.
+;Proof
-         proof: By F5 , <math>(a + b) \times (a - b) = a \times (a + (-b)) + b \times (a + (-b))</math>
-                                                <math>= a \times a + a \times (-b) + b \times a + (-b) \times b</math>
+:By F5 , <math>(a + b) \times (a - b) = a \times (a + (-b)) + b \times (a + (-b))</math>
-                                                <math>= a^2 - b^2</math>
+:<math>= a^2 - b^2</math>
-Theorem :
+==Theorem==
-         <math>\exists! \iota : \Z \rightarrow F</math>  s.t.
-. <math>\iota(0) = 0 , \iota(1) = 1</math>;
+<math>\exists! \iota : \Z \rightarrow F</math>  s.t.
-. <math>\forall m ,n \in \Z, \iota(m+n) = \iota(m) + \iota(n)</math>;
+:1. <math>\iota(0) = 0 , \iota(1) = 1</math>;
-. <math>\forall m ,n \in \Z, \iota(m\times n) = \iota(m) \times \iota(n)</math>.
+:2. <math>\forall m ,n \in \Z, \iota(m+n) = \iota(m) + \iota(n)</math>;
+:3. <math>\forall m ,n \in \Z, \iota(m\times n) = \iota(m) \times \iota(n)</math>.
+;Examples
-         <math>\iota(2) = \iota(1+1) = \iota(1) + \iota(1) = 1 + 1;</math>
-         <math>\iota(3) = \iota(2+1) = \iota(2) + \iota(1) = \iota(2) + 1;</math>
+<math>\iota(2) = \iota(1+1) = \iota(1) + \iota(1) = 1 + 1;</math>
+<math>\iota(3) = \iota(2+1) = \iota(2) + \iota(1) = \iota(2) + 1;</math>
-         ......
+......
+In F2:
-         In F2 , <math>27 ----> \iota(27) = \iota(26 + 1)</math>
+<math>
-                                         <math>= \iota(26) + \iota(1)</math>
+\begin{align}
-                                         <math>= \iota(26) + 1</math>
+----> \iota(27) &= \iota(26 + 1)\\
-                                         <math>= \iota(13 \times 2) + 1</math>
-                                         <math>= \iota(2) \times \iota(13) + 1</math>
+&= \iota(26) + \iota(1)\\
+&= \iota(26) + 1\\
-                                         <math>= (1 + 1) \times \iota(13) + 1</math>
+&= \iota(13 \times 2) + 1\\
-                                         <math>= 0 \times \iota(13) + 1</math>
+&= \iota(2) \times \iota(13) + 1\\
-                                         <math>= 1</math>
+&= (1 + 1) \times \iota(13) + 1\\
+&= 0 \times \iota(13) + 1\\
+&= 1
+\end{align}
+</math>
 http://drorbn.net/images/c/cd/MAT_240_lecture_3_%281_of_2%29.pdf (Lecture 3 notes by AM part 1 of 2)
 http://drorbn.net/images/6/6a/MAT240_lectuire_3_%282_of_2%29.pdf (Lecture 3 notes by AM part 2 of 2)

14-240/Classnotes for Monday September 15: Difference between revisions

Revision as of 10:05, 21 September 2014

Definition of Subtraction and Division

Basic Properties of a Field (cont'd)

Theorem

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