<s>[INST] Proving a set of 2x2 matrices are linearly independent. I'm trying to prove the following matrices are linearly independent.
$ \beta = \left\{\begin{bmatrix} 1 & -1 \\ 0 & 2 \end{bmatrix},\begin{bmatrix} 0 & 1 \\ 3 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \right\} $. 
My lecture notes say that because the matrix below is row equivalent to the identity matrix, the vectors in $\beta$ must be linearly independent. 
$ \begin{bmatrix} 1 & -1 & 0 & 2 \\ 0 & 1 & 3 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 \end{bmatrix}$
Could somebody please explain why this is?
Thanks,
Jack
 [/INST] Hint
The basis of $\mathbb{M}_{2\times 2}(\mathbb{R})$ is as follow
$$B=\left\{e_1=\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix},e_2=\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, e_3=\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}, e_4=\begin{bmatrix} 0 & 0 \\0 & 1 \end{bmatrix}\right\}$$ 
and 
$$\begin{bmatrix} 1 & -1 \\ 0 & 2 \end{bmatrix}=e_1-e_2+0\times e_3+2e_4\\
\qquad\begin{bmatrix} 0 & 1 \\ 3 & 0 \end{bmatrix}=0\times e_1+e_2+3e_3+0\times e_4\\
 \qquad\begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}=e_1+0\times e_2+e_3+0\times e_4\\
\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}=e_1+e_2+e_3+e_4$$
Edit
let 
$$\alpha\begin{bmatrix} 1 & -1 \\ 0 & 2 \end{bmatrix}+\beta\begin{bmatrix} 0 & 1 \\ 3 & 0 \end{bmatrix}+\gamma\begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}+\theta\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}=0_{2\times 2}$$
therefore
$$(\alpha+\gamma+\theta)e_1+(-\alpha+\beta+\theta)e_2+(3\beta+\gamma+\theta)e_3+(2\alpha+\theta)e_4=0_{2\times 2}$$
in other words
$$ \begin{bmatrix} 1 & 0 & 1 & 1 \\ -1 & 1 & 0 & 1 \\ 0 & 3& 1 & 1 \\ 2 & 0 & 0 & 1 \end{bmatrix}\begin{bmatrix}\alpha\\\beta\\\gamma\\\theta\end{bmatrix}=0$$
</s>