Drorbn - User contributions [en]

12-240/Classnotes for Thursday October 11

2012-10-12T01:50:24Z

Celtonmcgrath:

{{12-240/Navigation}}

== Annoucements ==
TA Office Hours (Still pending!) @ 215 Huron St., 10th floor.

Peter - 11am - 1pm

Brandon 1pm - 3pm

Topic: Replacement Theorem

== Lecture notes scanned by [[User:Oguzhancan|Oguzhancan]] ==
<gallery>
Image:12-240-1011-1.jpg|Page 1
Image:12-240-1011-2.jpg|Page 2
Image:12-240-1011-3.jpg|Page 3
</gallery>

12-240/Classnotes for Thursday October 11

2012-10-12T01:49:54Z

Celtonmcgrath:

== Lecture notes scanned by [[User:Oguzhancan|Oguzhancan]] ==
<gallery>
Image:12-240-1011-1.jpg|Page 1
Image:12-240-1011-2.jpg|Page 2
Image:12-240-1011-3.jpg|Page 3
</gallery>

12-240/Classnotes for Thursday October 11

2012-10-12T01:49:26Z

Celtonmcgrath:

12-240/Classnotes for Thursday October 11

2012-10-12T01:48:19Z

Celtonmcgrath:

== Annoucements ==
TA Office Hours (Still pending!) @ 215 Huron St., 10th floor.

Peter - 11am - 1pm

Brandon 1pm - 3pm

Topic: Replacement Theorem

== Lecture notes scanned by [[User:Oguzhancan|Oguzhancan]] ==
<gallery>
Image:12-240-1011-1.jpg|Page 1
Image:12-240-1011-2.jpg|Page 2
Image:12-240-1011-3.jpg|Page 3
</gallery>

12-240/Classnotes for Thursday October 4

2012-10-05T00:07:43Z

Celtonmcgrath: /* Reminders */

== Reminders ==
Web Fact: No link, doesn't exist!

Life Fact: Dror doesn't do email math!

Riddle: Professor and lion in a ring with V_p = V_l, help the professor live as long as possible.

== Recap ==

Base - what were doing today

Linear combination (lc) - We say v is a linear combination of a set S = {u1 ... un} if v = a1u1 ... anun for scalars from a field F.

Span - span(S) is the set of all linear combination of set S

Generate - We say S generates a vector space V is span(S) = V

== Pre - Basis ==

'''Linear dependance'''

'''Definition''' A set S ⊂ V is called linearly dependant if you can express the zero vector as a linear combination of distinct vectors from S, excluding the non-trivial linear combination where all of the scalars are 0.

Otherwise, we call S '''linearly independant.'''

=== Examples ===

1. In '''R'''^3, take S = {u1 = (1,4,7), u2 = (2,5,8), u3 = (3,6,9)}

u1 - 2u2 + u3 = 0

S is linearly dependant.

2. In R^n, take {e_i} = {0, 0, ... 1, 0, 0, 0} where 1 is in the ith position, and (ei) is a vector with n entries.

Claim: This set is linearly independent.

Proof: Suppose (∑ ai*ei) = 0 ({ei} is linearly dependant.)

(∑ ai*ei) = 0 ⇔ a1(1, 0, ..., 0) + ... + an(0, ..., 0, 1) = 0 ⇔ (a1, 0, ... , 0) + ... + (0, ... , an) = 0

⇒ a1 = a2 = ... = an = 0!

===Comments ===

1. {u} is linearly independant.
Proof:
⇐ If u≠0, suppose au =0
By property (a*u = 0, a = 0 or u = 0), a = 0. Thus, {u} is linearly independent.

⇒ By definition, au = 0 for {u} only when a = 0.

2. ∅ is linearly independant.

Exercise: Prove: '''Theorem''' Suppose S1 ⊂ S2 ⊂ V.

-> If S1 is linearly dependant, then S2 is dependant.

-> If S2 is linearly independent, then S1 is linearly independent. (Hint: contrapositive)

== Basis ==

'''Definition''': A subset β is called a basis if 1. β generates V → span(β) = V and 2. β is linearly independent.

===Examples===

1. V = {0}, β = {}

2. {ei} for F^n, this is what we call the '''standard basis'''

3. B = {(1,1),(1, -1)} is a basis for R^2

4. P_n(F) β = {x^n, x^n-1, ... , x^1, x^0}

5. P(F), β = (x^0, x^1 ... and on} ('''Infinite basis'''!)

== Interesting inequality ==

This holds is true if the field does not have characteristic 2. Can you see why?

(a,b) = (a+b)/2 * (1, 1) + (a-b)/2 * (1, -1)

12-240/Classnotes for Thursday October 4

2012-10-05T00:07:32Z

Celtonmcgrath: /* Comments */

== Reminders ==
Web Fact: No link, doesn't exist!
Life Fact: Dror doesn't do email math!
Riddle: Professor and lion in a ring with V_p = V_l, help the professor live as long as possible.

== Recap ==

Base - what were doing today

Linear combination (lc) - We say v is a linear combination of a set S = {u1 ... un} if v = a1u1 ... anun for scalars from a field F.

Span - span(S) is the set of all linear combination of set S

Generate - We say S generates a vector space V is span(S) = V

== Pre - Basis ==

'''Linear dependance'''

'''Definition''' A set S ⊂ V is called linearly dependant if you can express the zero vector as a linear combination of distinct vectors from S, excluding the non-trivial linear combination where all of the scalars are 0.

Otherwise, we call S '''linearly independant.'''

=== Examples ===

1. In '''R'''^3, take S = {u1 = (1,4,7), u2 = (2,5,8), u3 = (3,6,9)}

u1 - 2u2 + u3 = 0

S is linearly dependant.

2. In R^n, take {e_i} = {0, 0, ... 1, 0, 0, 0} where 1 is in the ith position, and (ei) is a vector with n entries.

Claim: This set is linearly independent.

Proof: Suppose (∑ ai*ei) = 0 ({ei} is linearly dependant.)

(∑ ai*ei) = 0 ⇔ a1(1, 0, ..., 0) + ... + an(0, ..., 0, 1) = 0 ⇔ (a1, 0, ... , 0) + ... + (0, ... , an) = 0

⇒ a1 = a2 = ... = an = 0!

===Comments ===

1. {u} is linearly independant.
Proof:
⇐ If u≠0, suppose au =0
By property (a*u = 0, a = 0 or u = 0), a = 0. Thus, {u} is linearly independent.

⇒ By definition, au = 0 for {u} only when a = 0.

2. ∅ is linearly independant.

Exercise: Prove: '''Theorem''' Suppose S1 ⊂ S2 ⊂ V.

-> If S1 is linearly dependant, then S2 is dependant.

-> If S2 is linearly independent, then S1 is linearly independent. (Hint: contrapositive)

== Basis ==

'''Definition''': A subset β is called a basis if 1. β generates V → span(β) = V and 2. β is linearly independent.

===Examples===

1. V = {0}, β = {}

2. {ei} for F^n, this is what we call the '''standard basis'''

3. B = {(1,1),(1, -1)} is a basis for R^2

4. P_n(F) β = {x^n, x^n-1, ... , x^1, x^0}

5. P(F), β = (x^0, x^1 ... and on} ('''Infinite basis'''!)

== Interesting inequality ==

This holds is true if the field does not have characteristic 2. Can you see why?

(a,b) = (a+b)/2 * (1, 1) + (a-b)/2 * (1, -1)

12-240/Classnotes for Thursday October 4

2012-10-05T00:07:14Z

Celtonmcgrath:

== Reminders ==
Web Fact: No link, doesn't exist!
Life Fact: Dror doesn't do email math!
Riddle: Professor and lion in a ring with V_p = V_l, help the professor live as long as possible.

== Recap ==

Base - what were doing today

Linear combination (lc) - We say v is a linear combination of a set S = {u1 ... un} if v = a1u1 ... anun for scalars from a field F.

Span - span(S) is the set of all linear combination of set S

Generate - We say S generates a vector space V is span(S) = V

== Pre - Basis ==

'''Linear dependance'''

'''Definition''' A set S ⊂ V is called linearly dependant if you can express the zero vector as a linear combination of distinct vectors from S, excluding the non-trivial linear combination where all of the scalars are 0.

Otherwise, we call S '''linearly independant.'''

=== Examples ===

1. In '''R'''^3, take S = {u1 = (1,4,7), u2 = (2,5,8), u3 = (3,6,9)}

u1 - 2u2 + u3 = 0

S is linearly dependant.

2. In R^n, take {e_i} = {0, 0, ... 1, 0, 0, 0} where 1 is in the ith position, and (ei) is a vector with n entries.

Claim: This set is linearly independent.

Proof: Suppose (∑ ai*ei) = 0 ({ei} is linearly dependant.)

(∑ ai*ei) = 0 ⇔ a1(1, 0, ..., 0) + ... + an(0, ..., 0, 1) = 0 ⇔ (a1, 0, ... , 0) + ... + (0, ... , an) = 0

⇒ a1 = a2 = ... = an = 0!

===Comments ===

1. {u} is linearly independant.
Proof:
⇐ If u≠0, suppose au =0
By property (a*u = 0, a = 0 or u = 0), a = 0. Thus, {u} is linearly independent.

⇒ By definition, au = 0 for {u} only when a = 0.

2. ∅ is linearly independant.

Exercise: Prove: '''Theorem''' Suppose S1 ⊂ S2 ⊂ V.

-> If S1 is linearly dependant, then S2 is dependant.
-> If S2 is linearly independent, then S1 is linearly independent. (Hint: contrapositive)

== Basis ==

'''Definition''': A subset β is called a basis if 1. β generates V → span(β) = V and 2. β is linearly independent.

===Examples===

1. V = {0}, β = {}

2. {ei} for F^n, this is what we call the '''standard basis'''

3. B = {(1,1),(1, -1)} is a basis for R^2

4. P_n(F) β = {x^n, x^n-1, ... , x^1, x^0}

5. P(F), β = (x^0, x^1 ... and on} ('''Infinite basis'''!)

== Interesting inequality ==

This holds is true if the field does not have characteristic 2. Can you see why?

(a,b) = (a+b)/2 * (1, 1) + (a-b)/2 * (1, -1)

12-240/Classnotes for Thursday October 4

2012-10-05T00:04:04Z

Celtonmcgrath:

== Reminders ==
Web Fact: No link, doesn't exist!
Life Fact: Dror doesn't do email math!
Riddle: Professor and lion in a ring with V_p = V_l, help the professor live as long as possible.

== Recap == '

Base - what were doing today

Linear combination (lc) - We say v is a linear combination of a set S = {u1 ... un} if v = a1u1 ... anun for scalars from a field F.

Span - span(S) is the set of all linear combination of set S

Generate - We say S generates a vector space V is span(S) = V

== Pre - Basis ==

'''Linear dependance'''

'''Definition''' A set S ⊂ V is called linearly dependant if you can express the zero vector as a linear combination of distinct vectors from S, excluding the non-trivial linear combination where all of the scalars are 0.

Otherwise, we call S '''linearly independant.'''

=== Examples ===

1. In '''R'''^3, take S = {u1 = (1,4,7), u2 = (2,5,8), u3 = (3,6,9)}

u1 - 2u2 + u3 = 0

S is linearly dependant.

2. In R^n, take {e_i} = {0, 0, ... 1, 0, 0, 0} where 1 is in the ith position, and (ei) is a vector with n entries.

Claim: This set is linearly independent.

Proof: Suppose (∑ ai*ei) = 0 ({ei} is linearly dependant.)

(∑ ai*ei) = 0 ⇔ a1(1, 0, ..., 0) + ... + an(0, ..., 0, 1) = 0 ⇔ (a1, 0, ... , 0) + ... + (0, ... , an) = 0

⇒ a1 = a2 = ... = an = 0!

Proof:
⇐ If u≠0, suppose au =0
By property (a*u = 0, a = 0 or u = 0), a = 0. Thus, {u} is linearly independent.

⇒ By definition, au = 0 for {u} only when a = 0.

Exercise: Prove: '''Theorem''' Suppose S1 ⊂ S2 ⊂ V.

-> If S1 is linearly dependant, then S2 is dependant.
-> If S2 is linearly independent, then S1 is linearly independent. (Hint: contrapositive)

== Basis ==

'''Definition''': A subset β is called a basis if 1. β generates V → span(β) = V and 2. β is linearly independent.

===Examples===

1. V = {0}, β = {}

2. {ei} for F^n, this is what we call the '''standard basis'''

3. B = {(1,1),(1, -1)} is a basis for R^2

4. P_n(F) β = {x^n, x^n-1, ... , x^1, x^0}

5. P(F), β = (x^0, x^1 ... and on} ('''Infinite basis'''!)

12-240/Classnotes for Thursday October 4

2012-10-04T23:41:28Z

Celtonmcgrath: /* Introduction to Basis */

'''Recap'''

Base - what were doing today

Linear combination (lc) - We say v is a linear combination of a set S = {u1 ... un} if v = a1u1 ... anun for scalars from a field F.

Span - span(S) is the set of all linear combination of set S

Generate - We say S generates a vector space V is span(S) = V

==== Introduction to Basis ====

'''Linear dependance'''

'''Definition''' A set S subsetof V is called linearly dependant if you can express the zero vector as a linear combination of distinct vectors from S, excluding the non-trivial linear combination where all of the scalars are 0.

Otherwise, we call S '''linearly independant.'''

=== Examples ===

1. In '''R'''^3, take S = {u1 = (1,4,7), u2 = (2,5,8), u3 = (3,6,9)}

u1 - 2u2 + u3 = 0

S is linearly dependant.

2. In R^n, take {e_i} = {0, 0, ... 1, 0, 0, 0} where 1 is in the ith position, and (ei) is a vector with n entries.

Claim: This set is linearly independent.

Proof: Suppose (∑ ai*ei) = 0 ({ei} is linearly dependant.)

(∑ ai*ei) = 0 ⇔ a1(1, 0, ..., 0) + ... + an(0, ..., 0, 1) = 0 ⇔ (a1, 0, ... , 0) + ... + (0, ... , an) = 0

⇒ a1 = a2 = ... = an = 0!

=== Comments ===
1. {∅} is linearly independent. (Think about logical statements like "all elements of the empty set are purple")
2. {u} is linearly independent when u≠0.

Proof:

12-240/Classnotes for Thursday October 4

2012-10-04T23:40:44Z

Celtonmcgrath: /* Examples */

'''Recap'''

Base - what were doing today

Linear combination (lc) - We say v is a linear combination of a set S = {u1 ... un} if v = a1u1 ... anun for scalars from a field F.

Span - span(S) is the set of all linear combination of set S

Generate - We say S generates a vector space V is span(S) = V

== Introduction to Basis ==

'''Linear dependance'''

'''Definition''' A set S subsetof V is called linearly dependant if you can express the zero vector as a linear combination of distinct vectors from S, excluding the non-trivial linear combination where all of the scalars are 0.

Otherwise, we call S '''linearly independant.'''

=== Examples ===

1. In '''R'''^3, take S = {u1 = (1,4,7), u2 = (2,5,8), u3 = (3,6,9)}

u1 - 2u2 + u3 = 0

S is linearly dependant.

2. In R^n, take {e_i} = {0, 0, ... 1, 0, 0, 0} where 1 is in the ith position, and (ei) is a vector with n entries.

Claim: This set is linearly independent.

Proof: Suppose (∑ ai*ei) = 0 ({ei} is linearly dependant.)

(∑ ai*ei) = 0 ⇔ a1(1, 0, ..., 0) + ... + an(0, ..., 0, 1) = 0 ⇔ (a1, 0, ... , 0) + ... + (0, ... , an) = 0

⇒ a1 = a2 = ... = an = 0!

=== Comments ===
1. {∅} is linearly independent. (Think about logical statements like "all elements of the empty set are purple")
2. {u} is linearly independent when u≠0.

Proof:

12-240/Classnotes for Thursday October 4

2012-10-04T23:34:22Z

Celtonmcgrath:

'''Recap'''

Base - what were doing today

Linear combination (lc) - We say v is a linear combination of a set S = {u1 ... un} if v = a1u1 ... anun for scalars from a field F.

Span - span(S) is the set of all linear combination of set S

Generate - We say S generates a vector space V is span(S) = V

== Introduction to Basis ==

'''Linear dependance'''

'''Definition''' A set S subsetof V is called linearly dependant if you can express the zero vector as a linear combination of distinct vectors from S, excluding the non-trivial linear combination where all of the scalars are 0.

Otherwise, we call S '''linearly independant.'''

=== Examples ===

1. In '''R'''^3, take S = {u1 = (1,4,7), u2 = (2,5,8), u3 = (3,6,9)}

u1 - 2u2 + u3 = 0

S is linearly dependant.

2. In R^n, take {e_i} = {0, 0, ... 1, 0, 0, 0} where 1 is in the ith position, and (ei) is a vector with n entries.

Claim: This set is linearly independent.

12-240/Classnotes for Thursday October 4

2012-10-04T23:24:59Z

Celtonmcgrath:

Recap
Base - what were doing today
Linear combination (lc) - We say v is a linear combination of a set S = {u1 ... un} if v = a1u1 ... anun for scalars from a field F.
Span - span(S) is the set of all linear combination of set S
Generate - We say S generates a vector space V is span(S) = V

== Introduction to Basis ==

12-240/Navigation

2012-10-04T23:19:31Z

Celtonmcgrath:

<noinclude>Back to [[12-240]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 10
|[[12-240/About This Class|About This Class]], [[12-240/Classnotes for Tuesday September 11|Tuesday]], [[12-240/Classnotes for Thursday September 13|Thursday]]
|- align=left
|align=center|2
|Sep 17
|[[12-240/Homework Assignment 1|HW1]], [[12-240/Classnotes for Tuesday September 18|Tuesday]], [[12-240/Classnotes for Thursday September 20|Thursday]]
|- align=left
|align=center|3
|Sep 24
|[[12-240/Homework Assignment 2|HW2]], [[12-240/Classnotes for Tuesday September 25|Tuesday]], [[12-240/Class Photo|Class Photo]], [[12-240/Classnotes for Thursday September 27|Thursday]]
|- align=left
|align=center|4
|Oct 1
|[[12-240/Homework Assignment 3|HW3]], [[12-240/Classnotes for Tuesday October 2|Tuesday]], [[12-240/Classnotes for Thursday October 4|Thursday]]
|- align=left
|align=center|5
|Oct 8
|HW4
|- align=left
|align=center|6
|Oct 15
|
|- align=left
|align=center|7
|Oct 22
|HW5, [[12-240/Term Test|Term Test]] on Thursday.
|- align=left
|align=center|8
|Oct 29
|HW6; Nov 4 is the last day to drop this class
|- align=left
|align=center|9
|Nov 5
|
|- align=left
|align=center|10
|Nov 12
|Monday-Tuesday is UofT November break, HW7 on web Wednesday
|- align=left
|align=center|11
|Nov 19
|HW8
|- align=left
|align=center|12
|Nov 26
|HW9
|- align=left
|align=center|13
|Dec 3
|UofT Fall Semester ends Wednesday
|- align=left
|align=center|F1
|Dec 10
|Finals
|- align=left
|align=center|F2
|Dec 17
|Finals
|- align=left
|colspan=3 align=center|[[12-240/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:12-240-ClassPhoto.jpg|310px]] [[12-240/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:12-240-Splash.png|310px]]
|}

12-240/Classnotes for Thursday September 27

2012-09-30T19:57:28Z

Celtonmcgrath: /* Subspaces */

{{12-240/Navigation}}

'''Vector Spaces'''

== Reminders ==

- Tag yourself in the photo!

- Read along textbook 1.1 to 1.4

- Riddle: Professor in ring with lion around the perimeter.
Consider this: http://mathforum.org/library/drmath/view/63421.html

== Vector space axioms ==

''(Quick recap)''

VS1. x + y = y + x

VS2. (x + y) + z = x + (y + z)

VS3. 0 vector

VS4. + inverse → -

VS5. 1x = x

VS6. a(bx) = (ab)x

VS7. a(x + y) = ax + ay

VS8. (a+b)x = ax + bx

== Theorems ==

1.a x + z = y + z ⇒ x = y

1.b ax = ay, a ≠ 0, ⇒ x = y

1.c ax = bx, x ≠ 0, ⇒ a = b

2. 0 is unique.

3. Additive inverse is unique.

4. 0_F ∙ x = 0_V

5. a ∙ 0_V = 0_V

6. (-a) x = -(ax) = a(-x)

7. cx = 0 ⇔ c = 0 or x = 0_V

=== Hints for proofs ===

1.a Same as for fields

1.b. Use similar proof as for fields, but use VS6 NOT F2b. F2b guarantees existence, but VS6 allows algebraic manipulation.

1.c Discussed after proof of 7, harder than you think at first glance.

2. Same as F.

3. Same as F

4. 0_F + 0_F = 0_F => by [VS8]: 0x + 0x = (0+0)x = 0x = 0x + 0 [VS3] = 0 + 0x [VS1]
⇒ 0x + 0x = 0 + 0x ⇒ [Cancellation property] 0x = 0

5. Same as 4 except using 0_V + 0_V = 0_V and using VS7

6. Skip

7. Prove both ways: Easy way is to the left, show left is 0 if either on right is 0.
To the right, Suppose c not= 0, then show x must equal 0.

1.c Add (-bx) to each side, use VS8 then VS6 -> (a-b)x =0, use property 7.

== Subspaces ==

Definition: Let V be a vector space over a field F. A ''subspace'' W of V is a subset of V, has the operations inherited from V and 0_V of V, is itself a vector space.

=== Examples of subspaces ===
Look at scanned notes for examples of subspaces!

Theorem: A subset W ⊂ V, W ≠ ∅, is a subspace iff it is closed under the operations of V.

1. ∀ x, y ∈ W, x + y ∈ W

2. ∀ c ∈ F, ∀ x ∈ W, cx ∈ W

== Scanned notes upload by [[User:Starash|Starash]] ==

<gallery>
Image:12-240-0927-1.jpg|Page 1
Image:12-240-0927-2.jpg|Page 2
</gallery>

12-240/Classnotes for Thursday September 27

2012-09-28T02:46:39Z

Celtonmcgrath: /* Rough sketches for proofs */

{{12-240/Navigation}}

'''Vector Spaces'''

== Reminders ==

- Tag yourself in the photo!

- Read along textbook 1.1 to 1.4

- Riddle: Professor in ring with lion around the perimeter.
Consider this: http://mathforum.org/library/drmath/view/63421.html

== Vector space axioms ==

''(Quick recap)''

VS1. x + y = y + x

VS2. (x + y) + z = x + (y + z)

VS3. 0 vector

VS4. + inverse → -

VS5. 1x = x

VS6. a(bx) = (ab)x

VS7. a(x + y) = ax + ay

VS8. (a+b)x = ax + bx

== Theorems ==

1.a x + z = y + z ⇒ x = y

1.b ax = ay, a ≠ 0, ⇒ x = y

1.c ax = bx, x ≠ 0, ⇒ a = b

2. 0 is unique.

3. Additive inverse is unique.

4. 0_F ∙ x = 0_V

5. a ∙ 0_V = 0_V

6. (-a) x = -(ax) = a(-x)

7. cx = 0 ⇔ c = 0 or x = 0_V

=== Hints for proofs ===

1.a Same as for fields

1.b. Use similar proof as for fields, but use VS6 NOT F2b. F2b guarantees existence, but VS6 allows algebraic manipulation.

1.c Discussed after proof of 7, harder than you think at first glance.

2. Same as F.

3. Same as F

4. 0x + 0x = (0+0)x [VS8] = 0x = 0x + 0 [VS3] = 0 + 0x [VS1]
⇒ 0x + 0x = 0 + 0x ⇒ [Cancellation property] 0x = 0

5. Same as 4 except using 0_V + 0_V = 0_V and using VS7

6. Skip

7. Prove both ways: Easy way is to the left, show left is 0 if either on right is 0.
To the right, Suppose c not= 0, then show x must equal 0.

1.c Add (-bx) to each side, use VS8 then VS6 -> (a-b)x =0, use property 7.

== Subspaces ==

Definition: Let V be a vector space over a field F. A ''subspace'' W of V is a subset of V, has the operations inherited from V and 0_V of V, is itself a vector space.

Theorem: A subset W ⊂ V, W ≠ {∅}, is a subspace iff it is closed under the operations of V.

1. ∀ x, y ∈ W, x + y ∈ W

2. ∀ c ∈ F, ∀ x ∈ W, cx ∈ W

12-240/Classnotes for Thursday September 27

2012-09-28T02:40:54Z

Celtonmcgrath:

{{12-240/Navigation}}

'''Vector Spaces'''

== Reminders ==

- Tag yourself in the photo!

- Read along textbook 1.1 to 1.4

- Riddle: Professor in ring with lion around the perimeter.
Consider this: http://mathforum.org/library/drmath/view/63421.html

== Vector space axioms ==

''(Quick recap)''

VS1. x + y = y + x

VS2. (x + y) + z = x + (y + z)

VS3. 0 vector

VS4. + inverse → -

VS5. 1x = x

VS6. a(bx) = (ab)x

VS7. a(x + y) = ax + ay

VS8. (a+b)x = ax + bx

== Theorems ==

1.a x + z = y + z ⇒ x = y

1.b ax = ay, a ≠ 0, ⇒ x = y

1.c ax = bx, x ≠ 0, ⇒ a = b

2. 0 is unique.

3. Additive inverse is unique.

4. 0_F ∙ x = 0_V

5. a ∙ 0_V = 0_V

6. (-a) x = -(ax) = a(-x)

7. cx = 0 ⇔ c = 0 or x = 0_V

=== Rough sketches for proofs ===

1.a Same as for fields

1.b. Use similar proof as for fields, but use VS6 NOT F2b. F2b guarantees existence, but VS6 allows algebraic manipulation.

1.c Discussed after proof of 7, harder than you think at first glance.

2. Same as F.

3. Same as F

4. 0x + 0x = (0+0)x [VS8] = 0x = 0x + 0 [VS3] = 0 + 0x [VS1]
⇒ 0x + 0x = 0 + 0x ⇒ [Cancellation property] 0x = 0

5. Same as 4 except using 0_V + 0_V = 0_V and using VS7

6. Skip

7. Prove both ways: Easy way is to the left, show left is 0 if either on right is 0.
To the right, Suppose c not= 0, then show x must equal 0.

1.c

== Subspaces ==

Definition: Let V be a vector space over a field F. A ''subspace'' W of V is a subset of V, has the operations inherited from V and 0_V of V, is itself a vector space.

Theorem: A subset W ⊂ V, W ≠ {∅}, is a subspace iff it is closed under the operations of V.

1. ∀ x, y ∈ W, x + y ∈ W

2. ∀ c ∈ F, ∀ x ∈ W, cx ∈ W

12-240/Classnotes for Thursday September 27

2012-09-28T02:40:18Z

Celtonmcgrath: /* Rough sketches for proofs */

{{12-240/Navigation}}

'''Vector Spaces'''

== Reminders ==

- Tag yourself in the photo!

- Read along textbook 1.1 to 1.4

- Riddle: Professor in ring with lion around the perimeter.
Consider this: http://mathforum.org/library/drmath/view/63421.html

== Vector space axioms ==

''(Quick recap)''

VS1. x + y = y + x

VS2. (x + y) + z = x + (y + z)

VS3. 0 vector

VS4. + inverse → -

VS5. 1x = x

VS6. a(bx) = (ab)x

VS7. a(x + y) = ax + ay

VS8. (a+b)x = ax + bx

== Theorems ==

1.a x + z = y + z ⇒ x = y

1.b ax = ay, a ≠ 0, ⇒ x = y

1.c ax = bx, x ≠ 0, ⇒ a = b

2. 0 is unique.

3. Additive inverse is unique.

4. 0_F ∙ x = 0_V

5. a ∙ 0_V = 0_V

6. (-a) x = -(ax) = a(-x)

7. cx = 0 ⇔ c = 0 or x = 0_V

=== Rough sketches for proofs ===

1.a Same as for fields

1.b. Use similar proof as for fields, but use VS6 NOT F2b. F2b guarantees existence, but VS6 allows algebraic manipulation.

1.c Discussed after proof of 7, harder than you think at first glance.

2. Same as F.

3. Same as F

4. 0x + 0x = (0+0)x [VS8] = 0x = 0x + 0 [VS3] = 0 + 0x [VS1]

=> 0x + 0x = 0 + 0x => [Cancellation property] 0x = 0

5. Same as 4 except using 0_V + 0_V = 0_V and using VS7

6. Skip

7. Prove both ways: Easy way is to the left, show left is 0 if either on right is 0.
To the right, Suppose c not= 0, then show x must equal 0.

1.c

== Subspaces ==

Definition: Let V be a vector space over a field F. A ''subspace'' W of V is a subset of V, has the operations inherited from V and 0_V of V, is itself a vector space.

Theorem: A subset W ⊂ V, W ≠ {∅}, is a subspace iff it is closed under the operations of V.

1. ∀ x, y ∈ W, x + y ∈ W

2. ∀ c ∈ F, ∀ x ∈ W, cx ∈ W

12-240/Classnotes for Thursday September 27

2012-09-28T02:40:02Z

Celtonmcgrath: /* Rough sketches for proofs */

{{12-240/Navigation}}

'''Vector Spaces'''

== Reminders ==

- Tag yourself in the photo!

- Read along textbook 1.1 to 1.4

- Riddle: Professor in ring with lion around the perimeter.
Consider this: http://mathforum.org/library/drmath/view/63421.html

== Vector space axioms ==

''(Quick recap)''

VS1. x + y = y + x

VS2. (x + y) + z = x + (y + z)

VS3. 0 vector

VS4. + inverse → -

VS5. 1x = x

VS6. a(bx) = (ab)x

VS7. a(x + y) = ax + ay

VS8. (a+b)x = ax + bx

== Theorems ==

1.a x + z = y + z ⇒ x = y

1.b ax = ay, a ≠ 0, ⇒ x = y

1.c ax = bx, x ≠ 0, ⇒ a = b

2. 0 is unique.

3. Additive inverse is unique.

4. 0_F ∙ x = 0_V

5. a ∙ 0_V = 0_V

6. (-a) x = -(ax) = a(-x)

7. cx = 0 ⇔ c = 0 or x = 0_V

=== Rough sketches for proofs ===

1.a Same as for fields

1.b. Use similar proof as for fields, but use VS6 NOT F2b. F2b guarantees existence, but VS6 allows algebraic manipulation.

1.c Discussed after proof of 7, harder than you think at first glance.

2. Same as F.

3. Same as F

4. 0x + 0x = (0+0)x [VS8] = 0x = 0x + 0 [VS3] = 0 + 0x [VS1]
=> 0x + 0x = 0 + 0x => [Cancellation property] 0x = 0

5. Same as 4 except using 0_V + 0_V = 0_V and using VS7

6. Skip

7. Prove both ways: Easy way is to the left, show left is 0 if either on right is 0.
To the right, Suppose c not= 0, then show x must equal 0.

1.c

== Subspaces ==

Definition: Let V be a vector space over a field F. A ''subspace'' W of V is a subset of V, has the operations inherited from V and 0_V of V, is itself a vector space.

Theorem: A subset W ⊂ V, W ≠ {∅}, is a subspace iff it is closed under the operations of V.

1. ∀ x, y ∈ W, x + y ∈ W

2. ∀ c ∈ F, ∀ x ∈ W, cx ∈ W

12-240/Class Photo

2012-09-28T01:48:20Z

Celtonmcgrath: /* Who We Are... */

Our class on September 25, 2012:

[[Image:12-240-ClassPhoto.jpg|thumb|centre|800px|Class Photo: click to enlarge]]
{{12-240/Navigation}}

Please identify yourself in this photo! There are two ways to do that:

* [[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.
* Send [[User:Drorbn|Dror]] an email message with this information.

The first option is more fun but less private.

===Who We Are...===

{| align=center border=1 cellspacing=0
|-
!First name
!Last name
!User ID
!Email
!Place in photo
!Comments

{{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.}}
{{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=}}
{{Photo Entry|last=Hoover|first=Ken|userid=Khoover|email=ken.hoover@ mail.utoronto.ca|location=First row, fourth from the right.|comments=}}
{{Photo Entry|last=Klingspor|first=Josefine|userid=Josefine|email=josefine. klingspor@ mail. utoronto. ca|location=First row, second from left.|comments=}}
{{Photo Entry|last=Liu|first=Zhaowei|userid=tod|email=tod.liu@mail.utoronto .ca|location=First row, third from the right|comments=}}
{{Photo Entry|last=Millson|first=Richard|userid=Richardm|email=r.millson@ mail. utoronto. ca|location=Seventh row from the front, fourth from the right, blue sweater|comments=}}
{{Photo Entry|last=McGrath|first=Celton|userid=CeltonMcGrath|email=celton.mcgrath@.mail.utoronto.ca|location=4th row front from, centre right, brown sweater|comments=}}
{{Photo Entry|last=Morenz|first=Karen|userid=KJMorenz|email=kjmorenz@ gmail.com|location=3rd-ish row from the back, centre right, purple shirt|comments=}}
{{Photo Entry|last=Vicencio-Heap|first=Felipe|userid=Heapfeli|email=felipe. vicencio. heap@ mail. utoronto. ca|location=Second row from the front, furthest to the right.|comments=}}
{{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=}}
{{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=}}
{{Photo Entry|last=Zhang|first=BingZhen|userid=Zetalda|email=bingzhen.zhang@ mail. utoronto. ca|location=Second last row, third from left.|comments=}}
{{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=}}
{{Photo Entry|last=Zoghi|first=Sina|userid=sina.zoghi|email=sina.zoghi@ utoronto .ca|location=First row, leftest left.|comments=}}

|}

12-240/Class Photo

2012-09-28T01:46:14Z

Celtonmcgrath: /* Who We Are... */

Our class on September 25, 2012:

[[Image:12-240-ClassPhoto.jpg|thumb|centre|800px|Class Photo: click to enlarge]]
{{12-240/Navigation}}

Please identify yourself in this photo! There are two ways to do that:

* [[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.
* Send [[User:Drorbn|Dror]] an email message with this information.

The first option is more fun but less private.

===Who We Are...===

{| align=center border=1 cellspacing=0
|-
!First name
!Last name
!User ID
!Email
!Place in photo
!Comments

{{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.}}
{{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=}}
{{Photo Entry|last=Hoover|first=Ken|userid=Khoover|email=ken.hoover@ mail.utoronto.ca|location=First row, fourth from the right.|comments=}}
{{Photo Entry|last=Klingspor|first=Josefine|userid=Josefine|email=josefine. klingspor@ mail. utoronto. ca|location=First row, second from left.|comments=}}
{{Photo Entry|last=Liu|first=Zhaowei|userid=tod|email=tod.liu@mail.utoronto .ca|location=First row, third from the right|comments=}}
{{Photo Entry|last=Millson|first=Richard|userid=Richardm|email=r.millson@ mail. utoronto. ca|location=Seventh row from the front, fourth from the right, blue sweater|comments=}}
{{Photo Entry|last=McGrath|first=Celton|userid=CeltonMcGrath|email=celton.mcgrath@utoronto.ca|location=4th row front from, centre right, brown sweater|comments=}}
{{Photo Entry|last=Morenz|first=Karen|userid=KJMorenz|email=kjmorenz@ gmail.com|location=3rd-ish row from the back, centre right, purple shirt|comments=}}
{{Photo Entry|last=Vicencio-Heap|first=Felipe|userid=Heapfeli|email=felipe. vicencio. heap@ mail. utoronto. ca|location=Second row from the front, furthest to the right.|comments=}}
{{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=}}
{{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=}}
{{Photo Entry|last=Zhang|first=BingZhen|userid=Zetalda|email=bingzhen.zhang@ mail. utoronto. ca|location=Second last row, third from left.|comments=}}
{{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=}}
{{Photo Entry|last=Zoghi|first=Sina|userid=sina.zoghi|email=sina.zoghi@ utoronto .ca|location=First row, leftest left.|comments=}}

|}

12-240/Class Photo

2012-09-28T01:45:55Z

Celtonmcgrath: /* Who We Are... */

Our class on September 25, 2012:

[[Image:12-240-ClassPhoto.jpg|thumb|centre|800px|Class Photo: click to enlarge]]
{{12-240/Navigation}}

Please identify yourself in this photo! There are two ways to do that:

* [[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.
* Send [[User:Drorbn|Dror]] an email message with this information.

The first option is more fun but less private.

===Who We Are...===

{| align=center border=1 cellspacing=0
|-
!First name
!Last name
!User ID
!Email
!Place in photo
!Comments

{{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.}}
{{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=}}
{{Photo Entry|last=Hoover|first=Ken|userid=Khoover|email=ken.hoover@ mail.utoronto.ca|location=First row, fourth from the right.|comments=}}
{{Photo Entry|last=Klingspor|first=Josefine|userid=Josefine|email=josefine. klingspor@ mail. utoronto. ca|location=First row, second from left.|comments=}}
{{Photo Entry|last=Liu|first=Zhaowei|userid=tod|email=tod.liu@mail.utoronto .ca|location=First row, third from the right|comments=}}
{{Photo Entry|last=Millson|first=Richard|userid=Richardm|email=r.millson@ mail. utoronto. ca|location=Seventh row from the front, fourth from the right, blue sweater|comments=}}
{{Photo Entry|last=McGrath|first=Celton|userid=CeltonMcGrath|email=celton.mcgrath@utoronto.ca|location=4th row front from, centre right, brown sweater|comments=}}
{{Photo Entry|last=Morenz|first=Karen|userid=KJMorenz|email=kjmorenz@ gmail.com|location=3rd-ish row from the back, centre right, purple shirt|comments=}}
{{Photo Entry|last=McGrath|first=Celton|userid=CeltonMcGrath|email=celton.mcgrath@utoronto.ca|location=4th row front from, centre right, brown sweater|comments=}}
{{Photo Entry|last=Vicencio-Heap|first=Felipe|userid=Heapfeli|email=felipe. vicencio. heap@ mail. utoronto. ca|location=Second row from the front, furthest to the right.|comments=}}
{{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=}}
{{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=}}
{{Photo Entry|last=Zhang|first=BingZhen|userid=Zetalda|email=bingzhen.zhang@ mail. utoronto. ca|location=Second last row, third from left.|comments=}}
{{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=}}
{{Photo Entry|last=Zoghi|first=Sina|userid=sina.zoghi|email=sina.zoghi@ utoronto .ca|location=First row, leftest left.|comments=}}

|}

12-240/Classnotes for Thursday September 27

2012-09-28T01:43:53Z

Celtonmcgrath:

12-240/Classnotes for Thursday September 27

2012-09-28T01:42:23Z

Celtonmcgrath:

12-240/Classnotes for Thursday September 27

2012-09-28T01:39:54Z

Celtonmcgrath:

{{12-240/Navigation}}

'''Vector Spaces'''

== Vector space axioms ==

''(Quick recap)''

VS1. x + y = y + x

VS2. (x + y) + z = x + (y + z)

VS3. 0 vector

VS4. + inverse → -

VS5. 1x = x

VS6. a(bx) = (ab)x

VS7. a(x + y) = ax + ay

VS8. (a+b)x = ax + bx

== Theorems ==

1.a x + z = y + z ⇒ x = y

1.b ax = ay, a ≠ 0, ⇒ x = y

1.c ax = bx, x ≠ 0, ⇒ a = b

2. 0 is unique.

3. Additive inverse is unique.

4. 0_F ∙ x = 0_V

5. a ∙ 0_V = 0_V

6. (-a) x = -(ax) = a(-x)

7. cx = 0 ⇔ c = 0 or x = 0_V

=== Rough sketches for proofs ===

== Subspaces ==

Definition: Let V be a vector space over a field F. A ''subspace'' W of V is a subset of V, has the operations inherited from V and 0_V of V, is itself a vector space.

Theorem: A subset W ⊂ V, W ≠ {∅}, is a subspace iff it is closed under the operations of V.

1. ∀ x, y ∈ W, x + y ∈ W

2. ∀ c ∈ F, ∀ x ∈ W, cx ∈ W

12-240/Classnotes for Thursday September 27

2012-09-28T01:34:00Z

Celtonmcgrath:

{{12-240/Navigation}}

'''Vector Spaces'''

== Vector space axioms ==

''(Quick recap)''

VS1. x + y = y + x

VS2. (x + y) + z = x + (y + z)

VS3. 0 vector

VS4. + inverse -> -

VS5. 1x = x

VS6. a(bx) = (ab)x

VS7. a(x + y) = ax + ay

VS8. (a+b)x = ax + bx

== Theorems ==

1.a x + z = y + z => x = y

1.b ax = ay, a != 0, => x = y

1.c ax = bx, x != 0, => a = b

2. 0 is unique.

3. Additive inverse is unique.

4. 0_F * x = 0_V

5. a * 0_V = 0_V

6. (-a) x = -(ax) = a(-x)

7. cx = 0 <=> c = 0 or x = 0_V

=== Rough sketches for proofs ===

== Subspaces ==

Definition: Let V be a vector space over a field F. A ''subspace'' W of V is a subset of V, has the operations inherited from V and 0_V of V, is itself a vector space.

Theorem: A subset W C V, W != {emptyset}, is a subspace iff it is closed under the operations of V.

1. forall x, y elementof W, x + y elementof W

2. forall c elementof F, forall x elementof W, cx elementof W

12-240/Classnotes for Thursday September 27

2012-09-28T01:32:49Z

Celtonmcgrath:

{{12-240/Navigation}}

'''Vector Spaces'''

== Vector space axioms ==

''(Quick recap)''

VS1. x + y = y + x

VS2. (x + y) + z = x + (y + z)

VS3. 0 vector

VS4. + inverse -> -

VS5. 1x = x

VS6. a(bx) = (ab)x

VS7. a(x + y) = ax + ay

VS8. (a+b)x = ax + bx

== Theorems ==

1.a x + z = y + z => x = y

1.b ax = ay, a != 0, => x = y

1.c ax = bx, x != 0, => a = b

2. 0 is unique.

3. Additive inverse is unique.

4. 0_F * x = 0_V

5. a * 0_V = 0_V

6. (-a) x = -(ax) = a(-x)

7. cx = 0 <=> c = 0 or x = 0_V

== Subspaces ==

Definition: Let V be a vector space over a field F. A ''subspace'' W of V is a subset of V, has the operations inherited from V and 0_V of V, is itself a vector space.

Theorem: A subset W C V, W != {emptyset}, is a subspace iff it is closed under the operations of V.

1. forall x, y elementof W, x + y elementof W

2. forall c elementof F, forall x elementof W, cx elementof W

12-240/Classnotes for Thursday September 27

2012-09-28T01:26:43Z

Celtonmcgrath:

12-240/Classnotes for Thursday September 27

2012-09-28T01:21:08Z

Celtonmcgrath:

{{12-240/Navigation}}

In this course, we will be focusing on both a practical side and a theoretical side.

== Vector space axioms ==
''(Quick recap)''
VS1. x + y = y + x
VS2. (x + y) + z = x + (y + z)
VS3. 0 vector
VS4. + inverse -> -
VS5. 1x = x
VS6. a(bx) = (ab)x
VS7. a(x + y) = ax + ay
VS8. (a+b)x = ax + bx

12-240/Navigation

2012-09-28T01:15:22Z

Celtonmcgrath:

<noinclude>Back to [[12-240]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 10
|[[12-240/About This Class|About This Class]], [[12-240/Classnotes for Tuesday September 11|Tuesday]], [[12-240/Classnotes for Thursday September 13|Thursday]]
|- align=left
|align=center|2
|Sep 17
|[[12-240/Homework Assignment 1|HW1]], [[12-240/Classnotes for Tuesday September 18|Tuesday]], [[12-240/Classnotes for Thursday September 20|Thursday]]
|- align=left
|align=center|3
|Sep 24
|[[12-240/Homework Assignment 2|HW2]], [[12-240/Classnotes for Tuesday September 25|Tuesday]], [[12-240/Class Photo|Class Photo]], [[12-240/Classnotes for Thursday September 27|Thursday]]
|- align=left
|align=center|4
|Oct 1
|HW3
|- align=left
|align=center|5
|Oct 8
|HW4
|- align=left
|align=center|6
|Oct 15
|
|- align=left
|align=center|7
|Oct 22
|HW5, [[12-240/Term Test|Term Test]] on Thursday.
|- align=left
|align=center|8
|Oct 29
|HW6; Nov 4 is the last day to drop this class
|- align=left
|align=center|9
|Nov 5
|
|- align=left
|align=center|10
|Nov 12
|Monday-Tuesday is UofT November break, HW7 on web Wednesday
|- align=left
|align=center|11
|Nov 19
|HW8
|- align=left
|align=center|12
|Nov 26
|HW9
|- align=left
|align=center|13
|Dec 3
|UofT Fall Semester ends Wednesday
|- align=left
|align=center|F1
|Dec 10
|Finals
|- align=left
|align=center|F2
|Dec 17
|Finals
|- align=left
|colspan=3 align=center|[[12-240/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:12-240-ClassPhoto.jpg|310px]] [[12-240/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:12-240-Splash.png|310px]]
|}

12-240/Classnotes for Tuesday September 18

2012-09-19T02:54:46Z

Celtonmcgrath: /* Complex number */

{{12-240/Navigation}}
In this class, the professor continued with some more theorems of field and introduced definition and theorems of complex number.

== Various properties of fields ==
'''Thrm 1''': 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 (∀ m,n ∈ Z):

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.

'''Rough proof''':

Test somes cases:

Ɩ(2) = Ɩ(1+1) = Ɩ(1) + Ɩ(1) = 1 + 1 ≠ 2

Ɩ(3) = Ɩ(2 +1)= Ɩ(2) + Ɩ(1) = 1+ 1+ 1 ≠ 3

.
.
.

Ɩ(n) = 1 + ... + 1 (n times)

Ɩ(-3) = ?

Ɩ(-3 + 3) = Ɩ(-3) + Ɩ(3) ⇒ Ɩ(-3) = -Ɩ(3) = -(1+1+1)

''What about uniqueness?'' Simply put, we had not choice in the definition of Ɩ. All followed from the given properties.

At this point, we will be lazy and simply denote Ɩ(3) = 3_f [3 with subscript f]

∃ m≠0, m ∈ '''N''', Ɩ(m) =0

In which case, there is a smallest m<0, for which Ɩ(m)=0. '''m' is the characteristic of F.'' Denoted char(F).
Examples: char(F_2)=2, char(F_3)=3... but NOTE: char('''R''')=0

'''Thrm:''' If F is a field and char(F) >0, then char(F) is a prime number.

Proof: Suppose char(F) =m, m>0. Suppose also m is not prime: m=ts, t,s ∈ '''N'''.

Then, Ɩ(m) = 0 = Ɩ(t)∙Ɩ(s) ⇒ Ɩ(t)=0 or Ɩ(s)=0 by P12.

If Ɩ(t)=0 ⇒ t≧m ⇒ m=t, s=1 or likewise for Ɩ(s)=0, and m=s, t=1

In any factorization of m, one of the factors is m and the other is 1. So m is prime. ∎

== Complex number==

'''Abstraction, generalization, definition, examples, properties, dream, implications, realization = formalization, PROOF.'''

Consider that fact that in '''R''', ∄ x s.t. x^2 = -1

Dream: Add new number element 𝒊 to '''R''', so as to still get a field & 𝒊^2 = -1

Implications: By adding 𝒊, we must add 7𝒊, and 2+7𝒊, 3+4𝒊, (2+7𝒊)∙(3+4𝒊), (2+7𝒊)^-1, etc.

So, how do we define this field?

'''Definition'''

'''C''' = {(a,b): a,b ∈ '''R'''}
Also, 0 (of the field) = (0,0); 1( of the field) = (1,0)

Define addition: (a,b)+(c,d) = (a+c, b+d)

Define multification: (a,b)(c,d) = (ad-bd, ad+bc)

'''Theorems:'''

'''Thrm. 1.''' ('''C''', 0, 1, +, ∙) is a field.

'''Thrm. 2.''' ∃ 𝒊 ∈ '''C''' s.t. 𝒊^2 = -1

'''Thrm. 3.''' '''C''' contains '''R'''

Proof (1): Show that each of the field axioms holds for '''C'''.

Ex. F1(a): ƶ1 + ƶ2 = ƶ2 + ƶ1, where ƶ1 = (a1, b1) and ƶ2 = (a2, b2)

LHS: (a1,b1)+(a2,b2) = (a1+a2, b1+b2)

RHS: (a2,b2)+(a1,b1) = (a2+a1, b2+b1)

LHS=RHS by F1 of '''R'''

F1(b) and so on...

Proof (2):
Given a field '''C''', define a map Ɩ: '''R''' → '''C''' with the properties (∀ m,n ∈ Z):

1) Ɩ(0) =0, Ɩ(1)=1

2) Ɩ(m+n) = Ɩ(m) +Ɩ(n)

3) Ɩ(mn) = Ɩ(m)∙Ɩ(n)

Finally, define Ɩ(x)=(x,0). Thus, it follows. ∎

So, what does a+b𝒊 mean? (a, b ∈ '''R''')

a+b𝒊= Ɩ(a) + Ɩ(b)+Ɩ(𝒊) = (a,0) + (b,0)∙(0,1) = (a,b)

Hence, (a,b) ~ a+b𝒊

Thus, we can use a+b𝒊 with less hesitation.

== Lecture 3, scanned notes upload by [[User:Starash|Starash]] ==

<gallery>
Image:12-240-0918-1.jpg|Page 1
Image:12-240-0918-2.jpg|Page 2
Image:12-240-0918-3.jpg|Page 3
</gallery>

12-240/Classnotes for Tuesday September 18

2012-09-19T02:50:21Z

Celtonmcgrath: /* Complex number */

{{12-240/Navigation}}
In this class, the professor continued with some more theorems of field and introduced definition and theorems of complex number.

== Various properties of fields ==
'''Thrm 1''': 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 (∀ m,n ∈ Z):

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.

'''Rough proof''':

Test somes cases:

Ɩ(2) = Ɩ(1+1) = Ɩ(1) + Ɩ(1) = 1 + 1 ≠ 2

Ɩ(3) = Ɩ(2 +1)= Ɩ(2) + Ɩ(1) = 1+ 1+ 1 ≠ 3

.
.
.

Ɩ(n) = 1 + ... + 1 (n times)

Ɩ(-3) = ?

Ɩ(-3 + 3) = Ɩ(-3) + Ɩ(3) ⇒ Ɩ(-3) = -Ɩ(3) = -(1+1+1)

''What about uniqueness?'' Simply put, we had not choice in the definition of Ɩ. All followed from the given properties.

At this point, we will be lazy and simply denote Ɩ(3) = 3_f [3 with subscript f]

∃ m≠0, m ∈ '''N''', Ɩ(m) =0

In which case, there is a smallest m<0, for which Ɩ(m)=0. '''m' is the characteristic of F.'' Denoted char(F).
Examples: char(F_2)=2, char(F_3)=3... but NOTE: char('''R''')=0

'''Thrm:''' If F is a field and char(F) >0, then char(F) is a prime number.

Proof: Suppose char(F) =m, m>0. Suppose also m is not prime: m=ts, t,s ∈ '''N'''.

Then, Ɩ(m) = 0 = Ɩ(t)∙Ɩ(s) ⇒ Ɩ(t)=0 or Ɩ(s)=0 by P12.

If Ɩ(t)=0 ⇒ t≧m ⇒ m=t, s=1 or likewise for Ɩ(s)=0, and m=s, t=1

In any factorization of m, one of the factors is m and the other is 1. So m is prime. ∎

== Complex number==

'''Abstraction, generalization, definition, examples, properties, dream, implications, realization = formalization, PROOF.'''

Consider that fact that in '''R''', ∄ x s.t. x^2 = -1

Dream: Add new number element 𝒊 to '''R''', so as to still get a field & 𝒊^2 = -1

Implications: By adding 𝒊, we must add 7𝒊, and 2+7𝒊, 3+4𝒊, (2+7𝒊)∙(3+4𝒊), (2+7𝒊)^-1, etc.

So, how do we define this field?

'''Definition'''

'''C''' = {(a,b): a,b ∈ '''R'''}
Also, 0 (of the field) = (0,0); 1( of the field) = (1,0)

Define addition: (a,b)+(c,d) = (a+c, b+d)

Define multification: (a,b)(c,d) = (ad-bd, ad+bc)

'''Theorems:'''

'''Thrm. 1.''' ('''C''', 0, 1, +, ∙) is a field.

'''Thrm. 2.''' ∃ 𝒊 ∈ '''C''' s.t. 𝒊^2 = -1

'''Thrm. 3.''' '''C''' contains '''R'''

Proof (1): Show that each of the field axioms holds for '''C'''.

Ex. F1(a): ƶ1 + ƶ2 = ƶ2 + ƶ1, where ƶ1 = (a1, b1) and ƶ2 = (a2, b2)

LHS: (a1,b1)+(a2,b2) = (a1+a2, b1+b2)

RHS: (a2,b2)+(a1,b1) = (a2+a1, b2+b1)

LHS=RHS by F1 of '''R'''

F1(b) and so on...

Proof (2):
Given a the field '''C''', define a map Ɩ: '''R''' → '''C''' with the properties (∀ m,n ∈ Z):

1) Ɩ(0) =0, Ɩ(1)=1

2) Ɩ(m+n) = Ɩ(m) +Ɩ(n)

3) Ɩ(mn) = Ɩ(m)∙Ɩ(n)

BUT, also define Ɩ(x)=(x,0). Thus, it follows. ∎

== Lecture 3, scanned notes upload by [[User:Starash|Starash]] ==

<gallery>
Image:12-240-0918-1.jpg|Page 1
Image:12-240-0918-2.jpg|Page 2
Image:12-240-0918-3.jpg|Page 3
</gallery>

12-240/Classnotes for Tuesday September 18

2012-09-19T02:47:41Z

Celtonmcgrath: /* Complex number */

{{12-240/Navigation}}
In this class, the professor continued with some more theorems of field and introduced definition and theorems of complex number.

== Various properties of fields ==
'''Thrm 1''': 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 (∀ m,n ∈ Z):

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.

'''Rough proof''':

Test somes cases:

Ɩ(2) = Ɩ(1+1) = Ɩ(1) + Ɩ(1) = 1 + 1 ≠ 2

Ɩ(3) = Ɩ(2 +1)= Ɩ(2) + Ɩ(1) = 1+ 1+ 1 ≠ 3

.
.
.

Ɩ(n) = 1 + ... + 1 (n times)

Ɩ(-3) = ?

Ɩ(-3 + 3) = Ɩ(-3) + Ɩ(3) ⇒ Ɩ(-3) = -Ɩ(3) = -(1+1+1)

''What about uniqueness?'' Simply put, we had not choice in the definition of Ɩ. All followed from the given properties.

At this point, we will be lazy and simply denote Ɩ(3) = 3_f [3 with subscript f]

∃ m≠0, m ∈ '''N''', Ɩ(m) =0

In which case, there is a smallest m<0, for which Ɩ(m)=0. '''m' is the characteristic of F.'' Denoted char(F).
Examples: char(F_2)=2, char(F_3)=3... but NOTE: char('''R''')=0

'''Thrm:''' If F is a field and char(F) >0, then char(F) is a prime number.

Proof: Suppoer char(F) =m, m>0. Suppose also m is not prime: m=ts, t,s ∈ '''N'''.

Then, Ɩ(m) = 0 = Ɩ(t)∙Ɩ(s) ⇒ Ɩ(t)=0 or Ɩ(s)=0 by P12.

If Ɩ(t)=0 ⇒ t≧m ⇒ m=t, s=1 or likewise for Ɩ(s)=0, and m=s, t=1

In any factorization of m, one of the factors is m and the other is 1. So m is prime. ∎

== Complex number==

'''Abstraction, generalization, definition, examples, properties, dream, implications, realization = formalization, PROOF.'''

Consider that fact that in '''R''', ∄ x s.t. x^2 = -1

Dream: Add new number element 𝒊 to '''R''', so as to still get a field & 𝒊^2 = -1

Implications: By adding 𝒊, we must add 7𝒊, and 2+7𝒊, 3+4𝒊, (2+7𝒊)∙(3+4𝒊), (2+7𝒊)^-1, etc.

So, how do we define this field?

'''Definition'''

'''C''' = {(a,b): a,b ∈ '''R'''}
Also, 0 (of the field) = (0,0); 1( of the field) = (1,0)

Define addition: (a,b)+(c,d) = (a+c, b+d)

Define multification: (a,b)(c,d) = (ad-bd, ad+bc)

'''Theorems:'''

'''Thrm. 1.''' ('''C''', 0, 1, +, ∙) is a field.

'''Thrm. 2.''' ∃ 𝒊 ∈ '''C''' s.t. 𝒊^2 = -1

'''Thrm. 3.''' '''C''' contains '''R'''

Proof (1): Show that each of the field axioms holds for '''C'''.

Ex. F1(a): ƶ1 + ƶ2 = ƶ2 + ƶ1, where ƶ1 = (a1, b1) and ƶ2 = (a2, b2)

LHS: (a1,b1)+(a2,b2) = (a1+a2, b1+b2)

RHS: (a2,b2)+(a1,b1) = (a2+a1, b2+b1)

LHS=RHS by F1 of '''R'''

F1(b) and so on...

Proof (2):

== Lecture 3, scanned notes upload by [[User:Starash|Starash]] ==

<gallery>
Image:12-240-0918-1.jpg|Page 1
Image:12-240-0918-2.jpg|Page 2
Image:12-240-0918-3.jpg|Page 3
</gallery>

12-240/Classnotes for Tuesday September 18

2012-09-19T02:44:28Z

Celtonmcgrath: /* Complex number */

{{12-240/Navigation}}

== Various properties of fields ==
'''Thrm 1''': 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 (∀ m,n ∈ Z):

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.

'''Rough proof''':

Test somes cases:

Ɩ(2) = Ɩ(1+1) = Ɩ(1) + Ɩ(1) = 1 + 1 ≠ 2

Ɩ(3) = Ɩ(2 +1)= Ɩ(2) + Ɩ(1) = 1+ 1+ 1 ≠ 3

.
.
.

Ɩ(n) = 1 + ... + 1 (n times)

Ɩ(-3) = ?

Ɩ(-3 + 3) = Ɩ(-3) + Ɩ(3) ⇒ Ɩ(-3) = -Ɩ(3) = -(1+1+1)

''What about uniqueness?'' Simply put, we had not choice in the definition of Ɩ. All followed from the given properties.

At this point, we will be lazy and simply denote Ɩ(3) = 3_f [3 with subscript f]

∃ m≠0, m ∈ '''N''', Ɩ(m) =0

In which case, there is a smallest m<0, for which Ɩ(m)=0. '''m' is the characteristic of F.'' Denoted char(F).
Examples: char(F_2)=2, char(F_3)=3... but NOTE: char('''R''')=0

'''Thrm:''' If F is a field and char(F) >0, then char(F) is a prime number.

Proof: Suppoer char(F) =m, m>0. Suppose also m is not prime: m=ts, t,s ∈ '''N'''.

Then, Ɩ(m) = 0 = Ɩ(t)∙Ɩ(s) ⇒ Ɩ(t)=0 or Ɩ(s)=0 by P12.

If Ɩ(t)=0 ⇒ t≧m ⇒ m=t, s=1 or likewise for Ɩ(s)=0, and m=s, t=1

In any factorization of m, one of the factors is m and the other is 1. So m is prime. ∎

== Complex number==

'''Abstraction, generalization, definition, examples, properties, dream, implications, realization = formalization, PROOF.'''

Consider that fact that in '''R''', ∄ x s.t. x^2 = -1

Dream: Add new number element 𝒊 to '''R''', so as to still get a field & 𝒊^2 = -1

Implications: By adding 𝒊, we must add 7𝒊, and 2+7𝒊, 3+4𝒊, (2+7𝒊)∙(3+4𝒊), (2+7𝒊)^-1, etc.

So, how do we define this field?

'''Definition'''

'''C''' = {(a,b): a,b ∈ '''R'''}
Also, 0 (of the field) = (0,0); 1( of the field) = (1,0)

Define addition: (a,b)+(c,d) = (a+c, b+d)

Define multification: (a,b)(c,d) = (ad-bd, ad+bc)

'''Theorems:'''

'''Thrm. 1.''' ('''C''', 0, 1, +, ∙) is a field.

'''Thrm. 2.''' ∃ 𝒊 ∈ '''C''' s.t. 𝒊^2 = -1

'''Thrm. 3.''' '''C''' contains '''R'''

Proof (1): Show that each of the field axioms holds for '''C'''.

Ex. F1(a): ƶ1 + ƶ2 = ƶ2 + ƶ1, where ƶ1 = (a1, b1) and ƶ2 = (a2, b2)

LHS: (a1,b1)+(a2,b2) = (a1+a2, b1+b2)

RHS: (a2,b2)+(a1,b1) = (a2+a1, b2+b1)

LHS=RHS by F1 of '''R'''

F1(b) and so on...

== Lecture 3, scanned notes upload by [[User:Starash|Starash]] ==

<gallery>
Image:12-240-0918-1.jpg|Page 1
Image:12-240-0918-2.jpg|Page 2
Image:12-240-0918-3.jpg|Page 3
</gallery>

12-240/Classnotes for Tuesday September 18

2012-09-19T02:44:00Z

Celtonmcgrath: /* Complex number */

{{12-240/Navigation}}

== Various properties of fields ==
'''Thrm 1''': 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 (∀ m,n ∈ Z):

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.

'''Rough proof''':

Test somes cases:

Ɩ(2) = Ɩ(1+1) = Ɩ(1) + Ɩ(1) = 1 + 1 ≠ 2

Ɩ(3) = Ɩ(2 +1)= Ɩ(2) + Ɩ(1) = 1+ 1+ 1 ≠ 3

.
.
.

Ɩ(n) = 1 + ... + 1 (n times)

Ɩ(-3) = ?

Ɩ(-3 + 3) = Ɩ(-3) + Ɩ(3) ⇒ Ɩ(-3) = -Ɩ(3) = -(1+1+1)

''What about uniqueness?'' Simply put, we had not choice in the definition of Ɩ. All followed from the given properties.

At this point, we will be lazy and simply denote Ɩ(3) = 3_f [3 with subscript f]

∃ m≠0, m ∈ '''N''', Ɩ(m) =0

In which case, there is a smallest m<0, for which Ɩ(m)=0. '''m' is the characteristic of F.'' Denoted char(F).
Examples: char(F_2)=2, char(F_3)=3... but NOTE: char('''R''')=0

'''Thrm:''' If F is a field and char(F) >0, then char(F) is a prime number.

Proof: Suppoer char(F) =m, m>0. Suppose also m is not prime: m=ts, t,s ∈ '''N'''.

Then, Ɩ(m) = 0 = Ɩ(t)∙Ɩ(s) ⇒ Ɩ(t)=0 or Ɩ(s)=0 by P12.

If Ɩ(t)=0 ⇒ t≧m ⇒ m=t, s=1 or likewise for Ɩ(s)=0, and m=s, t=1

In any factorization of m, one of the factors is m and the other is 1. So m is prime. ∎

== Complex number==

'''Abstraction, generalization, definition, examples, properties, dream, implications, realization = formalization, PROOF.'''

Consider that fact that in '''R''', ∄ x s.t. x^2 = -1

Dream: Add new number element 𝒊 to '''R''', so as to still get a field & 𝒊^2 = -1

Implications: By adding 𝒊, we must add 7𝒊, and 2+7𝒊, 3+4𝒊, (2+7𝒊)∙(3+4𝒊), (2+7𝒊)^-1, etc.

So, how do we define this field?

'''Definition'''

'''C''' = {(a,b): a,b ∈ '''R'''}
Also, 0 (of the field) = (0,0); 1( of the field) = (1,0)

Define addition: (a,b)+(c,d) = (a+c, b+d)

Define multification: (a,b)(c,d) = (ad-bd, ad+bc)

'''Theorems:'''

'''Thrm. 1.''' ('''C''', 0, 1, +, ∙) is a field.

'''Thrm. 2.''' ∃ 𝒊 ∈ '''C''' s.t. 𝒊^2 = -1

'''Thrm. 3.''' '''C''' contains '''R'''

Proof (1): Show that each of the field axioms holds for '''C'''.

Ex. F1(a): ƶ1 + ƶ2 = ƶ2 + ƶ1, where ƶ1 = (a1, b1) and ƶ2 = (a2, b2)

LHS: (a1,b1)+(a2,b2) = (a1+a2, b1+b2)

RHS: (a2,b2)+(a1,b1) = (a2+a1, b2+b1)

LHS=RHS by F1 of '''R'''

== Lecture 3, scanned notes upload by [[User:Starash|Starash]] ==

<gallery>
Image:12-240-0918-1.jpg|Page 1
Image:12-240-0918-2.jpg|Page 2
Image:12-240-0918-3.jpg|Page 3
</gallery>

12-240/Classnotes for Tuesday September 18

2012-09-19T02:42:54Z

Celtonmcgrath: /* Complex number */

{{12-240/Navigation}}

== Various properties of fields ==
'''Thrm 1''': 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 (∀ m,n ∈ Z):

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.

'''Rough proof''':

Test somes cases:

Ɩ(2) = Ɩ(1+1) = Ɩ(1) + Ɩ(1) = 1 + 1 ≠ 2

Ɩ(3) = Ɩ(2 +1)= Ɩ(2) + Ɩ(1) = 1+ 1+ 1 ≠ 3

.
.
.

Ɩ(n) = 1 + ... + 1 (n times)

Ɩ(-3) = ?

Ɩ(-3 + 3) = Ɩ(-3) + Ɩ(3) ⇒ Ɩ(-3) = -Ɩ(3) = -(1+1+1)

''What about uniqueness?'' Simply put, we had not choice in the definition of Ɩ. All followed from the given properties.

At this point, we will be lazy and simply denote Ɩ(3) = 3_f [3 with subscript f]

∃ m≠0, m ∈ '''N''', Ɩ(m) =0

In which case, there is a smallest m<0, for which Ɩ(m)=0. '''m' is the characteristic of F.'' Denoted char(F).
Examples: char(F_2)=2, char(F_3)=3... but NOTE: char('''R''')=0

'''Thrm:''' If F is a field and char(F) >0, then char(F) is a prime number.

Proof: Suppoer char(F) =m, m>0. Suppose also m is not prime: m=ts, t,s ∈ '''N'''.

Then, Ɩ(m) = 0 = Ɩ(t)∙Ɩ(s) ⇒ Ɩ(t)=0 or Ɩ(s)=0 by P12.

If Ɩ(t)=0 ⇒ t≧m ⇒ m=t, s=1 or likewise for Ɩ(s)=0, and m=s, t=1

In any factorization of m, one of the factors is m and the other is 1. So m is prime. ∎

== Complex number==

'''Abstraction, generalization, definition, examples, properties, dream, implications, realization = formalization, PROOF.'''

Consider that fact that in '''R''', ∄ x s.t. x^2 = -1

Dream: Add new number element 𝒊 to '''R''', so as to still get a field & 𝒊^2 = -1

Implications: By adding 𝒊, we must add 7𝒊, and 2+7𝒊, 3+4𝒊, (2+7𝒊)∙(3+4𝒊), (2+7𝒊)^-1, etc.

So, how do we define this field?

'''Definition'''

'''C''' = {(a,b): a,b ∈ '''R'''}
Also, 0 (of the field) = (0,0); 1( of the field) = (1,0)

Define addition: (a,b)+(c,d) = (a+c, b+d)

Define multification: (a,b)(c,d) = (ad-bd, ad+bc)

'''Theorems:'''

'''Thrm. 1.''' ('''C''', 0, 1, +, ∙) is a field.

'''Thrm. 2.''' ∃ 𝒊 ∈ '''C''' s.t. 𝒊^2 = -1

'''Thrm. 3.''' '''C''' contains '''R'''

Proof (1): Show that each of the field axioms hold for '''C'''.

Ex. F1(a): ƶ1 + ƶ2 = ƶ2 + ƶ1, where ƶ1 = (a1, b1) and ƶ2 = (a2, b2)
LHS: (a1,b1)+(a2,b2) = (a1+a2, b1+b2)
RHS: (a2,b2)+(a1,b1) = (a2+a1, b2+b1)
LHS=RHS by F1 of '''R'''

== Lecture 3, scanned notes upload by [[User:Starash|Starash]] ==

<gallery>
Image:12-240-0918-1.jpg|Page 1
Image:12-240-0918-2.jpg|Page 2
Image:12-240-0918-3.jpg|Page 3
</gallery>

12-240/Classnotes for Tuesday September 18

2012-09-19T02:36:58Z

Celtonmcgrath: /* Various properties of fields */

{{12-240/Navigation}}

== Various properties of fields ==
'''Thrm 1''': 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 (∀ m,n ∈ Z):

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.

'''Rough proof''':

Test somes cases:

Ɩ(2) = Ɩ(1+1) = Ɩ(1) + Ɩ(1) = 1 + 1 ≠ 2

Ɩ(3) = Ɩ(2 +1)= Ɩ(2) + Ɩ(1) = 1+ 1+ 1 ≠ 3

.
.
.

Ɩ(n) = 1 + ... + 1 (n times)

Ɩ(-3) = ?

Ɩ(-3 + 3) = Ɩ(-3) + Ɩ(3) ⇒ Ɩ(-3) = -Ɩ(3) = -(1+1+1)

''What about uniqueness?'' Simply put, we had not choice in the definition of Ɩ. All followed from the given properties.

At this point, we will be lazy and simply denote Ɩ(3) = 3_f [3 with subscript f]

∃ m≠0, m ∈ '''N''', Ɩ(m) =0

In which case, there is a smallest m<0, for which Ɩ(m)=0. '''m' is the characteristic of F.'' Denoted char(F).
Examples: char(F_2)=2, char(F_3)=3... but NOTE: char('''R''')=0

'''Thrm:''' If F is a field and char(F) >0, then char(F) is a prime number.

Proof: Suppoer char(F) =m, m>0. Suppose also m is not prime: m=ts, t,s ∈ '''N'''.

Then, Ɩ(m) = 0 = Ɩ(t)∙Ɩ(s) ⇒ Ɩ(t)=0 or Ɩ(s)=0 by P12.

If Ɩ(t)=0 ⇒ t≧m ⇒ m=t, s=1 or likewise for Ɩ(s)=0, and m=s, t=1

In any factorization of m, one of the factors is m and the other is 1. So m is prime. ∎
== Complex number==

'''Abstraction, generalization, definition, examples, properties, dream, implications, realization = formalization, PROOF.'''

In '''R''', ∄ x s.t. x^2 = -1

Dream: Add new number element 𝒊 to '''R''', so as to still get a field & 𝒊^2 = -1

Implications: By adding 𝒊, we must add 7𝒊, and 2+7𝒊, 3+4𝒊, (2+7𝒊)∙(3+4𝒊), (2+7𝒊)^-1, etc.

So, how do we define this field?

'''C''' = {(a,b): a,b ∈ '''R'''}
Also, 0 (of the field) = (0,0); 1 = (1,0)
Define +: (a,b)+(c,d) = (a+c, b+d)
Define ∙: (a,b)∙(c,d) = (ad-bd, ad+bc)

'''Thrm. 1.''' ('''C''', 0, 1, +, ∙) is a field.

'''Thrm. 2.''' ∃ 𝒊 ∈ '''C''' s.t. 𝒊^2 = -1

'''Thrm. 3.''' '''C''' contains '''R'''

Proof (1): Show that each of the field axioms hold for '''C'''.

== Lecture 3, scanned notes upload by [[User:Starash|Starash]] ==

<gallery>
Image:12-240-0918-1.jpg|Page 1
Image:12-240-0918-2.jpg|Page 2
Image:12-240-0918-3.jpg|Page 3
</gallery>

12-240/Classnotes for Tuesday September 18

2012-09-19T02:34:49Z

Celtonmcgrath: /* Various properties of fields */

{{12-240/Navigation}}

== Various properties of fields ==
'''Thrm 1''': 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 (∀ m,n ∈ Z):

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.

'''Rough proof''':

Test somes cases:

Ɩ(2) = Ɩ(1+1) = Ɩ(1) + Ɩ(1) = 1 + 1 ≠ 2

Ɩ(3) = Ɩ(2 +1)= Ɩ(2) + Ɩ(1) = 1+ 1+ 1 ≠ 3

.
.
.

Ɩ(n) = 1 + ... + 1 (n times)

Ɩ(-3) = ?

Ɩ(-3 + 3) = Ɩ(-3) + Ɩ(3) ⇒ Ɩ(-3) = -Ɩ(3) = -(1+1+1)

''What about uniqueness?'' Simply put, we had not choice in the definition of Ɩ. All followed from the given properties.

At this point, we will be lazy and simply denote Ɩ(3) = 3_f [3 with subscript f]

∃ m≠0, m ∈ '''N''', Ɩ(m) =0

In which case, there is a smallest m<0, for which Ɩ(m)=0. '''m' is the characteristic of F.'' Denoted char(F).
Examples: char(F_2)=2, char(F_3)=3... but NOTE: char('''R''')=0

'''Thrm:''' If F is a field and char(F) >0, then char(F) is a prime number.

Proof: Suppoer char(F) =m, m>0. Suppose also m is not prime: m=ts, t,s ∈ '''N'''.

Then, Ɩ(m) = 0 = Ɩ(t)∙Ɩ(s) ⇒ Ɩ(t)=0 or Ɩ(s)=0 by P12.

If Ɩ(t)=0 ⇒ t≧m ⇒ m=t, s=1 or likewise for Ɩ(s)=0, and m=s, t=1

In any factorization of m, one of the factors is m and the other is 1. So m is prime. ∎

'''Abstraction, generalization, definition, examples, properties, dream, implications, realization = formalization, PROOF.'''

In '''R''', ∄ x s.t. x^2 = -1

Dream: Add new number element 𝒊 to '''R''', so as to still get a field & 𝒊^2 = -1

Implications: By adding 𝒊, we must add 7𝒊, and 2+7𝒊, 3+4𝒊, (2+7𝒊)∙(3+4𝒊), (2+7𝒊)^-1, etc.

So, how do we define this field?

'''C''' = {(a,b): a,b ∈ '''R'''}
Also, 0 (of the field) = (0,0); 1 = (1,0)
Define +: (a,b)+(c,d) = (a+c, b+d)
Define ∙: (a,b)∙(c,d) = (ad-bd, ad+bc)

'''Thrm. 1.''' ('''C''', 0, 1, +, ∙) is a field.
'''Thrm. 2.''' ∃ 𝒊 ∈ '''C''' s.t. 𝒊^2 = -1
'''Thrm. 3.''' '''C''' contains '''R'''

== Lecture 3, scanned notes upload by [[User:Starash|Starash]] ==

<gallery>
Image:12-240-0918-1.jpg|Page 1
Image:12-240-0918-2.jpg|Page 2
Image:12-240-0918-3.jpg|Page 3
</gallery>

12-240/Classnotes for Tuesday September 18

2012-09-19T02:31:34Z

Celtonmcgrath: /* Various properties of fields */

{{12-240/Navigation}}

== Various properties of fields ==
'''Thrm 1''': 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 (∀ m,n ∈ Z):

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.

'''Rough proof''':

Test somes cases:

Ɩ(2) = Ɩ(1+1) = Ɩ(1) + Ɩ(1) = 1 + 1 ≠ 2

Ɩ(3) = Ɩ(2 +1)= Ɩ(2) + Ɩ(1) = 1+ 1+ 1 ≠ 3

.
.
.

Ɩ(n) = 1 + ... + 1 (n times)

Ɩ(-3) = ?

Ɩ(-3 + 3) = Ɩ(-3) + Ɩ(3) ⇒ Ɩ(-3) = -Ɩ(3) = -(1+1+1)

''What about uniqueness?'' Simply put, we had not choice in the definition of Ɩ. All followed from the given properties.

At this point, we will be lazy and simply denote Ɩ(3) = 3_f [3 with subscript f]

∃ m≠0, m ∈ '''N''', Ɩ(m) =0

In which case, there is a smallest m<0, for which Ɩ(m)=0. '''m' is the characteristic of F.'' Denoted char(F).
Examples: char(F_2)=2, char(F_3)=3... but NOTE: char('''R''')=0

'''Thrm:''' If F is a field and char(F) >0, then char(F) is a prime number.

Proof: Suppoer char(F) =m, m>0. Suppose also m is not prime: m=ts, t,s ∈ '''N'''.

Then, Ɩ(m) = 0 = Ɩ(t)∙Ɩ(s) ⇒ Ɩ(t)=0 or Ɩ(s)=0 by P12.

If Ɩ(t)=0 ⇒ t≧m ⇒ m=t, s=1 or likewise for Ɩ(s)=0, and m=s, t=1

In any factorization of m, one of the factors is m and the other is 1. So m is prime. ∎

'''Abstraction, generalization, definition, examples, properties, dream, implications, realization = formalization, PROOF.'''

In '''R''', ∄ x s.t. x^2 = -1

Dream: Add new number element 𝒊 to '''R''', so as to still get a field & 𝒊^2 = -1

Implications: By adding 𝒊, we must add 7𝒊, and 2+7𝒊, 3+4𝒊, (2+7𝒊)∙(3+4𝒊), (2+7𝒊)^-1, etc.

So, how do we define this field?

'''C''' = {(a,b): a,b ∈ '''R'''}
Also, 0 (of the field) = (0,0); 1 = (1,0)
Define +: (a,b)+(c,d) = (a+c, b+d)
Define ∙: (a,b)∙(c,d) = (ad-bd, ad+bc)

'''Thrm. 1.''' ('''C''', 0, 1, +, ∙) is a field.

== Lecture 3, scanned notes upload by [[User:Starash|Starash]] ==

<gallery>
Image:12-240-0918-1.jpg|Page 1
Image:12-240-0918-2.jpg|Page 2
Image:12-240-0918-3.jpg|Page 3
</gallery>

12-240/Classnotes for Tuesday September 18

2012-09-19T02:23:40Z

Celtonmcgrath: /* Various properties of fields */

12-240/Classnotes for Tuesday September 18

2012-09-19T02:18:49Z

Celtonmcgrath:

12-240/Classnotes for Tuesday September 18

2012-09-19T02:12:27Z

Celtonmcgrath:

12-240/Classnotes for Tuesday September 18

2012-09-19T02:11:57Z

Celtonmcgrath:

12-240/Classnotes for Tuesday September 18

2012-09-19T02:03:39Z

Celtonmcgrath: /* Recap: */

{{12-240/Navigation}}

== Various properties of fields ==
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.

12-240/Classnotes for Tuesday September 18

2012-09-19T01:57:21Z

Celtonmcgrath: /* Recap: */

{{12-240/Navigation}}

== Recap: ==
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

12-240/Classnotes for Tuesday September 18

2012-09-19T01:56:14Z

Celtonmcgrath:

{{12-240/Navigation}}

== Recap: ==
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

12-240/Classnotes for Tuesday September 18

2012-09-19T01:55:44Z

Celtonmcgrath:

{{12-240/Navigation}}

Recap:
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

12-240/Classnotes for Tuesday September 18

2012-09-19T01:48:33Z

Celtonmcgrath:

{{12-240/Navigation}}

Recap:
Thrm: In a field F:
1. a+b = c+b

12-240/Classnotes for Tuesday September 18

2012-09-19T01:45:55Z

Celtonmcgrath:

12-240/Navigation

2012-09-19T01:44:19Z

Celtonmcgrath:

<noinclude>Back to [[12-240]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 10
|[[12-240/About This Class|About This Class]], [[12-240/Classnotes for Tuesday September 11|Tuesday]], [[12-240/Classnotes for Thursday September 13|Thursday]]
|- align=left
|align=center|2
|Sep 17
|[[12-240/Homework Assignment 1|HW1]], [[12-240/Classnotes for Tuesday September 18|Tuesday]]
|- align=left
|align=center|3
|Sep 24
|HW2, [[12-240/Class Photo|Class Photo]]
|- align=left
|align=center|4
|Oct 1
|HW3
|- align=left
|align=center|5
|Oct 8
|HW4
|- align=left
|align=center|6
|Oct 15
|
|- align=left
|align=center|7
|Oct 22
|HW5, [[12-240/Term Test|Term Test]] on Thursday.
|- align=left
|align=center|8
|Oct 29
|HW6; Nov 4 is the last day to drop this class
|- align=left
|align=center|9
|Nov 5
|
|- align=left
|align=center|10
|Nov 12
|Monday-Tuesday is UofT November break, HW7 on web Wednesday
|- align=left
|align=center|11
|Nov 19
|HW8
|- align=left
|align=center|12
|Nov 26
|HW9
|- align=left
|align=center|13
|Dec 3
|UofT Fall Semester ends Wednesday
|- align=left
|align=center|F1
|Dec 10
|Finals
|- align=left
|align=center|F2
|Dec 17
|Finals
|- align=left
|colspan=3 align=center|[[12-240/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:12-240-ClassPhoto.jpg|310px]] [[12-240/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:12-240-Splash.png|310px]]
|}