12-240/Classnotes for Tuesday September 11

2012-09-12T02:32:30Z

Jonmorrispocock:

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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
|}

*<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>

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12-240/Classnotes for Tuesday September 11