<s>[INST] Specify a basis and determine the dimension for the sub space of $2\times2$ matrices of the form... Specify a basis and determine the dimension for the sub space of $2\times2$ matrices of the form 
$$
        \begin{pmatrix}
         a & a+b\\
         0 & 2b \\
        \end{pmatrix}
$$
What are the coordinates for :
$$
        \begin{pmatrix}
         2 & -1\\
         0 & -6 \\
        \end{pmatrix}
$$
in the basis.
I'm honestly completely lost on this one and any tips would be greatly appreciated
 [/INST] Notice that anything of that form is equal to:
$$ a \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix} + b\begin{pmatrix} 0 & 1 \\ 0 & 2\end{pmatrix}.$$
So $\left\{\begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 0 & 2\end{pmatrix} \right\}$ is a spanning set. Since these matrices are not multiples of each other, they are linearly independent and hence a basis.
Therefore the dimension of the subspace is $2$.
For the second part, what values of $a$ and $b$ give you the matrix $\begin{pmatrix}2 & -1 \\ 0 & -6 \end{pmatrix}$?
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