# 14-240/Classnotes for Monday September 22

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Polar coordinates:

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

The Fundamantal Theorem of Algebra: ${\displaystyle a_{n}\times z^{n}+a_{n}-1\times z^{n-1}+\dots +a_{0}}$ where ${\displaystyle a_{i}\in C}$and${\displaystyle a_{i}!=0}$ has a soluion ${\displaystyle z\in C}$ In particular, ${\displaystyle z^{2}-1=0}$ 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 ${\displaystyle O_{v}\in V}$ and two binary operations:

• ${\displaystyle +:V\times V->V}$
• ${\displaystyle \times :V\times V->V}$

s.t.

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