How to calculate coordinates of a vector in relation to a basis Let's say I have a basis and a vector:
$$ \mathcal B_1=\{M_1,M_2,M_3,M_4 \} \ \ M_{2\times2}(\mathbb R)\\ v=\begin{pmatrix}a & b\\ c &d \end{pmatrix}$$
Suppose I have numeral values in all of the above matrices, how do I calculate coordinates of $v$ in relation to the basis ?
Is this the general approach:
$x_1M_1+x_2M_2+x_3M_3+x_4M_4=v$
This will yield a $4\times 5$ matrix, should I get it to RREF and find all of the $x_i$ ? Will these be the coordinates $[v]_{\mathcal B_1}=(x_1,x_2,x_3,x_4)^T$?

If you write out $x_1M_1+x_2M_2+x_3M_3+x_4M_4=v$ in terms of the given numbers in the $M_i$ and $v$, and indeterminates $x_1,x_2,x_3,x_4$, you will have a system of four linear equations in those four indeterminates, and you can solve for them however you like. The solution is guaranteed to be unique, and it is the coordinates of that particular $v$.
If you are given a different system of $M_i'$, $i=1\dots 4$, then you could take the same approach. It's just that the coefficients in your linear system will change, and so will the solution for most $v$.
