14-240/Classnotes for Monday September 22

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.

• $VS_{1}:\forall x,y\in V,x+y=y+x$.
• $VS_{2}:\forall x,y,z\in V,x+(y+z)=(x+y)+z$.
• $VS_{3}:\forall x\in V,x+0=x$.
• $VS_{4}:\forall x\in V,\exists y\in V,x+y=0$.
• $VS_{5}:\forall x\in V,1\times x=x$.
• $VS_{6}:\forall a,b\in F,\forall x\in V,a(bx)=(ab)x$.
• $VS_{7}:\forall a\in F,\forall x,y\in V,a(x+y)=ax+ay$.
• $VS_{8}:\forall a,b\in F,\forall x\in V,(a+b)x=ax+bx$.