I have three matrices $$\begin{Bmatrix} 1 & 1 & 1 \\ 2 & 0 & -1 \end{Bmatrix},\begin{Bmatrix} 1 & 1 & 0 \\ -1 & 0 & 1 \end{Bmatrix},\begin{Bmatrix} 2 & -2 & 1 \\ 1 & 0 & 0 \end{Bmatrix} \in M_{2,3}(Q)$$ and I have to determine if they are linearly independent.
$$\alpha_1*\begin{Bmatrix} 1 & 1 & 1 \\ 2 & 0 & -1 \end{Bmatrix}+\alpha_2*\begin{Bmatrix} 1 & 1 & 0 \\ -1 & 0 & 1 \end{Bmatrix}+\alpha_3*\begin{Bmatrix} 2 & -2 & 1 \\ 1 & 0 & 0 \end{Bmatrix}=\begin{Bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{Bmatrix}$$
$$\begin{Bmatrix} \alpha_1+\alpha_2+2*\alpha_3 & \alpha_1+\alpha_2-2*\alpha_3 & \alpha_1+\alpha_3 \\ 2*\alpha_1-\alpha_2+\alpha_3 & 0 & \alpha_1+\alpha_2 \end{Bmatrix}=\begin{Bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{Bmatrix}$$
$\left\{\begin{array}{l}\alpha_1+\alpha_2+2*\alpha_3=0\\\alpha_1+\alpha_2-2*\alpha_3=0\\\alpha_1+\alpha_3=0\\2*\alpha_1-\alpha_2+\alpha_3=0\\\alpha_1+\alpha_2=0\end{array}\right.$
$$\begin{pmatrix}1&1&2\\1&1&-2\\1&0&1\\2&-1&1\\1&1&0 \end{pmatrix}$$ Eliminating all the rows that are linearly combinations of the others, this matrix becomes $$\begin{pmatrix}1&0&0\\0&1&0\\0&0&1 \end{pmatrix}$$ The three initial matrices are independent but they aren't a base of $M_{2,3}(Q)$ because $dim_{M_{2,3}(Q)}=2*3=6$
I'm not sure if my attempt is correct