14-240/Classnotes for Monday September 22

DBN: Classes: 2014-15: 14-240 / Navigation Welcome to Math 240! (additions to this web site no longer count towards good deed points) # Week of... Notes and Links 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 Register of Good Deeds Add your name / see who's in!

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.

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

Scanned Lecture Notes by AM

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File:W31.pdf