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
 A: 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 $\begin{pmatrix}2 & -1 \\ 0 & -6 \end{pmatrix}$?
A: Hints:
You have two variables, so matrices of the given form can be written as
$$
\begin{bmatrix}a&a+b\\0&2b\end{bmatrix}=a\begin{bmatrix}1&1\\0&0\end{bmatrix}+b\begin{bmatrix}0&1\\0&2\end{bmatrix}.
$$
Therefore, you've written the matrices of the desired form as a linear combination of two matrices.  Do these matrices form a basis?
In order to answer the second part of the question, you are trying to solve 
$$
a\begin{bmatrix}1&1\\0&0\end{bmatrix}+b\begin{bmatrix}0&1\\0&2\end{bmatrix}=\begin{bmatrix}
2&-1\\0&-6
\end{bmatrix}.
$$
Turn this into four linear equations (one for each entry of the matrix), row reduce, and solve.
A: Note that you have exactly $2$ parameteres which determine your matrix, namely $a$ and $b$. In other words, once you fix those, your matrix is completely determined.
On the other hand, from the top-left entry, you can read the coefficient $a$, and from the bottom-right, you read $b$.
The basis is then naturally given by
$$A=\begin{pmatrix}1 & 1\\
 0 & 0\end{pmatrix} \quad B=\begin{pmatrix}0 & 1\\
 0 & 2\end{pmatrix}$$
A: The fact that there are two parameters, $a$ and $b$, should suggest that this is a $2$-dimensional space.
Set $a=1,b=0$ to get one member of a basis.
Set $a=0, b=1$ to get another.
