### Teorija

Lineāru izteiksmju vienkāršošana
Vektora reizināšanai ar skaitli piemīt šādas īpašības:
1) ${k}_{1}\left({k}_{2}\stackrel{\to }{a}\right)=\left({k}_{1}{k}_{2}\right)\stackrel{\to }{a}$
2) ${k}_{1}\stackrel{\to }{a}+{k}_{2}\stackrel{\to }{a}=\left({k}_{1}+{k}_{2}\right)\stackrel{\to }{a}$
3) $k\stackrel{\to }{a}+k\stackrel{\to }{b}=k\left(\stackrel{\to }{a}+\stackrel{\to }{b}\right)$

Tāpēc ar izteiksmēm, kuros ir vektoru saskaitīšana, atņemšana un reizināšana, var darboties tāpat kā parastajā algebrā.

Piemērs:
$2\left(\stackrel{\to }{a}-\stackrel{\to }{b}\right)+\left(2\stackrel{\to }{b}-\stackrel{\to }{a}\right)=2\stackrel{\to }{a}-2\stackrel{\to }{b}+2\stackrel{\to }{b}-\stackrel{\to }{a}=\stackrel{\to }{a}$

Vispirms atver abas iekavas:
$2\left(\stackrel{\to }{a}-\stackrel{\to }{b}\right)=2\stackrel{\to }{a}-2\stackrel{\to }{b}$ (3. īpašība) un $\left(2\stackrel{\to }{b}-\stackrel{\to }{a}\right)=2\stackrel{\to }{b}-\stackrel{\to }{a}$.

$2\stackrel{\to }{a}-2\stackrel{\to }{b}+2\stackrel{\to }{b}-\stackrel{\to }{a}=\left(2-1\right)\stackrel{\to }{a}+\left(-2+2\right)\stackrel{\to }{b}=1\stackrel{\to }{a}+0\stackrel{\to }{b}=\stackrel{\to }{a}+\stackrel{\to }{0}=\stackrel{\to }{a}$
$\stackrel{\to }{a}+\frac{1}{2}\left(\stackrel{\to }{b}-\mathit{2}\stackrel{\to }{a}\right)=\stackrel{\to }{a}+\frac{1}{2}\stackrel{\to }{b}-\stackrel{\to }{a}=\frac{1}{2}\stackrel{\to }{b}$
$\frac{1}{2}\left(\stackrel{\to }{b}-2\stackrel{\to }{a}\right)=\frac{1}{2}\stackrel{\to }{b}-\frac{1}{2}\cdot 2\stackrel{\to }{a}=\frac{1}{2}\stackrel{\to }{b}-\left(\frac{1}{2}\cdot 2\right)\stackrel{\to }{a}=\frac{1}{2}\stackrel{\to }{b}-\mathit{1}\stackrel{\to }{a}=\frac{1}{2}\stackrel{\to }{b}-\stackrel{\to }{a}$.
$\stackrel{\to }{a}+\frac{1}{2}\stackrel{\to }{b}-\stackrel{\to }{a}=\left(1-1\right)\stackrel{\to }{a}+\frac{1}{2}\stackrel{\to }{b}=0\stackrel{\to }{a}+\frac{1}{2}\stackrel{\to }{b}=\stackrel{\to }{0}+\frac{1}{2}\stackrel{\to }{b}=\frac{1}{2}\stackrel{\to }{b}$.