14-240/Classnotes for Monday September 22

DBN: Classes: 2014-15: 14-240 / Navigation

#	Week of...	Notes and Links
Welcome to Math 240! (additions to this web site no longer count towards good deed points)
1	Sep 8	About This Class, What is this class about? (PDF, HTML), Monday, Wednesday
2	Sep 15	HW1, Monday, Wednesday, TheComplexField.pdf,HW1_solutions.pdf
3	Sep 22	HW2, Class Photo, Monday, Wednesday, HW2_solutions.pdf
4	Sep 29	HW3, Wednesday, Tutorial, HW3_solutions.pdf
5	Oct 6	HW4, Monday, Wednesday, Tutorial, HW4_solutions.pdf
6	Oct 13	No Monday class (Thanksgiving), Wednesday, Tutorial
7	Oct 20	HW5, Term Test at tutorials on Tuesday, Wednesday
8	Oct 27	HW6, Monday, Why LinAlg?, Wednesday, Tutorial
9	Nov 3	Monday is the last day to drop this class, HW7, Monday, Wednesday, Tutorial
10	Nov 10	HW8, Monday, Tutorial
11	Nov 17	Monday-Tuesday is UofT November break
12	Nov 24	HW9
13	Dec 1	Wednesday is a "makeup Monday"! End-of-Course Schedule, Tutorial
F	Dec 8	The Final Exam
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Polar coordinates:

$r\times e^{i\theta }=r\times cos\theta +i\times rsin\theta$
$r_{1}\times e^{i\theta _{2}}=r_{1}\times (cos\theta +sin\theta$

The Fundamantal Theorem of Algebra: $a_{n}\times z^{n}+a_{n}-1\times z^{n-1}+\dots +a_{0}$ where $a_{i}\in C$ and $a_{i}!=0$ has a soluion $z\in C$ In particular, $z^{2}-1=0$ has a solution.

Forces can multiple by a "scalar"(number).

No "multiplication" of forces.

Definition of Vector Space: A "Vector Space" over a field F is a set V with a special element $O_{v}\in V$ and two binary operations:

$+:V\times V->V$
$\times :V\times V->V$

s.t.

$VS_{1}:\forall x,y\in V,x+y=y+x$ .
$VS_{2}:\forall x,y,z\in V,x+(y+z)=(x+y)+z$ .
$VS_{3}:\forall x\in V,x+0=x$ .
$VS_{4}:\forall x\in V,\exists y\in V,x+y=0$ .
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle VS_5 : \forall x \in V, 1 \times x = x} .
$VS_{6}:\forall a,b\in F,\forall x\in V,a(bx)=(ab)x$ .
$VS_{7}:\forall a\in F,\forall x,y\in V,a(x+y)=ax+ay$ .
$VS_{8}:\forall a,b\in F,\forall x\in V,(a+b)x=ax+bx$ .