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!


NSERC  CMS Math in Moscow Scholarships
The Natural Sciences and Engineering Research Council (NSERC) and the Canadian Mathematical Society (CMS) support scholarships at $9,000 each. Canadian students registered in a mathematics or computer science program are eligible.
The scholarships are to attend a semester at the small elite Moscow Independent University.
Math in Moscow Program http://www.mccme.ru/mathinmoscow/
Application details http://www.cms.math.ca/Scholarships/Moscow
For additional information please see your department or call the CMS at 6137332662.
Deadline September 30, 2009 to attend the Winter 2010 semester.
Some links
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.
Class notes for today
• Convention for today: $x,y,a,b,c,d,...$ will be real numbers; $z,w,u,v,...$ will be complex numbers
Dream: Find a field $\mathbb {C}$ that contains $\mathbb {R}$ and also contains an element $i$ such that $i^{2}=1$
Implications:
• $b\in \mathbb {R} \Rightarrow bi\in \mathbb {C}$
• $a\in \mathbb {R} \Rightarrow a+bi\in \mathbb {C}$
• $c,d\in \mathbb {R} \Rightarrow c+di\in \mathbb {C}$
• $\Rightarrow (a+bi)+(c+di)$ must be in $\mathbb {C}$
 $=(a+c)+(bi+di)$
 $=(a+c)+(b+d)i$
 $=e+fi$
$(a+bi)(c+di)=(a+c)+(b+d)i$
 $=a(c+di)+bi(c+di)$
 $=ac+adi+bic+bidi$
 $=ac+bdi^{2}+adi+bci$
 $=(acbd)+(ad+bc)i$
 $=e+fi$
 $0_{C}=0+0i$
 $1_{C}=1+0i$
 $(a+bi)+(c+di)=0+0i$
 $(a+bi)=(a)+(b)i$
 $a+bi\neq 0\Rightarrow (a,b)\neq 0$
• Find another element of $\mathbb {C}$, $x+yi$ such that $(a+bi)(x+yi)=(1+0i)$
 $(a+bi)(x+yi)=(axby)+(ay+bx)i=1+0i$
 $axby=1$ (1)
 $bx+ay=0$ (2)
 $a,b$ are given
 $x,y$ unknowns
• $b\times (1)$ $abxb^{2}y=b$
• $a\times (2)$ $abx+a^{2}y=0$
 $\Rightarrow a^{2}y+b^{2}y=b$
 $y={\frac {b}{a^{2}+b^{2}}}$
 $x={\frac {a}{a^{2}+b^{2}}}$
• (Note: We can divide since we assumed that $(a,b)\neq 0$
 $(a+bi)^{1}={\frac {a}{a^{2}+b^{2}}}+{\frac {b}{a^{2}+b^{2}}}i={\frac {abi}{a^{2}+b^{2}}}={\frac {\overline {a+bi}}{a+bi^{2}}}={\frac {\mbox{conjugate}}{\mbox{norm squared }}}$
Def: Let $\mathbb {C}$ be the set of all pairs of real numbers ${(a,b)}={a+bi}$
with $+:(a,b)+(c,d)=(a+c,b+d)$
 $(a+bi)+(c+di)=(a+c)+(b+d)i$
$\times :(a+bi)(c+di)=$...you know what
• 0 = you know what
• 1 = you know what
Theorem:
 $\mathbb {C}$ is a field
 $(0+1i)^{2}=(0,1)^{2}=i^{2}=1_{C}=(1,0)$
 $\mathbb {R} \rightarrow \mathbb {C}$ by $a\rightarrow a+0i$
Proof: $F_{1},F_{2},F_{3},...$
Example: $F_{5}$ (distributivity)
• Show that $z(u+v)=zu+zv$
Let $z=(a+bi)$
 $u=(c+di)$
 $v=(e+fi)$
When $a,b,c,d,e,f\in \mathbb {R}$
 $(a+bi)[(c+di)+(e+fi)]=(a+bi)(c+di)+(a+bi)(e+fi)=(acbd)+\ldots$
• NEXT WEEK: Complex numbers have geometric meaning, geometric interpretation (waves)