Give algebraic and geometric descriptions of the $\operatorname{Span} \{ a_1, a_2, a_3, a_4 \}$ Give algebraic and geometric descriptions of $\operatorname{Span} \{ a_1, a_2, a_3, a_4 \}$ where
$a_1 = (1, -1, -2), a_2 = (3, -3, -1), a_3 = (2, -2, -4), a_4 = (2, -2, 1)$
So far, I have:
$$
\begin{matrix}
 \;\;\,1 & \;\;\,3 & \;\;\,2 & \;\;\,2 \\ 
-1 &-3 &-2 &-2\\
-2 & -1 &-4 & \;\;\,1 \\
\end{matrix}
$$
Though, I feel like I'm missing a column. What should this system be equal to?

$\begin{pmatrix}
1 & 3 & 2 & 2\\ 
 -1&  -3&  -2& -2\\ 
 -2&  -2&  4& 1
\end{pmatrix}$
only use elementary row operation,we can get
$
\begin{pmatrix}
1 &0  &2  &-1 \\ 
 0&1  & 0 &1 \\ 
 0&0  &0  &0 
\end{pmatrix}$
then,$a_1=2a_3$,and $a_4=-a_1+a_2$
$\operatorname{Span} \{ a_1, a_2, a_3, a_4 \}=\operatorname{Span} \{ a_1, a_2\}$
