# 06-240/Classnotes For Thursday, September 21

## Force Vectors

A force has a direction and a magnitude.

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

#### Properties

(convention: ${\displaystyle x,y,z}$ are vectors; ${\displaystyle a,b,c}$ are scalars)

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

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

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

9. ${\displaystyle x\mapsto \vert x\vert \in \mathbb {R} \ \vert x+y\vert \leq \vert x\vert +\vert y\vert }$

#### Examples

Ex.1. ${\displaystyle F^{n}=\lbrace (a_{1},a_{2},a_{3},\ldots ,a_{n-1},a_{n}):\forall i\ a_{i}\in F\rbrace }$
${\displaystyle n\in \mathbb {Z} \ ,n\geq 0}$
${\displaystyle x=(a_{1},\ldots ,a_{2})\ y=(b_{1},\ldots ,b_{2})\ }$
${\displaystyle x+y:=(a_{1}+b_{1},a_{2}+b_{2},\ldots ,a_{n}+b_{n})\ }$
${\displaystyle 0_{F^{n}}=(0,\ldots ,0)}$
${\displaystyle a\in F\ ax=(aa_{1},aa_{2},\ldots ,aa_{n})}$
${\displaystyle {\mbox{In }}\mathbb {Q} ^{3}\ \left({\frac {3}{2}},-2,7\right)+\left({\frac {-3}{2}},{\frac {1}{3}},240\right)=\left(0,{\frac {-5}{3}},247\right)}$
${\displaystyle 7\left({\frac {1}{5}},{\frac {1}{7}},{\frac {1}{9}}\right)=\left({\frac {7}{5}},1,{\frac {7}{9}}\right)}$
Ex.2. ${\displaystyle V=M_{m\times n}(F)=\left\lbrace {\begin{pmatrix}a_{11}&\cdots &a_{1n}\\\vdots &&\vdots \\a_{m1}&\cdots &a_{mn}\end{pmatrix}}:a_{ij}\in F\right\rbrace }$
${\displaystyle M_{3\times 2}(\mathbb {R} )\ni {\begin{pmatrix}7&-7\\\pi &{\mathit {e}}\\-5&2\end{pmatrix}}}$
${\displaystyle {\mbox{Addition by adding entry by entry:}}}$

${\displaystyle M_{2\times 2}\ {\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}+{\begin{pmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{pmatrix}}={\begin{pmatrix}{a_{11}+b_{11}}&{a_{12}+b_{12}}\\{a_{21}+b_{21}}&{a_{22}+b_{22}}\end{pmatrix}}}$

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

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

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

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