09-240/Classnotes for Thursday September 10

09-240/Navigation Panel   Additions to the MAT 240 web site no longer count towards good deed points # Week of... Notes and Links 1 Sep 7 Tue, About, Thu 2 Sep 14 Tue, HW1, HW1 Solution, Thu 3 Sep 21 Tue, HW2, HW2 Solution, Thu, Photo 4 Sep 28 Tue, HW3, HW3 Solution, Thu 5 Oct 5 Tue, HW4, HW4 Solution, Thu, 6 Oct 12 Tue, Thu 7 Oct 19 Tue, HW5, HW5 Solution, Term Test on Thu 8 Oct 26 Tue, Why LinAlg?, HW6, HW6 Solution, Thu 9 Nov 2 Tue, MIT LinAlg, Thu 10 Nov 9 Tue, HW7, HW7 Solution Thu 11 Nov 16 Tue, HW8, HW8 Solution, Thu 12 Nov 23 Tue, HW9, HW9 Solution, Thu 13 Nov 30 Tue, On the final, Thu S Dec 7 Office Hours F Dec 14 Final on Dec 16 To Do List The Algebra Song! Register of Good Deeds Misplaced Material Add your name / see who's in!

WARNING: The notes below, written for students and by students, are provided "as is", with absolutely no warranty. They can not be assumed to be complete, correct, reliable or relevant. If you don't like them, don't read them. It is a bad idea to stop taking your own notes thinking that these notes can be a total replacement - there's nothing like one's own handwriting! Visit this pages' history tab to see who added what and when.

Date: Thurs. Sept. 10, 2009

• Professor's name: Dror Bar-Natan

• Solve systems of equations

$5x_{1}-2x_{2}+x_{3}=9$

$-x_{1}+x_{2}-x_{3}=2$

$2x_{1}+9x_{2}-3x_{3}=-4$

• how? when? one/many?

• This describes the small-scale behaviour of almost everything that has a mathematical description.

1. A matrix is a square or rectangular array of numbers.

${\begin{pmatrix}5&-2&1\\-1&1&-1\\2&9&-3\end{pmatrix}}$

• we will learn addition, multiplication, and powers of matrices

$\mathbf {A} ={\begin{pmatrix}5&-2&1\\-1&1&-1\\2&9&-3\end{pmatrix}},\mathbf {B} =\cdots$

${\begin{pmatrix}5&-2&1\\-1&1&-1\\2&9&-3\end{pmatrix}}+\mathbf {B}$

$\mathbf {AB} \neq \mathbf {BA}$

$\mathbf {A} ^{2009}$

• describes the approximate long-term behaviour of almost anything...

• Do all this without choosing coordinates.

2. Do everything over other “systems of numbers”

1. real numbers
2. rational numbers
3. complex numbers (things like alternating current, circuit)
4. {0,1} (binary, computer science)

3. Hidden Agenda

• Learn the basic pure-math processes of: abstraction, generalizations, definitions, theorems, proofs, notation logic

4. Administration

• can add things to wiki (so long as relevant to course material)

• any page added to wiki must start with 09-240- or 09-240/

• HW assigned on Tuesday, due in tutorial 9 days later.

• HW graded and returned by following tutorial

5. Classwork done today

• The Real Numbers: a set $\mathbb {R}$ with two binary operations $\,\!+$, $\times$(2 inputs, one output) and also with two distinguished elements $0,1\epsilon \mathbb {R}$ with the following properties:

R1 $\forall a,b$

1. $\,\!a+b=b+a$
2. $a\cdot b=b\cdot a$

Aside: The $\perp$ character used for additon:

• Prof. Dror asked why $+$ is sometimes written as $\perp$?

• This is a Jewish tradition that dates back to at least the 19th century, and is still used today in Israeli elementary schools. It avoids the writing of the $+$ symbol, which resembles a Christian cross. (reference: http://en.wikipedia.org/wiki/Plus_and_minus_signs#Alternative_plus_sign)