06-240/Classnotes For Thursday, September 21

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Force Vectors

A force has a direction and a magnitude.

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

Properties

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

1. $xy=yx\$
2. $x(yz)=(xy)z\$
3. $x0=x\$
4. $\forall x\;\exists \ y\ {\mbox{ s.t. }}xy=0$
5. $1\cdot x=x\$
6. $a(bx)=(ab)x\$
7. $a(xy)=axay\$
8. $(ab)x=axbx\$

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. $\forall x,y\in V\ xy=yx$
2. $\forall x,y\in V\ x(yz)=(xy)z$
3. $\exists \ 0\in Vs.t.\;\forall x\in V\ x0=x$
4. $\forall x\in V\;\exists \ y\in V\ s.t.\ xy=0$
5. $1\cdot x=x\$
6. $a(bx)=(ab)x\$
7. $a(xy)=axay\$
8. $\forall x\in V\ ,\forall a,b\in F\ (ab)x=axbx$

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

Examples

Ex.1. $F^{n}=\lbrace (a_{1},a_{2},a_{3},\ldots ,a_{n-1},a_{n}):\forall i\ a_{i}\in F\rbrace$
$n\in \mathbb {Z} \ ,n\geq 0$
$x=(a_{1},\ldots ,a_{2})\ y=(b_{1},\ldots ,b_{2})\$
$xy:=(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. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V=M_{m\times n}(F)=\left\lbrace\begin{pmatrix} a_{11}}