Dimension von Span mit HIlfe von Gauß

Hallo,

ich weiß zwar wie ich an die folgende Aufgabe herangehe (denke ich zumindest) nur kommt bei mir evtl. etwas falsches heraus.

Gegeben seien Vektoren ${\displaystyle {\mathrm{v<mspace\; width="0.12em"></mspace>}}_1,{\mathrm{v<mspace\; width="0.12em"></mspace>}}_2,{\mathrm{v<mspace\; width="0.12em"></mspace>}}_3,{\mathrm{v<mspace\; width="0.12em"></mspace>}}_4,{\mathrm{v<mspace\; width="0.12em"></mspace>}}_5\in {\mathrm{R<mspace\; width="0.12em"></mspace>}}^5}$ mit

${\displaystyle {\mathrm{v<mspace\; width="0.12em"></mspace>}}_1\left(\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}-2\\ 2\end{array}\\ -1\end{array}\\ 1\end{array}\\ 3\end{array}\right){\mathrm{v<mspace\; width="0.12em"></mspace>}}_2\left(\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}-4\\ 0\end{array}\\ -3\end{array}\\ 1\end{array}\\ 1\end{array}\right){\mathrm{v<mspace\; width="0.12em"></mspace>}}_3\left(\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}3\\ -1\end{array}\\ 2\end{array}\\ -1\end{array}\\ -1\end{array}\right){\mathrm{v<mspace\; width="0.12em"></mspace>}}_4\left(\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}3\\ -1\end{array}\\ 2\end{array}\\ -1\end{array}\\ -1\end{array}\right){\mathrm{v<mspace\; width="0.12em"></mspace>}}_5\left(\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}1\\ 1\end{array}\\ 1\end{array}\\ 0\end{array}\\ 2\end{array}\right)}$ 

Bestimmen Sie die Dimension von Span {${\displaystyle {\mathrm{v<mspace\; width="0.12em"></mspace>}}_1,{\mathrm{v<mspace\; width="0.12em"></mspace>}}_2,{\mathrm{v<mspace\; width="0.12em"></mspace>}}_3,{\mathrm{v<mspace\; width="0.12em"></mspace>}}_4,{\mathrm{v<mspace\; width="0.12em"></mspace>}}_5}$} ( die Anzahl von linear unabhängigen Vektoren) mit Hilfe des Gaußschen Algorithmus.

Lösungansatz:

${\displaystyle \left|\begin{array}{ccc}-2& -4& \begin{array}{cc}\begin{array}{cc}3& 3\end{array}& 1\end{array}\\ 2& 0& \begin{array}{cc}\begin{array}{cc}-1& -1\end{array}& 1\end{array}\\ \begin{array}{c}\begin{array}{c}-1\\ 1\end{array}\\ 3\end{array}& \begin{array}{c}\begin{array}{c}-3\\ 1\end{array}\\ 1\end{array}& \begin{array}{cc}\begin{array}{cc}\begin{array}{c}\begin{array}{c}2\end{array}\\ \begin{array}{c}-1\\ -1\end{array}\end{array}& \begin{array}{c}\begin{array}{c}2\\ -1\end{array}\\ -1\end{array}\end{array}& \begin{array}{c}\begin{array}{c}1\\ 0\end{array}\\ 2\end{array}\end{array}\end{array}\right|\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}0\\ 0\end{array}\\ 0\end{array}\\ 0\end{array}\\ 0\end{array}}$

=> ${\displaystyle {\mathrm{v<mspace\; width="0.12em"></mspace>}}_1={\mathrm{v<mspace\; width="0.12em"></mspace>}}_2={\mathrm{v<mspace\; width="0.12em"></mspace>}}_3={\mathrm{v<mspace\; width="0.12em"></mspace>}}_4={\mathrm{v<mspace\; width="0.12em"></mspace>}}_5=0}$

${\displaystyle \left|\begin{array}{ccc}\begin{array}{cc}\begin{array}{cc}1& 0\end{array}& 0\end{array}& 0& 1\\ \begin{array}{cc}\begin{array}{cc}0& 1\end{array}& 0\end{array}& 0& 0\\ \begin{array}{cc}\begin{array}{cc}0& 0\end{array}& 1\end{array}& 1& 1\end{array}\right|\begin{array}{c}\begin{array}{c}0\\ 0\end{array}\\ 0\end{array}}$

Ich glaube aber irgentwie nicht, dass das stimmt :(

Ich bitte um Hilfe :)

Gruß W0DKA