Drorbn - User contributions [en]

12-240/Classnotes for Thursday October 18

2012-10-19T12:31:03Z

Starash:

{{12-240/Navigation}}
===Riddle Along===
The game of 15 is played as follows. Two players alternate choosing cards numbered between 1 and 9, with repetitions forbidden, so that the game ends at most after 9 moves (or four and a half rounds). The first player to have within her/his cards a set of precisely 3 cards that add up to 15 wins.

Does this game has a winning strategy? What is it? Who wins, the first to move or the second? Why am I asking this question at this particular time?

[[Image:12-240-DeckOfCards.png|center]]

See also [https://media.library.utoronto.ca/play.php?DJ6CPFxByy2J&id=8503 a video] and the [https://cmc.math.ca/home/videos/game-of-15-and-isomorphisms/ transcript] of that video.

{{12-240:Dror/Students Divider}}
== Theorems ==
1. If G generates, |G| <math>\ge \!\,</math> n and G contains a basis, |G|=n then G is a basis

2. If L is linearly independent, |L| <math>\le \!\,</math> n and L can be extended to be a basis. |L|=n => L is a basis.

3.W <math>\subset \!\,</math> V a subspace then W is finite dimensioned and dim W <math>\le \!\,</math> dim V

If dim W = dim V, then V = W
If dim W < dim V, then any basis of W can be extended to be a basis of V

Proof of W is finite dimensioned:

Let L be a linearly independent subset of W which is of maximal size.

Fact about '''N'''
: Every subset A of '''N''', which is:

1. Non empty

2. Bounded : <math>\exist \!\,</math> N <math>\in \!\,</math> '''N''', <math>\forall \!\,</math> a <math>\in \!\,</math> A, a <math>\le \!\,</math> N

has a maximal element: an element m <math>\in \!\,</math> A, <math>\forall\!\,</math> a <math>\in \!\,</math> A, a <math>\le \!\,</math> m ( m + 1 <math>\notin \!\,</math> A )

== class note ==

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Image:12-240-Oct-15-Page-1.jpg |page1
Image:12-240-Oct-15-Page-2.jpg |page2
Image:12-240-Oct-15-Page-3.jpg |page3
</gallery>

== lecture note on oct 18, uploaded by [[User:starash|starash]]==

<gallery>
Image:12-240-1018-1.jpg |page1
Image:12-240-1018-2.jpg |page2
</gallery>

File:12-240-1018-2.jpg

2012-10-19T12:29:07Z

Starash:

File:12-240-1018-1.jpg

2012-10-19T12:29:05Z

Starash:

12-240/Classnotes for Thursday October 4

2012-10-05T00:17:56Z

Starash:

== 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)

== Lecture notes scanned by [[User:starash|starash]] ==
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2012-10-05T00:16:35Z

Starash:

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2012-10-05T00:16:32Z

Starash:

12-240/Classnotes for Thursday September 27

2012-09-28T04:22:43Z

Starash:

{{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

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

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12-240/Navigation

2012-09-28T00:31:38Z

Starash:

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12-240/Navigation

2012-09-28T00:29:57Z

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2012-09-28T00:28:25Z

Starash:

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2012-09-28T00:28:11Z

Starash:

12-240/Classnotes for Thursday September 20

2012-09-22T00:39:15Z

Starash:

In this class, the professor completes the lecture about complex number and then introduces vector space.

== Complex number ==

'''Definition and properties'''

''' C'''={(a,b): a, b <math>\in\!\,</math> '''R'''}

1 ( of '''C''') = (1,0); 0 ( of '''C''')= (0,0)

i=(0,1)

(a,b)+(c,d)=(a+c,b+d); (a,b)x(c,d)=(ac-bd,ad+bc)

i^2=-1

'''C''' contains '''R''' as {(a,0)} ( actually, this is not the set of real number but a copy of it )

'''Political statement'''

The professor totally disagrees with the name complex number because, indeed, the construction of '''C''' is much easier than the construction of '''R'''.

From '''Q''' ( set of quotient number) we can also construct a set containing i, which has a square equal to -1, and this construction is considered relatively easy
Meanwhile, from '''Q''', the construction of R is extremely hard and hence, of course, much more complicated.

'''Interpretation of complex number'''

Since complex number has two elements, it can be express in geometric form in coordinate plane

== Scan of class note ==

[[Image:IMG.jpg]]
[[Image:IMG2.jpg]]
[[Image:IMG3.jpg]]

== Lecture 4, scanned notes upload by [[User:Starash|Starash]] ==

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2012-09-22T00:37:54Z

Starash:

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2012-09-22T00:37:54Z

Starash:

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2012-09-22T00:37:48Z

Starash:

12-240/Classnotes for Tuesday September 18

2012-09-19T02:14:40Z

Starash:

{{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]

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

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2012-09-18T19:23:51Z

Starash:

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2012-09-18T19:23:37Z

Starash:

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2012-09-18T19:23:35Z

Starash:

12-240/Classnotes for Tuesday September 11

2012-09-14T12:34:50Z

Starash:

{{12-240/Navigation}}

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

== Practical Side ==

1.
Solving complicated systems of equations, such as:

:<math> 5x_1 - 2x_2 + x_3 = 9\!</math>
:<math>x_1 + x_2 - x_3 = -2\!</math>
:<math>2x_1 + 9x_2 - 3x_3 = -4\!</math>

2.
We can turn the above into a matrix!
:<math>
\begin{pmatrix}
5 & -2 & 1 \\
-1 & 1 & -1 \\
2 & 9 & -3
\end{pmatrix} = A
</math>

== Theory Side ==

3.
"The world doesn't come with coordinates."
We will learn to do all of this in a coordinate-free way.

4.
We'll learn to do all of this over other sets of numbers and fields.

== Hidden Agenda ==

5.
We'll learn the process of pure mathematics by doing it.
We'll learn about:
*Abstraction
*Generalization
*Definitions
*Theorems
*Proofs
*Notation
*Logic

----

A number system has specific properties of the real numbers.

== Real Numbers ==

A set, <math>\mathbb{R}\!</math>, with:
*Two binary operations, addition and multiplication.
*Two special elements, 0 and 1.

The real numbers have some special properties:

=== Commutative Laws ===
<math>\mathbb{R}1</math>
:<math>\forall\ a, b\ \epsilon\ \mathbb{R} \quad a+b = b+a\!</math>
:<math>\forall\ a, b\ \epsilon\ \mathbb{R} \quad ab = ba\!</math>

=== Associative Laws ===
<math>\mathbb{R}2</math>
:<math>\forall\ a, b, c\ \epsilon\ \mathbb{R} \quad (a + b) + c = a + (b + c)\!</math>
:<math>\forall\ a, b, c\ \epsilon\ \mathbb{R} \quad (ab) \cdot c = a \cdot (bc)\!</math>

=== Existence of "Units" ===
<math>\mathbb{R}3</math>
:<math>\forall\ a\ \epsilon\ \mathbb{R} \quad a + 0 = a\!</math>
:<math>\forall\ a\ \epsilon\ \mathbb{R} \quad a \cdot 1 = a\!</math>

=== Existence of Negatives/Inverses ===
<math>\mathbb{R}4</math>
:<math>\forall\ a\ \epsilon\ \mathbb{R}\ \exists\ b\ \epsilon\ \mathbb{R} \quad a + b = 0\!</math>
:<math>\forall\ a\ \epsilon\ \mathbb{R}\ \exists\ b\ \epsilon\ \mathbb{R} \quad a \cdot b = 1\!</math>

=== Distributive Law ===
<math>\mathbb{R}5</math>
:<math>\forall\ a, b, c\ \epsilon\ \mathbb{R} \quad (a+b) \cdot c = ac + bc\!</math>

==== An example of a property that follows from the earlier ones: ====
:<math>a^2 - b^2 = (a + b)(a - b)\!</math>
We can define subtraction and squaring from the properties covered above.

==== An example of a property that does not follow from the earlier ones: ====
The existence of square roots:

:<math>\forall\ a\ \exists\ b\ \quad b^2 = a\ or\ b^2 = -a\!</math>
We can construct a set that has all of the 5 properties described above, but for which this property does not follow.

This set is the rational numbers.

There is a rational number <math>a\!</math> where there is no <math>b</math> in the set.

This is because<math>\sqrt{2}</math> is irrational.

== Fields ==

The properties we have been discussing aren't restricted to only the real numbers.

They are also properties of:
*Rational numbers
*Complex numbers
*Others

Let's construct an abstract universe where these properties hold.

Definition: Field
*A field is a set, <math>\mathbb{F}</math>, with:
**Two binary operations, addition and multiplication.
**Two special elements, 0 and 1, where 0 does not equal 1.
**All of the above mentioned properties hold.

Now, instead of speaking of <math>\mathbb{R}1,\ \mathbb{R}2,\ \mathbb{R}3,\ \mathbb{R}4,\ \mathbb{R}5</math>, we can speak of <math>\mathbb{F}1,\ \mathbb{F}2,\ \mathbb{F}3,\ \mathbb{F}4,\ \mathbb{F}5</math>.

We have abstracted!

== Examples of Fields ==
*Take <math>\mathbb{F} = \mathbb{R}</math>

*Take <math>\mathbb{F} = \mathbb{Q}</math> (Rational numbers)

*The complex numbers. <math>\mathbb{C} = \lbrace a + bi \quad a, b\ \epsilon\ \mathbb{R} \rbrace</math>

The above fields have an infinite number of elements. We can also have finite fields:

*<math>\mathbb{F} = \mathbb{F}_2 = \mathbb{Z}/2 = \lbrace 0, 1 \rbrace</math>
**There are only 2 elements.
**You can think of 0 as even and 1 as odd, which will help you construct the tables below.
**You can also think of the results below as the remainder of the operations when divided by 2. (mod 2)

::{| border="1"
! scope="col" | +
! scope="col" | 0
! scope="col" | 1
|-
! scope="row" | 0
| 0 || 1
|-
! scope="row" | 1
| 1
| 0
|}

::{| border="1"
! scope="col" | x
! scope="col" | 0
! scope="col" | 1
|-
! scope="row" | 0
| 0 || 0
|-
! scope="row" | 1
| 0
| 1
|}

*Question: Are addition and multiplication defined here only arbitrary? Can we define many other ways to add or multiply, for a set, as long as the result satisfies F1-5 to show that F is indeed a field?
** Answer by [[User:Drorbn|Drorbn]] 16:51, 13 September 2012 (EDT): A "field" is a set with two operations and 0 and 1 so that some properties hold. In principle, the same set can be made into a field in many different ways - by choosing different operations (so long as they satisfy F1-5). In practice though, there is essentially only one field with 5 elements (but I the word "essentially" here requires an explanation). Many other sets can be considered as fields in many "genuinely different" ways, depending in how you choose the operations <math>+</math> and <math>\times</math>.

*<math>\mathbb{F} = \mathbb{F}_3 = \mathbb{Z}/3 = \lbrace 0, 1, 2 \rbrace</math>
::{| border="1"
! scope="col" | +
! scope="col" | 0
! scope="col" | 1
! scope="col" | 2
|-
! scope="row" | 0
| 0 || 1 || 2
|-
! scope="row" | 1
| 1
| 2
| 0
|-
! scope="row" | 2
| 2
| 0
| 1
|}

::{| border="1"
! scope="col" | x
! scope="col" | 0
! scope="col" | 1
! scope="col" | 2
|-
! scope="row" | 0
| 0 || 0 || 0
|-
! scope="row" | 1
| 0
| 1
| 2
|-
! scope="row" | 2
| 0
| 2
| 1
|}

*<math>\mathbb{F} = \mathbb{F}_5 = \mathbb{Z}/5 = \lbrace 0, 1, 2, 3, 4 \rbrace</math>
**Not going to bother making the tables here.

*<math>\mathbb{F}_4</math> is '''not a field.'''
**It does not have the property <math>\mathbb{R}5</math>.
:::<math>2 \cdot 0 = 0</math>
:::<math>2 \cdot 1 = 2</math>
:::<math>2 \cdot 2 = 0</math>
:::<math>2 \cdot 3 = 2</math>
:::We never got a 1.

*If the subscript is a prime number, it is a field.

----

Theorem:

1.

:Let F be a field.
:<math>\forall a, b, c\ \epsilon\ \mathbb{F} \quad a+b = c+b</math>
::"Cancellation Lemma"
:<math>\Rightarrow a = c</math>

2.

:<math>ab = cb, b \ne 0</math>
:<math>\Rightarrow a = c</math>

We'll cover 3-11 next class!

Proof of 1:

:Let <math>a, b, c\ \epsilon\ \mathbb{F}</math>
:by <math>\mathbb{F} 4\ \exists\ d\ \epsilon\ \mathbb{F} \quad b+d = 0</math>
:so with this d, <math>a+b = c+b\!</math>
:and so <math>(a+b)+d = (c+b)+d\!</math>
:so by <math>\mathbb{F} 2</math>, <math>a+(b+d) = c+(b+d)\!</math>
:so <math>a+0 = c+0\!</math>
:so by <math>\mathbb{F} 3 \quad a = c\!</math>
:<math>\Box</math>

==Scanned Notes by [[User:Sina.zoghi|Sina.zoghi]]==
[[User:Sina.zoghi|Sina.zoghi]] - Thanks for improving on the previously-uploaded scans - though there is still too much "white space" around each page. It is probably not worth your while to fix it for these scans, but it is something to keep in mind for later ones. [[User:Drorbn|Drorbn]] 10:50, 13 September 2012 (EDT)

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== Lecture notes upload by [[User:Starash|Starash]] ==
Another upload of lecture 1 notes.
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12-240/Classnotes for Thursday September 13

2012-09-14T01:21:42Z

Starash:

{{12-240/Navigation}}
In the second day of the class, the professor continues on the definition of a field.

== Definition of a field ==

Combined with a part from the first class, we have a complete definition as follow:

A field is a set "F' with two binary operations +,x defind on it, and two special elements 0 ≠ 1 such that

'''F1:''' commutative law

<math>\forall \!\,</math> a, b <math>\in \!\,</math> F: a+b=b+a and a.b=b.a

'''F2:''' associative law

<math>\forall \!\,</math> a, b, c <math>\in \!\,</math> F: (a+b)+c=a+(b+c) and (a.b).c= a.(b.c)

'''F3:''' the existence of identity elements

<math>\forall \!\,</math> a <math>\in \!\,</math> F, a+o=a and a.1=a

'''F4:''' existence of inverses

<math>\forall \!\,</math> a <math>\in \!\,</math> F ,<math> \exists \!\, c, d \in \!\ </math> F such that a+c=o and a.d=1

'''F5:''' contributive law

<math>\forall \!\,</math> a, b, c <math>\in \!\,</math> F, a.(b+c)=a.b + a.c
== Examples ==

== Lecture Notes, upload by [[User:Starash|Starash]] ==

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2012-09-14T01:14:51Z

Starash:

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2012-09-14T01:13:58Z

Starash:

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2012-09-13T12:37:34Z

Starash:

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2012-09-13T12:36:39Z

Starash:

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2012-09-13T12:34:21Z

Starash: