Lineaar algebra

Bispal

Aatj an dring san tuup 62 juar ual. Föör 6 juaren wiar di aatj 4-sis so ual üs san dring. Hü ual san's?

Diar koon am son liknangsüsteem faan maage (a=aatj, d=dring):

${\displaystyle {\begin{matrix}(1)&a+d&=&62\\(2)&a-6&=&4\cdot (d-6)\end{matrix}}}$

Nü wurd a liknangen amsteld, so dat jo detsalew furem fu. Lachter ääg a wariaabeln, rochter ääg a taalen:

${\displaystyle {\begin{matrix}(1)&a&+&d&=&62\\(2)&a&-&4d&=&-18\end{matrix}}}$

Tu't liasin jaft at flook metooden. Heer woort det aditjuunsmetood brükt, an det boowenst liknang faan det onerst uftaanj:

${\displaystyle {\begin{aligned}a-4d-(a+d)&=-18-62\\-5d&=-80\\\end{aligned}}}$

Do woort efter d apliaset, ${\displaystyle d=16}$ , di dring as 16 juar ual. Det liasang woort uun't iarst liknang iinsaat:

${\displaystyle a+16=62}$

An diar komt do ${\displaystyle a=46}$ bi ütj, di aatj as 46 juar ual.

En linear liknangsüsteem koon uk uun furem faan en maatriks skrewen wurd. Diar stun do bluas noch taalen uun. Det algemian furem sjocht so ütj:

${\displaystyle \left({\begin{array}{c|c}A&b\end{array}}\right)=\left({\begin{array}{cccc|c}a_{11}&a_{12}&\cdots &a_{1n}&b_{1}\\a_{21}&a_{22}&\cdots &a_{2n}&b_{2}\\\vdots &\vdots &\ddots &\vdots &\\a_{m1}&a_{m2}&\cdots &a_{mn}&b_{m}\end{array}}\right)}$

Uun üüs faal (boowen) wiar det:

${\displaystyle \left({\begin{array}{cc|c}1&1&62\\1&-4&-18\end{array}}\right)}$

An do reegenst dü efter Cramer sin reegel:

${\displaystyle a={\frac {(62\cdot (-4))-(-18\cdot 1)}{(1\cdot (-4))-(1\cdot 1)}}={\frac {-248+18}{-4-1}}={\frac {-230}{-5}}=46}$

${\displaystyle d={\frac {(1\cdot (-18))-(1\cdot 62)}{(1\cdot (-4))-(1\cdot 1)}}={\frac {-18-62}{-4-1}}={\frac {-80}{-5}}=16}$

 Commonskategorii: Lineaar algebra – Saamlang faan bilen of filmer

Wikibooks: Mathematik 3, Lineaar algebra (sjiisk)