06-240/Classnotes For Thursday, September 21

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Force Vectors

A force has a direction and a magnitude.

  1. There is a special force vector called 0.
  2. They can be added.
  3. They can be multiplied by any scalar.

Properties

(convention: x,y,z are vectors; a,b,c are scalars)

  1.  x+y=y+x
  2.  x+(y+z)=(x+y)+z \
  3.  x+0=x \
  4.  \forall x\; \exists\ y \ \mbox{ s.t. }x+y=0
  5.  1\cdot x=x \
  6.  a(bx)=(ab)x \
  7.  a(x+y)=ax+ay \
  8.  (a+b)x=ax+bx \
Definition

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

 +: V \times V \to V
  \cdot: F \times V \to V \mbox{, so that:}
  1.  \forall x,y \in V\ x+y=y+x
  2.  \forall x,y \in V\ x+(y+z)=(x+y)+z
  3.  \exists\ 0 \in V s.t.\; \forall x \in V\ x+0=x
  4.  \forall x \in V\; \exists\ y \in V\  s.t. \ x+y=0
  5.   1\cdot x=x\
  6.  a(bx)=(ab)x\
  7.  a(x+y)=ax+ay\
  8.  \forall x \in V\ ,\forall a,b \in F\ (a+b)x=ax+bx

9.  x \mapsto \vert x\vert \in \mathbb{R} \ \vert x+y\vert \le \vert x\vert+\vert y\vert

Examples

Ex.1.  F^n= \lbrace(a_1,a_2,a_3,\ldots,a_{n-1},a_n):\forall i\ a_i \in F \rbrace
 n \in \mathbb{Z}\ , n \ge 0
 x=(a_1,\ldots,a_2)\ y=(b_1,\ldots, b_2)\
 x+y:=(a_1+b_1,a_2+b_2,\ldots,a_n+b_n)\
 0_{F^n}=(0,\ldots,0)
 a\in F\ ax=(aa_1,aa_2,\ldots,aa_n)
 \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)
 7\left( \frac{1}{5},\frac{1}{7},\frac{1}{9}\right)=\left( \frac{7}{5},1,\frac{7}{9}\right)
Ex.2.  V=M_{m\times n}(F)=\left\lbrace\begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & 
 & \vdots \\ a_{m1} & \cdots & a_{mn}\end{pmatrix}: a_{ij} \in F \right\rbrace
 M_{3\times 2}( \mathbb{R})\ni \begin{pmatrix} 7 & -7 \\ \pi & \mathit{e} \\ -5 & 2 \end{pmatrix}
\mbox{Addition by adding entry by entry:}

 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}

\mbox{Multiplication by multiplying scalar c to all entries by M.}

 c\cdot M_{2\times 2}\ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}=\begin{pmatrix} c\cdot a_{11} & c\cdot a_{12} \\ c\cdot a_{21} & c\cdot a_{22} \end{pmatrix}

\mbox{Zero matrix has all entries = 0:}

 0_{M_{m\times n}}=\begin{pmatrix} 0 & \cdots & 0 \\ \vdots & 
 & \vdots \\ 0 & \cdots & 0\end{pmatrix}
Ex.3.  \mathbb{C} form a vector space over  \mathbb{R}.
Ex.4. \mbox{F is a vector space over itself.}
Ex.5.  \mathbb{R} is a vector space over  \mathbb{Q}.
Ex.6. \mbox{Let S be a set. Let}
 \mathcal{F}(S,\mathbb{R})=\big\{f:S\to \mathbb{R} \big\}
 f,g \in \mathcal{F}(S,\mathbb{R})
 (f+g)(t)=f(t)+g(t)\ for\ any\ t\in S
 (af)(t)=a\cdot f(t)\