06-240/Classnotes For Thursday, September 21

A force has a direction & a magnitude.

Force Vectors

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-vectors; a,b,c-scalars)

1. $x+y=y+x\$
2. $x+(y+z)=(x+y)+z\$
3. $x+0=x\$
4. $\forall x\;\exists \ y\ 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:

$+:V\times V\to V$

$\cdot :F\times V\to V$, 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. $\exists \ 0\in Vs.t.\;\forall x\in V\ x+0=x$
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. $\forall x\in V\ ,\forall a,b\in F\ (a+b)x=ax+bx$

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

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

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

Multiplication by multiplying scalar c to all entries by M.

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

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. F is a vector space over itself.
Ex.5. $\mathbb {R}$ is a vector space over $\mathbb {Q}$.
Ex.6. 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)\$