<s>[INST] Basis of a $2\times2$ matrix How would I find the basis for an arbitrary matrix W such that:
$$
W =\left\{ 
\begin{pmatrix} a & b \\ c & a +b +c\end{pmatrix} \ \big| \ \  a ,b ,c \in \mathbb{R} 
\right\}
$$
 [/INST] Hint
$$\begin{pmatrix}a&b\\c&d\end{pmatrix}\in W\iff d=a+b+c\iff\begin{pmatrix}a&b\\c&d\end{pmatrix}=a\begin{pmatrix}1&0\\0&1\end{pmatrix}+b\begin{pmatrix}0&1\\0&1\end{pmatrix}+c\begin{pmatrix}0&0\\1&1\end{pmatrix}$$
</s>