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12-240/Classnotes for Tuesday September 11
2012-09-12T02:32:30Z
<p>Jonmorrispocock: </p>
<hr />
<div>{{12-240/Navigation}}<br />
<br />
In this course, we will be focusing on both a practical side and a theoretical side.<br />
<br />
== Practical Side ==<br />
<br />
1.<br />
Solving complicated systems of equations, such as:<br />
<br />
:<math> 5x_1 - 2x_2 + x_3 = 9\!</math><br />
:<math>x_1 + x_2 - x_3 = -2\!</math><br />
:<math>2x_1 + 9x_2 - 3x_3 = -4\!</math><br />
<br />
<br />
2.<br />
We can turn the above into a matrix!<br />
:<math><br />
\begin{pmatrix}<br />
5 & -2 & 1 \\<br />
-1 & 1 & -1 \\<br />
2 & 9 & -3<br />
\end{pmatrix} = A<br />
</math><br />
<br />
<br />
<br />
== Theory Side ==<br />
<br />
3.<br />
"The world doesn't come with coordinates."<br />
We will learn to do all of this in a coordinate-free way.<br />
<br />
4.<br />
We'll learn to do all of this over other sets of numbers and fields.<br />
<br />
<br />
== Hidden Agenda ==<br />
<br />
5.<br />
We'll learn the process of pure mathematics by doing it.<br />
We'll learn about:<br />
*Abstraction<br />
*Generalization<br />
*Definitions<br />
*Theorems<br />
*Proofs<br />
*Notation<br />
*Logic<br />
<br />
<br />
----<br />
<br />
<br />
A number system has specific properties of the real numbers.<br />
<br />
== Real Numbers ==<br />
<br />
A set, <math>\mathbb{R}\!</math>, with:<br />
*Two binary operations, addition and multiplication.<br />
*Two special elements, 0 and 1.<br />
<br />
The real numbers have some special properties:<br />
<br />
=== Commutative Laws ===<br />
<math>\mathbb{R}1</math><br />
:<math>\forall\ a, b\ \epsilon\ \mathbb{R} \quad a+b = b+a\!</math><br />
:<math>\forall\ a, b\ \epsilon\ \mathbb{R} \quad ab = ba\!</math><br />
<br />
=== Associative Laws ===<br />
<math>\mathbb{R}2</math><br />
:<math>\forall\ a, b, c\ \epsilon\ \mathbb{R} \quad (a + b) + c = a + (b + c)\!</math><br />
:<math>\forall\ a, b, c\ \epsilon\ \mathbb{R} \quad (ab) \cdot c = a \cdot (bc)\!</math><br />
<br />
=== Existence of "Units" ===<br />
<math>\mathbb{R}3</math><br />
:<math>\forall\ a\ \epsilon\ \mathbb{R} \quad a + 0 = a\!</math><br />
:<math>\forall\ a\ \epsilon\ \mathbb{R} \quad a \cdot 1 = a\!</math><br />
<br />
=== Existence of Negatives/Inverses ===<br />
<math>\mathbb{R}4</math><br />
:<math>\forall\ a\ \epsilon\ \mathbb{R}\ \exists\ b\ \epsilon\ \mathbb{R} \quad a + b = 0\!</math><br />
:<math>\forall\ a\ \epsilon\ \mathbb{R}\ \exists\ b\ \epsilon\ \mathbb{R} \quad a \cdot b = 1\!</math><br />
<br />
=== Distributive Law ===<br />
<math>\mathbb{R}5</math><br />
:<math>\forall\ a, b, c\ \epsilon\ \mathbb{R} \quad (a+b) \cdot c = ac + bc\!</math><br />
<br />
<br />
==== An example of a property that follows from the earlier ones: ====<br />
:<math>a^2 - b^2 = (a + b)(a - b)\!</math><br />
We can define subtraction and squaring from the properties covered above.<br />
<br />
<br />
==== An example of a property that does not follow from the earlier ones: ====<br />
The existence of square roots:<br />
<br />
:<math>\forall\ a\ \exists\ b\ \quad b^2 = a\ or\ b^2 = -a\!</math><br />
We can construct a set that has all of the 5 properties described above, but for which this property does not follow.<br />
<br />
This set is the rational numbers.<br />
<br />
There is a rational number <math>a\!</math> where there is no <math>b</math> in the set.<br />
<br />
This is because<math>\sqrt{2}</math> is irrational.<br />
<br />
<br />
== Fields ==<br />
<br />
The properties we have been discussing aren't restricted to only the real numbers.<br />
<br />
They are also properties of:<br />
*Rational numbers<br />
*Complex numbers<br />
*Others<br />
<br />
<br />
Let's construct an abstract universe where these properties hold.<br />
<br />
<br />
Definition: Field<br />
*A field is a set, <math>\mathbb{F}</math>, with:<br />
**Two binary operations, addition and multiplication.<br />
**Two special elements, 0 and 1, where 0 does not equal 1.<br />
**All of the above mentioned properties hold.<br />
<br />
<br />
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>.<br />
<br />
We have abstracted!<br />
<br />
<br />
== Examples of Fields ==<br />
*Take <math>\mathbb{F} = \mathbb{R}</math><br />
<br />
<br />
*Take <math>\mathbb{F} = \mathbb{Q}</math> (Rational numbers)<br />
<br />
<br />
*The complex numbers. <math>\mathbb{C} = \lbrace a + bi \quad a, b\ \epsilon\ \mathbb{R} \rbrace</math><br />
<br />
<br />
The above fields have an infinite number of elements. We can also have finite fields:<br />
<br />
<br />
*<math>\mathbb{F} = \mathbb{F}_2 = \mathbb{Z}/2 = \lbrace 0, 1 \rbrace</math><br />
**There are only 2 elements.<br />
**You can think of 0 as even and 1 as odd, which will help you construct the tables below.<br />
**You can also think of the results below as the remainder of the operations when divided by 2. (mod 2)<br />
<br />
::{| border="1"<br />
! scope="col" | + <br />
! scope="col" | 0<br />
! scope="col" | 1<br />
|-<br />
! scope="row" | 0<br />
| 0 || 1<br />
|-<br />
! scope="row" | 1<br />
| 1<br />
| 0<br />
|}<br />
<br />
::{| border="1"<br />
! scope="col" | x <br />
! scope="col" | 0<br />
! scope="col" | 1<br />
|-<br />
! scope="row" | 0<br />
| 0 || 0<br />
|-<br />
! scope="row" | 1<br />
| 0<br />
| 1<br />
|}<br />
<br />
<br />
*<math>\mathbb{F} = \mathbb{F}_3 = \mathbb{Z}/3 = \lbrace 0, 1, 2 \rbrace</math><br />
::{| border="1"<br />
! scope="col" | + <br />
! scope="col" | 0<br />
! scope="col" | 1<br />
! scope="col" | 2<br />
|-<br />
! scope="row" | 0<br />
| 0 || 1 || 2<br />
|-<br />
! scope="row" | 1<br />
| 1<br />
| 2<br />
| 0<br />
|-<br />
! scope="row" | 2<br />
| 2<br />
| 0<br />
| 1<br />
|}<br />
<br />
::{| border="1"<br />
! scope="col" | x <br />
! scope="col" | 0<br />
! scope="col" | 1<br />
! scope="col" | 2<br />
|-<br />
! scope="row" | 0<br />
| 0 || 0 || 0<br />
|-<br />
! scope="row" | 1<br />
| 0<br />
| 1<br />
| 2<br />
|-<br />
! scope="row" | 2<br />
| 0<br />
| 2<br />
| 1<br />
|}<br />
<br />
<br />
*<math>\mathbb{F} = \mathbb{F}_5 = \mathbb{Z}/5 = \lbrace 0, 1, 2, 3, 4 \rbrace</math><br />
**Not going to bother making the tables here.<br />
<br />
<br />
*<math>\mathbb{F}_4</math> is '''not a field.'''<br />
**It does not have the property <math>\mathbb{R}5</math>.<br />
:::<math>2 \cdot 0 = 0</math><br />
:::<math>2 \cdot 1 = 2</math><br />
:::<math>2 \cdot 2 = 0</math><br />
:::<math>2 \cdot 3 = 2</math><br />
:::We never got a 1.<br />
<br />
<br />
*If the subscript is a prime number, it is a field.<br />
<br />
<br />
----<br />
<br />
<br />
Theorem:<br />
<br />
1.<br />
<br />
:Let F be a field.<br />
:<math>\forall a, b, c\ \epsilon\ \mathbb{F} \quad a+b = c+b</math><br />
::"Cancellation Lemma"<br />
:<math>\Rightarrow a = c</math><br />
<br />
<br />
2.<br />
<br />
:<math>ab = cb, b \ne 0</math><br />
:<math>\Rightarrow a = c</math><br />
<br />
<br />
We'll cover 3-11 next class!<br />
<br />
<br />
Proof of 1:<br />
<br />
:Let <math>a, b, c\ \epsilon\ \mathbb{F}</math><br />
:by <math>\mathbb{F} 4\ \exists\ d\ \epsilon\ \mathbb{F} \quad b+d = 0</math><br />
:so with this d, <math>a+b = c+b\!</math><br />
:and so <math>(a+b)+d = (c+b)+d\!</math><br />
:so by <math>\mathbb{F} 2</math>, <math>a+(b+d) = c+(b+d)\!</math><br />
:so <math>a+0 = c+0\!</math><br />
:so by <math>\mathbb{F} 3 \quad a = c\!</math><br />
:<math>\Box</math></div>
Jonmorrispocock