http://drorbn.net/api.php?action=feedcontributions&user=KJMorenz&feedformat=atomDrorbn - User contributions [en]2024-03-29T09:24:28ZUser contributionsMediaWiki 1.21.1http://drorbn.net/index.php?title=12-240/Classnotes_for_Tuesday_October_2312-240/Classnotes for Tuesday October 232012-10-30T22:05:42Z<p>KJMorenz: /* Lecture notes scanned by KJMorenz */</p>
<hr />
<div>{{12-240/Navigation}}<br />
= ===<br />
Definition: L(V,W) is the set of all linear transformation L: V->W<br />
<br />
u <math>\in\,\!</math> V,<br />
0 of L(V,W) (u)=0 of W (this is a l.t.str)<br />
<br />
If L1 and L2 <math>\in\,\!</math> L(V,W),<br />
(L1 + L2) (u)= L1(u) +L2(u) (this is a l.t.str)<br />
<br />
If c <math>\in\,\!</math> F and L <math>\in\,\!</math> L(V,W),<br />
(c*L) (u)= c*L(u) (this is a l.t.str)<br />
<br />
Theorem: L(V,W) is a vector space<br />
<br />
Proof: "Distributivity" c(x+y)=cx+cy<br />
<br />
In our case need to show c(L1 + L2)= cL1 + cL2<br />
<br />
Where c <math>\in\,\!</math> F and L1 and L2 <math>\in\,\!</math> L(V,W)<br />
<br />
(LHS) (u)<br />
<br />
== Lecture notes scanned by [[User:Zetalda|Zetalda]] ==<br />
<gallery><br />
Image:12-240-Oct23-1.jpeg|Page 1<br />
Image:12-240-Oct23-2.jpeg|Page 2<br />
Image:12-240-Oct23-3.jpeg|Page 3<br />
Image:12-240-Oct23-4.jpeg|Page 4<br />
</gallery><br />
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== Lecture notes scanned by [[User:KJMorenz|KJMorenz]] ==<br />
<gallery><br />
Image:12-240-Oct30.jpg|Oct 30 Page 1<br />
Image:12-240-Oct30-2.jpg|Oct 30 Page 2<br />
Image:12-240-Oct2.jpg|Oct 2 Page 1<br />
Image:12-240-Oct2-2.jpg|Oct 2 Page 2<br />
Image:12-240-Oct2-3.jpg|Oct 2 Page 3<br />
Image:12-240-Oct2-4.jpg|Oct 2 Page 4<br />
Image:12-240-Basis.jpg|Basis of a Vector Space<br />
Image:12-240-TutOct4.jpg|Tutorial Oct 4<br />
</gallery></div>KJMorenzhttp://drorbn.net/index.php?title=12-240/Classnotes_for_Tuesday_October_2312-240/Classnotes for Tuesday October 232012-10-30T22:03:58Z<p>KJMorenz: /* Lecture notes scanned by KJMorenz */</p>
<hr />
<div>{{12-240/Navigation}}<br />
= ===<br />
Definition: L(V,W) is the set of all linear transformation L: V->W<br />
<br />
u <math>\in\,\!</math> V,<br />
0 of L(V,W) (u)=0 of W (this is a l.t.str)<br />
<br />
If L1 and L2 <math>\in\,\!</math> L(V,W),<br />
(L1 + L2) (u)= L1(u) +L2(u) (this is a l.t.str)<br />
<br />
If c <math>\in\,\!</math> F and L <math>\in\,\!</math> L(V,W),<br />
(c*L) (u)= c*L(u) (this is a l.t.str)<br />
<br />
Theorem: L(V,W) is a vector space<br />
<br />
Proof: "Distributivity" c(x+y)=cx+cy<br />
<br />
In our case need to show c(L1 + L2)= cL1 + cL2<br />
<br />
Where c <math>\in\,\!</math> F and L1 and L2 <math>\in\,\!</math> L(V,W)<br />
<br />
(LHS) (u)<br />
<br />
== Lecture notes scanned by [[User:Zetalda|Zetalda]] ==<br />
<gallery><br />
Image:12-240-Oct23-1.jpeg|Page 1<br />
Image:12-240-Oct23-2.jpeg|Page 2<br />
Image:12-240-Oct23-3.jpeg|Page 3<br />
Image:12-240-Oct23-4.jpeg|Page 4<br />
</gallery><br />
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== Lecture notes scanned by [[User:KJMorenz|KJMorenz]] ==<br />
<gallery><br />
Image:12-240-Oct30.jpg<br />
Image:12-240-Oct30-2.jpg<br />
Image:12-240-Oct2.jpg<br />
Image:12-240-Oct2-2.jpg<br />
Image:12-240-Oct2-3.jpg<br />
Image:12-240-Oct2-4.jpg<br />
Image:12-240-Basis.jpg<br />
Image:12-240-TutOct4.jpg<br />
</gallery></div>KJMorenzhttp://drorbn.net/index.php?title=File:12-240-TutOct4.jpgFile:12-240-TutOct4.jpg2012-10-30T22:03:43Z<p>KJMorenz: </p>
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<div></div>KJMorenzhttp://drorbn.net/index.php?title=File:12-240-Basis.jpgFile:12-240-Basis.jpg2012-10-30T22:02:46Z<p>KJMorenz: </p>
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<div></div>KJMorenzhttp://drorbn.net/index.php?title=12-240/Classnotes_for_Tuesday_October_2312-240/Classnotes for Tuesday October 232012-10-30T22:02:19Z<p>KJMorenz: /* Lecture notes scanned by KJMorenz */</p>
<hr />
<div>{{12-240/Navigation}}<br />
= ===<br />
Definition: L(V,W) is the set of all linear transformation L: V->W<br />
<br />
u <math>\in\,\!</math> V,<br />
0 of L(V,W) (u)=0 of W (this is a l.t.str)<br />
<br />
If L1 and L2 <math>\in\,\!</math> L(V,W),<br />
(L1 + L2) (u)= L1(u) +L2(u) (this is a l.t.str)<br />
<br />
If c <math>\in\,\!</math> F and L <math>\in\,\!</math> L(V,W),<br />
(c*L) (u)= c*L(u) (this is a l.t.str)<br />
<br />
Theorem: L(V,W) is a vector space<br />
<br />
Proof: "Distributivity" c(x+y)=cx+cy<br />
<br />
In our case need to show c(L1 + L2)= cL1 + cL2<br />
<br />
Where c <math>\in\,\!</math> F and L1 and L2 <math>\in\,\!</math> L(V,W)<br />
<br />
(LHS) (u)<br />
<br />
== Lecture notes scanned by [[User:Zetalda|Zetalda]] ==<br />
<gallery><br />
Image:12-240-Oct23-1.jpeg|Page 1<br />
Image:12-240-Oct23-2.jpeg|Page 2<br />
Image:12-240-Oct23-3.jpeg|Page 3<br />
Image:12-240-Oct23-4.jpeg|Page 4<br />
</gallery><br />
<br />
== Lecture notes scanned by [[User:KJMorenz|KJMorenz]] ==<br />
<gallery><br />
Image:12-240-Oct30.jpg<br />
Image:12-240-Oct30-2.jpg<br />
Image:12-240-Oct2.jpg<br />
Image:12-240-Oct2-2.jpg<br />
Image:12-240-Oct2-3.jpg<br />
Image:12-240-Oct2-4.jpg<br />
</gallery></div>KJMorenzhttp://drorbn.net/index.php?title=File:12-240-Oct2-4.jpgFile:12-240-Oct2-4.jpg2012-10-30T22:02:03Z<p>KJMorenz: </p>
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<div></div>KJMorenzhttp://drorbn.net/index.php?title=12-240/Classnotes_for_Tuesday_October_2312-240/Classnotes for Tuesday October 232012-10-30T22:00:48Z<p>KJMorenz: /* Lecture notes scanned by KJMorenz */</p>
<hr />
<div>{{12-240/Navigation}}<br />
= ===<br />
Definition: L(V,W) is the set of all linear transformation L: V->W<br />
<br />
u <math>\in\,\!</math> V,<br />
0 of L(V,W) (u)=0 of W (this is a l.t.str)<br />
<br />
If L1 and L2 <math>\in\,\!</math> L(V,W),<br />
(L1 + L2) (u)= L1(u) +L2(u) (this is a l.t.str)<br />
<br />
If c <math>\in\,\!</math> F and L <math>\in\,\!</math> L(V,W),<br />
(c*L) (u)= c*L(u) (this is a l.t.str)<br />
<br />
Theorem: L(V,W) is a vector space<br />
<br />
Proof: "Distributivity" c(x+y)=cx+cy<br />
<br />
In our case need to show c(L1 + L2)= cL1 + cL2<br />
<br />
Where c <math>\in\,\!</math> F and L1 and L2 <math>\in\,\!</math> L(V,W)<br />
<br />
(LHS) (u)<br />
<br />
== Lecture notes scanned by [[User:Zetalda|Zetalda]] ==<br />
<gallery><br />
Image:12-240-Oct23-1.jpeg|Page 1<br />
Image:12-240-Oct23-2.jpeg|Page 2<br />
Image:12-240-Oct23-3.jpeg|Page 3<br />
Image:12-240-Oct23-4.jpeg|Page 4<br />
</gallery><br />
<br />
== Lecture notes scanned by [[User:KJMorenz|KJMorenz]] ==<br />
<gallery><br />
Image:12-240-Oct30.jpg<br />
Image:12-240-Oct30-2.jpg<br />
Image:12-240-Oct2.jpg<br />
Image:12-240-Oct2-2.jpg<br />
Image:12-240-Oct2-3.jpg<br />
</gallery></div>KJMorenzhttp://drorbn.net/index.php?title=File:12-240-Oct2-3.jpgFile:12-240-Oct2-3.jpg2012-10-30T22:00:26Z<p>KJMorenz: </p>
<hr />
<div></div>KJMorenzhttp://drorbn.net/index.php?title=12-240/Classnotes_for_Tuesday_October_2312-240/Classnotes for Tuesday October 232012-10-30T21:59:54Z<p>KJMorenz: /* Lecture notes scanned by KJMorenz */</p>
<hr />
<div>{{12-240/Navigation}}<br />
= ===<br />
Definition: L(V,W) is the set of all linear transformation L: V->W<br />
<br />
u <math>\in\,\!</math> V,<br />
0 of L(V,W) (u)=0 of W (this is a l.t.str)<br />
<br />
If L1 and L2 <math>\in\,\!</math> L(V,W),<br />
(L1 + L2) (u)= L1(u) +L2(u) (this is a l.t.str)<br />
<br />
If c <math>\in\,\!</math> F and L <math>\in\,\!</math> L(V,W),<br />
(c*L) (u)= c*L(u) (this is a l.t.str)<br />
<br />
Theorem: L(V,W) is a vector space<br />
<br />
Proof: "Distributivity" c(x+y)=cx+cy<br />
<br />
In our case need to show c(L1 + L2)= cL1 + cL2<br />
<br />
Where c <math>\in\,\!</math> F and L1 and L2 <math>\in\,\!</math> L(V,W)<br />
<br />
(LHS) (u)<br />
<br />
== Lecture notes scanned by [[User:Zetalda|Zetalda]] ==<br />
<gallery><br />
Image:12-240-Oct23-1.jpeg|Page 1<br />
Image:12-240-Oct23-2.jpeg|Page 2<br />
Image:12-240-Oct23-3.jpeg|Page 3<br />
Image:12-240-Oct23-4.jpeg|Page 4<br />
</gallery><br />
<br />
== Lecture notes scanned by [[User:KJMorenz|KJMorenz]] ==<br />
<gallery><br />
Image:12-240-Oct30.jpg<br />
Image:12-240-Oct30-2.jpg<br />
Image:12-240-Oct2.jpg<br />
Image:12-240-Oct2-2.jpg<br />
</gallery></div>KJMorenzhttp://drorbn.net/index.php?title=File:12-240-Oct2.jpgFile:12-240-Oct2.jpg2012-10-30T21:59:32Z<p>KJMorenz: </p>
<hr />
<div></div>KJMorenzhttp://drorbn.net/index.php?title=12-240/Classnotes_for_Tuesday_October_2312-240/Classnotes for Tuesday October 232012-10-30T21:59:09Z<p>KJMorenz: /* Lecture notes scanned by KJMorenz */</p>
<hr />
<div>{{12-240/Navigation}}<br />
= ===<br />
Definition: L(V,W) is the set of all linear transformation L: V->W<br />
<br />
u <math>\in\,\!</math> V,<br />
0 of L(V,W) (u)=0 of W (this is a l.t.str)<br />
<br />
If L1 and L2 <math>\in\,\!</math> L(V,W),<br />
(L1 + L2) (u)= L1(u) +L2(u) (this is a l.t.str)<br />
<br />
If c <math>\in\,\!</math> F and L <math>\in\,\!</math> L(V,W),<br />
(c*L) (u)= c*L(u) (this is a l.t.str)<br />
<br />
Theorem: L(V,W) is a vector space<br />
<br />
Proof: "Distributivity" c(x+y)=cx+cy<br />
<br />
In our case need to show c(L1 + L2)= cL1 + cL2<br />
<br />
Where c <math>\in\,\!</math> F and L1 and L2 <math>\in\,\!</math> L(V,W)<br />
<br />
(LHS) (u)<br />
<br />
== Lecture notes scanned by [[User:Zetalda|Zetalda]] ==<br />
<gallery><br />
Image:12-240-Oct23-1.jpeg|Page 1<br />
Image:12-240-Oct23-2.jpeg|Page 2<br />
Image:12-240-Oct23-3.jpeg|Page 3<br />
Image:12-240-Oct23-4.jpeg|Page 4<br />
</gallery><br />
<br />
== Lecture notes scanned by [[User:KJMorenz|KJMorenz]] ==<br />
<gallery><br />
Image:12-240-Oct30.jpg<br />
Image:12-240-Oct30-2.jpg<br />
<br />
Image:12-240-Oct2-2.jpg<br />
</gallery></div>KJMorenzhttp://drorbn.net/index.php?title=File:12-240-Oct2-2.jpgFile:12-240-Oct2-2.jpg2012-10-30T21:58:43Z<p>KJMorenz: </p>
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<div></div>KJMorenzhttp://drorbn.net/index.php?title=12-240/Classnotes_for_Tuesday_October_2312-240/Classnotes for Tuesday October 232012-10-30T21:57:51Z<p>KJMorenz: /* Lecture notes scanned by KJMorenz */</p>
<hr />
<div>{{12-240/Navigation}}<br />
= ===<br />
Definition: L(V,W) is the set of all linear transformation L: V->W<br />
<br />
u <math>\in\,\!</math> V,<br />
0 of L(V,W) (u)=0 of W (this is a l.t.str)<br />
<br />
If L1 and L2 <math>\in\,\!</math> L(V,W),<br />
(L1 + L2) (u)= L1(u) +L2(u) (this is a l.t.str)<br />
<br />
If c <math>\in\,\!</math> F and L <math>\in\,\!</math> L(V,W),<br />
(c*L) (u)= c*L(u) (this is a l.t.str)<br />
<br />
Theorem: L(V,W) is a vector space<br />
<br />
Proof: "Distributivity" c(x+y)=cx+cy<br />
<br />
In our case need to show c(L1 + L2)= cL1 + cL2<br />
<br />
Where c <math>\in\,\!</math> F and L1 and L2 <math>\in\,\!</math> L(V,W)<br />
<br />
(LHS) (u)<br />
<br />
== Lecture notes scanned by [[User:Zetalda|Zetalda]] ==<br />
<gallery><br />
Image:12-240-Oct23-1.jpeg|Page 1<br />
Image:12-240-Oct23-2.jpeg|Page 2<br />
Image:12-240-Oct23-3.jpeg|Page 3<br />
Image:12-240-Oct23-4.jpeg|Page 4<br />
</gallery><br />
<br />
== Lecture notes scanned by [[User:KJMorenz|KJMorenz]] ==<br />
<gallery><br />
Image:12-240-Oct30-2.jpg<br />
Image:12-240-Oct30.jpg<br />
</gallery></div>KJMorenzhttp://drorbn.net/index.php?title=File:12-240-Oct30.jpgFile:12-240-Oct30.jpg2012-10-30T21:57:28Z<p>KJMorenz: </p>
<hr />
<div></div>KJMorenzhttp://drorbn.net/index.php?title=12-240/Classnotes_for_Tuesday_October_2312-240/Classnotes for Tuesday October 232012-10-30T21:56:52Z<p>KJMorenz: </p>
<hr />
<div>{{12-240/Navigation}}<br />
= ===<br />
Definition: L(V,W) is the set of all linear transformation L: V->W<br />
<br />
u <math>\in\,\!</math> V,<br />
0 of L(V,W) (u)=0 of W (this is a l.t.str)<br />
<br />
If L1 and L2 <math>\in\,\!</math> L(V,W),<br />
(L1 + L2) (u)= L1(u) +L2(u) (this is a l.t.str)<br />
<br />
If c <math>\in\,\!</math> F and L <math>\in\,\!</math> L(V,W),<br />
(c*L) (u)= c*L(u) (this is a l.t.str)<br />
<br />
Theorem: L(V,W) is a vector space<br />
<br />
Proof: "Distributivity" c(x+y)=cx+cy<br />
<br />
In our case need to show c(L1 + L2)= cL1 + cL2<br />
<br />
Where c <math>\in\,\!</math> F and L1 and L2 <math>\in\,\!</math> L(V,W)<br />
<br />
(LHS) (u)<br />
<br />
== Lecture notes scanned by [[User:Zetalda|Zetalda]] ==<br />
<gallery><br />
Image:12-240-Oct23-1.jpeg|Page 1<br />
Image:12-240-Oct23-2.jpeg|Page 2<br />
Image:12-240-Oct23-3.jpeg|Page 3<br />
Image:12-240-Oct23-4.jpeg|Page 4<br />
</gallery><br />
<br />
== Lecture notes scanned by [[User:KJMorenz|KJMorenz]] ==<br />
<gallery><br />
Image:12-240-Oct30-2.jpg|Page 1<br />
</gallery></div>KJMorenzhttp://drorbn.net/index.php?title=File:12-240-Oct30-2.jpgFile:12-240-Oct30-2.jpg2012-10-30T21:54:26Z<p>KJMorenz: </p>
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<div></div>KJMorenzhttp://drorbn.net/index.php?title=12-240/Classnotes_for_Tuesday_September_1812-240/Classnotes for Tuesday September 182012-10-30T19:53:23Z<p>KJMorenz: /* Lecture 3, scanned notes upload by Starash and {{User:KJMorenz|KJMorenz]] */</p>
<hr />
<div>{{12-240/Navigation}}<br />
<br />
Today's handout, "TheComplexField", can be had from {{Pensieve link|Classes/12-240|Pensieve: Classes: 12-240}}.<br />
<br />
{{Template:12-240:Dror/Students Divider}}<br />
<br />
In this class, the professor continued with some more theorems of field and introduced definition and theorems of complex number.<br />
<br />
== Various properties of fields ==<br />
'''Thrm 1''': In a field F:<br />
1. a+b = c+b ⇒ a=c<br />
<br />
2. b≠0, a∙b=c∙b ⇒ a=c<br />
<br />
3. 0 is unique.<br />
<br />
4. 1 is unique.<br />
<br />
5. -a is unique.<br />
<br />
6. a^-1 is unique (a≠0)<br />
<br />
7. -(-a)=a <br />
<br />
8. (a^-1)^-1 =a<br />
<br />
9. a∙0=0 **Surprisingly difficult, required distributivity. <br />
<br />
10. ∄ 0^-1, aka, ∄ b∈F s.t 0∙b=1<br />
<br />
11. (-a)∙(-b)=a∙b<br />
<br />
12. a∙b=0 ''iff'' a=0 or b=0<br />
<br />
.<br />
.<br />
.<br />
<br />
16. (a+b)∙(a-b)= a^2 - b^2 <br />
[Define a^2 = a∙a] <br />
Hint: Use distributive law<br />
<br />
<br />
<br />
'''Thrm 2''': Given a field F, there exists a map Ɩ: Z → F with the properties (∀ m,n ∈ Z): <br />
<br />
1) Ɩ(0) =0, Ɩ(1)=1<br />
<br />
2) Ɩ(m+n) = Ɩ(m) +Ɩ(n)<br />
<br />
3) Ɩ(mn) = Ɩ(m)∙Ɩ(n)<br />
<br />
Furthermore, Ɩ is unique.<br />
<br />
'''Rough proof''':<br />
<br />
Test somes cases:<br />
<br />
Ɩ(2) = Ɩ(1+1) = Ɩ(1) + Ɩ(1) = 1 + 1 ≠ 2<br />
<br />
Ɩ(3) = Ɩ(2 +1)= Ɩ(2) + Ɩ(1) = 1+ 1+ 1 ≠ 3<br />
<br />
.<br />
.<br />
.<br />
<br />
Ɩ(n) = 1 + ... + 1 (n times)<br />
<br />
Ɩ(-3) = ?<br />
<br />
Ɩ(-3 + 3) = Ɩ(-3) + Ɩ(3) ⇒ Ɩ(-3) = -Ɩ(3) = -(1+1+1)<br />
<br />
''What about uniqueness?'' Simply put, we had not choice in the definition of Ɩ. All followed from the given properties.<br />
<br />
At this point, we will be lazy and simply denote Ɩ(3) = 3_f [3 with subscript f]<br />
<br />
<br />
∃ m≠0, m ∈ '''N''', Ɩ(m) =0<br />
<br />
In which case, there is a smallest m>0, for which Ɩ(m)=0. '''m' is the characteristic of F.'' Denoted char(F).<br />
Examples: char(F_2)=2, char(F_3)=3... but NOTE: char('''R''')=0<br />
<br />
<br />
<br />
'''Thrm:''' If F is a field and char(F) >0, then char(F) is a prime number.<br />
<br />
Proof: Suppose char(F) =m, m>0. Suppose also m is not prime: m=ts, t,s ∈ '''N'''.<br />
<br />
Then, Ɩ(m) = 0 = Ɩ(t)∙Ɩ(s) ⇒ Ɩ(t)=0 or Ɩ(s)=0 by P12.<br />
<br />
If Ɩ(t)=0 ⇒ t≧m ⇒ m=t, s=1 or likewise for Ɩ(s)=0, and m=s, t=1<br />
<br />
In any factorization of m, one of the factors is m and the other is 1. So m is prime. ∎<br />
<br />
== Complex number==<br />
<br />
'''Abstraction, generalization, definition, examples, properties, dream, implications, realization = formalization, PROOF.'''<br />
<br />
Consider that fact that in '''R''', ∄ x s.t. x^2 = -1<br />
<br />
Dream: Add new number element 𝒊 to '''R''', so as to still get a field & 𝒊^2 = -1<br />
<br />
Implications: By adding 𝒊, we must add 7𝒊, and 2+7𝒊, 3+4𝒊, (2+7𝒊)∙(3+4𝒊), (2+7𝒊)^-1, etc.<br />
<br />
So, how do we define this field?<br />
<br />
'''Definition'''<br />
<br />
'''C''' = {(a,b): a,b ∈ '''R'''}<br />
Also, 0 (of the field) = (0,0); 1( of the field) = (1,0)<br />
<br />
Define addition: (a,b)+(c,d) = (a+c, b+d)<br />
<br />
Define multification: (a,b)(c,d) = (ad-bd, ad+bc)<br />
<br />
<br />
'''Theorems:'''<br />
<br />
'''Thrm. 1.''' ('''C''', 0, 1, +, ∙) is a field.<br />
<br />
'''Thrm. 2.''' ∃ 𝒊 ∈ '''C''' s.t. 𝒊^2 = -1<br />
<br />
'''Thrm. 3.''' '''C''' contains '''R'''<br />
<br />
Proof (1): Show that each of the field axioms holds for '''C'''.<br />
<br />
Ex. F1(a): ƶ1 + ƶ2 = ƶ2 + ƶ1, where ƶ1 = (a1, b1) and ƶ2 = (a2, b2)<br />
<br />
LHS: (a1,b1)+(a2,b2) = (a1+a2, b1+b2)<br />
<br />
RHS: (a2,b2)+(a1,b1) = (a2+a1, b2+b1)<br />
<br />
LHS=RHS by F1 of '''R'''<br />
<br />
F1(b) and so on...<br />
<br />
Proof (2):<br />
<br />
In '''C''', consider i=(0,1)<br />
<br />
By the definition i^2=i.i=(0.1-1.1,0.1+1.0)=(-1,0)<br />
<br />
We also have 1(of '''c''') + (-1,0)=(1,0)+(-1,0)=(0,0)=0 (of '''c''')<br />
<br />
Hence (-1,0) is the addictive inverse of 1, i.e, (-1,0)=-1<br />
<br />
Thus i^2=-1. ∎<br />
<br />
Proof 3: <br />
<br />
Given the field '''C''' : map J: '''R''' -> '''C'''<br />
<br />
1) J(0)=(0,0); J(1)=(1,0)<br />
<br />
2) J(x+y)=J(x)+J(y); J(x.y)=J(x)J(y)<br />
<br />
Define J(x)=(x,0), all will follow.<br />
<br />
From now on J(x) will be writen simply x<br />
<br />
EX: J(7)=7, J(3)=3<br />
<br />
So, what does a+b𝒊 mean? (a, b ∈ '''R''')<br />
<br />
a+b𝒊= Ɩ(a) + Ɩ(b)+Ɩ(𝒊) = (a,0) + (b,0)∙(0,1) = (a,b)<br />
<br />
Hence, (a,b) ~ a+b𝒊<br />
<br />
Thus, we can use a+b𝒊 with less hesitation.<br />
<br />
== Lecture 3, scanned notes upload by [[User:Starash|Starash]] and [[User:KJMorenz|KJMorenz]]==<br />
<br />
<gallery><br />
Image:12-240-0918-1.jpg|Page 1<br />
Image:12-240-0918-2.jpg|Page 2<br />
Image:12-240-0918-3.jpg|Page 3<br />
</gallery><br />
[[Image:12-240-CharacteristicOfField.jpg]]<br />
[[Image:12-240-ComplexNums.jpg]]</div>KJMorenzhttp://drorbn.net/index.php?title=File:12-240-ComplexNums.jpgFile:12-240-ComplexNums.jpg2012-09-28T01:17:21Z<p>KJMorenz: </p>
<hr />
<div></div>KJMorenzhttp://drorbn.net/index.php?title=File:12-240-CharacteristicOfField.jpgFile:12-240-CharacteristicOfField.jpg2012-09-28T01:14:11Z<p>KJMorenz: </p>
<hr />
<div></div>KJMorenzhttp://drorbn.net/index.php?title=12-240/Classnotes_for_Tuesday_September_1812-240/Classnotes for Tuesday September 182012-09-28T01:13:32Z<p>KJMorenz: /* Lecture 3, scanned notes upload by Starash */</p>
<hr />
<div>{{12-240/Navigation}}<br />
<br />
Today's handout, "TheComplexField", can be had from {{Pensieve link|Classes/12-240|Pensieve: Classes: 12-240}}.<br />
<br />
{{Template:12-240:Dror/Students Divider}}<br />
<br />
In this class, the professor continued with some more theorems of field and introduced definition and theorems of complex number.<br />
<br />
== Various properties of fields ==<br />
'''Thrm 1''': In a field F:<br />
1. a+b = c+b ⇒ a=c<br />
<br />
2. b≠0, a∙b=c∙b ⇒ a=c<br />
<br />
3. 0 is unique.<br />
<br />
4. 1 is unique.<br />
<br />
5. -a is unique.<br />
<br />
6. a^-1 is unique (a≠0)<br />
<br />
7. -(-a)=a <br />
<br />
8. (a^-1)^-1 =a<br />
<br />
9. a∙0=0 **Surprisingly difficult, required distributivity. <br />
<br />
10. ∄ 0^-1, aka, ∄ b∈F s.t 0∙b=1<br />
<br />
11. (-a)∙(-b)=a∙b<br />
<br />
12. a∙b=0 ''iff'' a=0 or b=0<br />
<br />
.<br />
.<br />
.<br />
<br />
16. (a+b)∙(a-b)= a^2 - b^2 <br />
[Define a^2 = a∙a] <br />
Hint: Use distributive law<br />
<br />
<br />
<br />
'''Thrm 2''': Given a field F, there exists a map Ɩ: Z → F with the properties (∀ m,n ∈ Z): <br />
<br />
1) Ɩ(0) =0, Ɩ(1)=1<br />
<br />
2) Ɩ(m+n) = Ɩ(m) +Ɩ(n)<br />
<br />
3) Ɩ(mn) = Ɩ(m)∙Ɩ(n)<br />
<br />
Furthermore, Ɩ is unique.<br />
<br />
'''Rough proof''':<br />
<br />
Test somes cases:<br />
<br />
Ɩ(2) = Ɩ(1+1) = Ɩ(1) + Ɩ(1) = 1 + 1 ≠ 2<br />
<br />
Ɩ(3) = Ɩ(2 +1)= Ɩ(2) + Ɩ(1) = 1+ 1+ 1 ≠ 3<br />
<br />
.<br />
.<br />
.<br />
<br />
Ɩ(n) = 1 + ... + 1 (n times)<br />
<br />
Ɩ(-3) = ?<br />
<br />
Ɩ(-3 + 3) = Ɩ(-3) + Ɩ(3) ⇒ Ɩ(-3) = -Ɩ(3) = -(1+1+1)<br />
<br />
''What about uniqueness?'' Simply put, we had not choice in the definition of Ɩ. All followed from the given properties.<br />
<br />
At this point, we will be lazy and simply denote Ɩ(3) = 3_f [3 with subscript f]<br />
<br />
<br />
∃ m≠0, m ∈ '''N''', Ɩ(m) =0<br />
<br />
In which case, there is a smallest m>0, for which Ɩ(m)=0. '''m' is the characteristic of F.'' Denoted char(F).<br />
Examples: char(F_2)=2, char(F_3)=3... but NOTE: char('''R''')=0<br />
<br />
<br />
<br />
'''Thrm:''' If F is a field and char(F) >0, then char(F) is a prime number.<br />
<br />
Proof: Suppose char(F) =m, m>0. Suppose also m is not prime: m=ts, t,s ∈ '''N'''.<br />
<br />
Then, Ɩ(m) = 0 = Ɩ(t)∙Ɩ(s) ⇒ Ɩ(t)=0 or Ɩ(s)=0 by P12.<br />
<br />
If Ɩ(t)=0 ⇒ t≧m ⇒ m=t, s=1 or likewise for Ɩ(s)=0, and m=s, t=1<br />
<br />
In any factorization of m, one of the factors is m and the other is 1. So m is prime. ∎<br />
<br />
== Complex number==<br />
<br />
'''Abstraction, generalization, definition, examples, properties, dream, implications, realization = formalization, PROOF.'''<br />
<br />
Consider that fact that in '''R''', ∄ x s.t. x^2 = -1<br />
<br />
Dream: Add new number element 𝒊 to '''R''', so as to still get a field & 𝒊^2 = -1<br />
<br />
Implications: By adding 𝒊, we must add 7𝒊, and 2+7𝒊, 3+4𝒊, (2+7𝒊)∙(3+4𝒊), (2+7𝒊)^-1, etc.<br />
<br />
So, how do we define this field?<br />
<br />
'''Definition'''<br />
<br />
'''C''' = {(a,b): a,b ∈ '''R'''}<br />
Also, 0 (of the field) = (0,0); 1( of the field) = (1,0)<br />
<br />
Define addition: (a,b)+(c,d) = (a+c, b+d)<br />
<br />
Define multification: (a,b)(c,d) = (ad-bd, ad+bc)<br />
<br />
<br />
'''Theorems:'''<br />
<br />
'''Thrm. 1.''' ('''C''', 0, 1, +, ∙) is a field.<br />
<br />
'''Thrm. 2.''' ∃ 𝒊 ∈ '''C''' s.t. 𝒊^2 = -1<br />
<br />
'''Thrm. 3.''' '''C''' contains '''R'''<br />
<br />
Proof (1): Show that each of the field axioms holds for '''C'''.<br />
<br />
Ex. F1(a): ƶ1 + ƶ2 = ƶ2 + ƶ1, where ƶ1 = (a1, b1) and ƶ2 = (a2, b2)<br />
<br />
LHS: (a1,b1)+(a2,b2) = (a1+a2, b1+b2)<br />
<br />
RHS: (a2,b2)+(a1,b1) = (a2+a1, b2+b1)<br />
<br />
LHS=RHS by F1 of '''R'''<br />
<br />
F1(b) and so on...<br />
<br />
Proof (2):<br />
<br />
In '''C''', consider i=(0,1)<br />
<br />
By the definition i^2=i.i=(0.1-1.1,0.1+1.0)=(-1,0)<br />
<br />
We also have 1(of '''c''') + (-1,0)=(1,0)+(-1,0)=(0,0)=0 (of '''c''')<br />
<br />
Hence (-1,0) is the addictive inverse of 1, i.e, (-1,0)=-1<br />
<br />
Thus i^2=-1. ∎<br />
<br />
Proof 3: <br />
<br />
Given the field '''C''' : map J: '''R''' -> '''C'''<br />
<br />
1) J(0)=(0,0); J(1)=(1,0)<br />
<br />
2) J(x+y)=J(x)+J(y); J(x.y)=J(x)J(y)<br />
<br />
Define J(x)=(x,0), all will follow.<br />
<br />
From now on J(x) will be writen simply x<br />
<br />
EX: J(7)=7, J(3)=3<br />
<br />
So, what does a+b𝒊 mean? (a, b ∈ '''R''')<br />
<br />
a+b𝒊= Ɩ(a) + Ɩ(b)+Ɩ(𝒊) = (a,0) + (b,0)∙(0,1) = (a,b)<br />
<br />
Hence, (a,b) ~ a+b𝒊<br />
<br />
Thus, we can use a+b𝒊 with less hesitation.<br />
<br />
== Lecture 3, scanned notes upload by [[User:Starash|Starash]] and {{User:KJMorenz|KJMorenz]]==<br />
<br />
<gallery><br />
Image:12-240-0918-1.jpg|Page 1<br />
Image:12-240-0918-2.jpg|Page 2<br />
Image:12-240-0918-3.jpg|Page 3<br />
</gallery><br />
[[Image:12-240-CharacteristicOfField.jpg]]<br />
[[Image:12-240-ComplexNums.jpg]]</div>KJMorenzhttp://drorbn.net/index.php?title=12-240/Class_Photo12-240/Class Photo2012-09-28T01:08:00Z<p>KJMorenz: /* Who We Are... */</p>
<hr />
<div>Our class on September 25, 2012:<br />
<br />
[[Image:12-240-ClassPhoto.jpg|thumb|centre|800px|Class Photo: click to enlarge]]<br />
{{12-240/Navigation}}<br />
<br />
Please identify yourself in this photo! There are two ways to do that:<br />
<br />
* [[Special:Userlogin|Log in]] to this Wiki and edit this page. Put your name, userid, email address and location in the picture in the alphabetical list below.<br />
* Send [[User:Drorbn|Dror]] an email message with this information.<br />
<br />
The first option is more fun but less private.<br />
<br />
===Who We Are...===<br />
<br />
{| align=center border=1 cellspacing=0<br />
|-<br />
!First name<br />
!Last name<br />
!User ID<br />
!Email<br />
!Place in photo<br />
!Comments<br />
<br />
{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn@ math. toronto. edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank. For better line-breaking, leave a space next to the "@" in email addresses.}}<br />
{{Photo Entry|last=Frailich|first=Rebecca|userid=Rebecca.frailich|email=rebecca.frailich@ mail.utoronto.ca|location=Last row, in between two guys standing at the back (one in red, one in black) |comments=}}<br />
{{Photo Entry|last=Hoover|first=Ken|userid=Khoover|email=ken.hoover@ mail.utoronto.ca|location=First row, fourth from the right.|comments=}}<br />
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{{Photo Entry|last=Morenz|first=Karen|userid=KJMorenz|email=kjmorenz@ gmail.com|location=3rd-ish row from the back, centre right, purple shirt|comments=}}<br />
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{{Photo Entry|last=Wamer|first=Kyle|userid=kylewamer|email=kyle. wamer @ mail. utoronto. ca|location=Second row, fifth from the left in the red shirt.|comments=}}<br />
{{Photo Entry|last=Yang|first=Chen|userid=chen|email=neochen. yang@ mail. utoronto. ca|location=sixth row, first from the right in the black pull-over.|comments=}}<br />
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{{Photo Entry|last=Zibert|first=Vincent|userid=vincezibert|email=vincent.zibert@mail.utoronto.ca|location=Directly beneath the white notice posted on the door on the right-hand side.|comments=}}<br />
{{Photo Entry|last=Zoghi|first=Sina|userid=sina.zoghi|email=sina.zoghi@ utoronto .ca|location=First row, leftest left.|comments=}}<br />
<!--PLEASE KEEP IN ALPHABETICAL ORDER, BY LAST NAME--><br />
|}<br />
<br />
<!--PLEASE KEEP IN ALPHABETICAL ORDER, BY LAST NAME--></div>KJMorenz