06-240/Classnotes For Thursday, September 21

2007-05-28T06:40:43Z

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==Scan of Lecture Notes==

* PDF file by [[User:Alla]]: [[Media:MAT_Lect004.pdf|Week 2 Lecture 2 notes]]
* PDF file by [[User:Gokmen]]: [[Media:06-240-Lecture-21-september.pdf|Week 2 Lecture 2 notes]]

==Scan of Tutorial notes==

* PDF file by [[User:Alla]]: [[Media:MAT_Tut002.pdf|Week 2 Tutorial notes]]
* PDF file by [[User:Gokmen]]: [[Media:06-240-tutorial-21-september.pdf|Week 2 Tutorial notes]]

==Force Vectors==
A force has a direction and a magnitude.
#<math>\mbox{There is a special force vector called 0.}</math>
#<math>\mbox{They can be added.}</math>
#<math>\mbox{They can be multiplied by any scalar.}</math>

====''Properties''====

<math>\mbox{(convention: }x,y,z\mbox{ }\mbox{ are vectors; }a,b,c\mbox{ }\mbox{ are scalars)}</math>
#<math> x y=y x \ </math>
#<math> x (y z)=(x y) z \ </math>
#<math> x 0=x \ </math>
#<math> \forall x\; \exists\ y \ \mbox{ s.t. }x y=0</math>
#<math> 1\cdot x=x \ </math>
#<math> a(bx)=(ab)x \ </math>
#<math> a(x y)=ax ay \ </math>
#<math> (a b)x=ax bx \ </math>

=====Definition=====

Let F be a field "of scalars". A vector space over F is a set V, of "vectors", along with two operations

: <math> : V \times V \to V </math>
: <math> \cdot: F \times V \to V \mbox{, so that:}</math>
#<math> \forall x,y \in V\ x y=y x </math>
#<math> \forall x,y \in V\ x (y z)=(x y) z </math>
#<math> \exists\ 0 \in V s.t.\; \forall x \in V\ x 0=x </math>
#<math> \forall x \in V\; \exists\ y \in V\ s.t. \ x y=0</math>
#<math> 1\cdot x=x\ </math>
#<math> a(bx)=(ab)x\ </math>
#<math> a(x y)=ax ay\ </math>
#<math> \forall x \in V\ ,\forall a,b \in F\ (a b)x=ax bx </math>
-----
9. <math> x \mapsto \vert x\vert \in \mathbb{R} \ \vert x y\vert \le \vert x\vert \vert y\vert </math>
====''Examples''====
'''Ex.1.'''
<math> F^n= \lbrace(a_1,a_2,a_3,\ldots,a_{n-1},a_n):\forall i\ a_i \in F \rbrace </math> 
<math> n \in \mathbb{Z}\ , n \ge 0 </math> 
<math> x=(a_1,\ldots,a_2)\ y=(b_1,\ldots, b_2)\ </math> 
<math> x y:=(a_1 b_1,a_2 b_2,\ldots,a_n b_n)\ </math> 
<math> 0_{F^n}=(0,\ldots,0) </math> 
<math> a\in F\ ax=(aa_1,aa_2,\ldots,aa_n) </math> 
<math> \mbox{In } \mathbb{Q}^3 \ \left( \frac{3}{2},-2,7\right) \left( \frac{-3}{2}, \frac{1}{3},240\right)=\left(0, \frac{-5}{3},247\right) </math> 
<math> 7\left( \frac{1}{5},\frac{1}{7},\frac{1}{9}\right)=\left( \frac{7}{5},1,\frac{7}{9}\right) </math> 
'''Ex.2.'''
<math> V=M_{m\times n}(F)=\left\lbrace\begin{pmatrix} a_{11}</math>

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06-240/Classnotes For Thursday, September 21