06-240/Classnotes For Thursday, September 21

2006-09-23T21:15:41Z

72.56.177.13:

A force has a direction & a magnitude.

==<center>'''Force Vectors'''</center>==
#There is a special force vector called 0.
#They can be added.
#They can be multiplied by any scalar.

====''Properties''==== (convention: x,y,z-vectors; a,b,c-scalars)
# <math> x+y=y+x \ </math>
#<math> x+(y+z)=(x+y)+z \ </math>
#<math> x+0=x \ </math>
#<math> \forall x\; \exists\ y \ s.t.\ x+y=0 \ </math>
#<math> 1.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 </math>, so that
#<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.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 |x| \in \mathbb{R} \ \ |x+y| \le |x|+|y| </math>
====''Examples''====
'''Ex.1.'''
<math> F^n= \big\{ (a_1,a_2,a_3,...,a_{n-1},a_n):\forall i\ a_i \in F \big\} </math> 
<math> n \in \mathbb{Z}\ , n \ge 0 </math> 
<math> x=(a_1,...,a_2)\ y=(b_1,...,b_2)\ </math> 
<math> x+y:=(a_1=b_1,a_2+b_2,...,a_n+b_n)\ </math> 
<math> 0_{F^n}=(0,...,0) </math> 
<math> a\in F\ ax=(aa_1,aa_2,...,aa_n) </math> 
<math> In \ \mathbb{Q}^3 \ ( \frac{3}{2},-2,7)+( \frac{-3}{2}, \frac{1}{3},240)=(0, \frac{-5}{3},247) </math> 
<math> 7( \frac{1}{5},\frac{1}{7},\frac{1}{9})=( \frac{7}{5},1,\frac{7}{9}) </math> 
'''Ex.2.'''
<math> V=M_{m \times n}(F)=\Bigg\{\begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots &
& \vdots \\ a_{m1} & \cdots & a_{mn}\end{pmatrix}: a_{ij} \in F \Bigg\} </math> 
<math> M_{3\times 2}( \mathbb{R})\ni \begin{pmatrix} 7 & -7 \\ \pi & \mathit{e} \\ -5 & 2 \end{pmatrix} </math> 
Add by adding entry by entry:<math> M_{2\times 2}\ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}+\begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}=\begin{pmatrix} {a_{11}+b_{11}} & {a_{12}+b_{12}} \\ {a_{21}+b_{21}} & {a_{22}+b_{22}} \end{pmatrix}</math> 
Multiplication by a is multiplication of all entries by a. 
<math> 0_{M_{m\times n}}=\begin{pmatrix} 0 & \cdots & 0 \\ \vdots &
& \vdots \\ 0 & \cdots & 0\end{pmatrix} </math> 
'''Ex.3.'''
<math> \mathbb{C}</math> form a vector space over <math> \mathbb{R}</math>. 
'''Ex.4.'''
F is a vector space over itself. 
'''Ex.5.'''
<math> \mathbb{R}</math> is a vector space over <math> \mathbb{Q}</math>. 
'''Ex.6.'''
Let S be a set. Let 
<math> \mathcal{F}(S,\mathbb{R})=\big\{f:S\to \mathbb{R} \big\} </math> 
<math> f,g \in \mathcal{F}(S,\mathbb{R}) </math> 
<math> (f+g)(t)=f(t)+g(t)\ for\ any\ t\in S </math> 
<math> (af)(t)=a.f(t)\ </math>

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06-240/Classnotes For Thursday, September 21